PHY132H1S Summer Practicals Manual - U of T Physicsjharlow/teaching/... · PHY132H1S Summer Practicals Manual Department of Physics July to August, 2009 University of Toronto Welcome

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PHY131 Practicals Manual Introduction

PHY132H1S Summer Practicals Manual

Department of

Physics

July to August, 2009 University of

Toronto Welcome to the Physics Practicals! We have devised a number of Activities and Projects which will help you to learn a lot of Physics. They will also help you to do well on the tests and exam of the course. We are very excited about this new way of helping you to learn Physics, and hope you find your time in the Practicals to be fun and productive. The course web-site has the most up-to-date contact information, handouts, schedules and information:

http://www.physics.utoronto.ca/~jharlow/ summerlab09.html The course coordinator and lecturer is

Jason Harlow, Office: MP129-A, Phone 416-946-4071. The materials in this book were mainly developed by

David Harrison, Office: MP121-B, Phone 416-978-2977 The course administrator is April Seeley, Office: MP129-E, Phone 416-946-0531. Email addresses are listed on the course web-site and on the Physics Department directory at http://www.physics.utoronto.ca/people . Course staff will endeavour to respond to email inquiries from students within 2 days. If you do not receive a reply within this period, please resubmit your question(s) and/or phone (leave a message if necessary). Please note that some servers (such as hotmail) can be unreliable in both sending and receiving messages.

Table of Contents Section Page

Introduction 1 Measurement Project 7

Waves Module 9 Ray Optics Module 24

Electricity and Magnetism Module 1 33 Electricity and Magnetism Module 2 43 Electricity and Magnetism Module 3 52 Electricity and Magnetism Module 4 72 Electricity and Magnetism Module 5 80 Electricity and Magnetism Module 6 87

Relativity Module 98

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Schedule (preliminary) Students attend two 2-hour practicals every week on Tuesdays and Thursdays: either 3:00-5:00 pm OR 8:00-10:00 pm.

Practical Dates Topics, Activities Tue. Jun 30 NO PRACTICALS today: First lecture 6:00pm

1 Thu. Jul 2 Waves Module – Activities TBA 2 Tue. Jul 7 Waves Module – Activities TBA 3 Thu. Jul 9 Ray Optics Module – Activities TBA 4 Tue. Jul 14 EM Module 1 – Activities TBA 5 Thu. Jul 16 EM Module 2 – Activities TBA

6 Tue. Jul 21 Scrambling teams EM Module 3 – Activities TBA

7 Thu. Jul 23 EM Module 4 – Activities TBA

8 Tue. Jul 28 EM Module 5 – Activities TBA (Measurement Project due)

9 Thu. Jul 30 EM Module 6 – Activities TBA

10 Tue. Aug 4 Relativity Module – Activities TBA Thu. Aug 6 NO PRACTICALS today: Last lecture 6:00pm

How the Practicals Work You will be meeting for 2 hours every Tuesday and Thursday in room MP125 (A, B, or C). Each Group will have a maximum of 36 students. You will be working in a Team with up to three of your classmates. There will be two Teaching Assistant Instructors present for each Practical. Your Team will keep a single lab book, which is to be a complete record of everything you did, what you and your teammates thought it meant, and what conclusions you have drawn from your work. Each Practical session will include time for student questions and discussion. However the “heart” of the Practicals will be a series of Activities. Every week you will be doing Activities based on the material currently being discussed in class. Often the Activities will be based on material that has already been discussed in class, but sometimes the Activities may be used to introduce material that has not yet been talking about in class. In addition, you will be doing three “value added” Modules,

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that we believe are important for your overall learning about science in general and Physics in particular. These are:

• A Module on the Scientific Method • A Module on effective Teamwork • A Module on Numerical Approximation

For each Practical session two members of each Team will serve the following roles:

Facilitator. This person, a different individual each week, is responsible for keeping the Team on track with the Activities. When the entire Practical group discusses some topic, the Facilitator will be the Team’s primary spokesperson.

Recorder. This person, also a different individual each week, takes primary responsibility for recording all work, speculations, conclusions etc. in the lab notebook.

Evaluation and Marks The Practicals will count for 15% of your mark in PHY131. All marks will be given on an integer scale from 0 to 4:

0. Missing work. 1. Seriously deficient. 2. Requires improvement. 3. The standard mark indicating good work 4. Exceptional. We will be very stingy in awarding marks of 4.

Each mark component has a weight, and the mark times the weight will be added to generate a Practical mark. The total number of weights of all components is 20. The one exception to this marking system is the Error Analysis Assignment. It is marked out of 100. Attendance at the Practical is vital for your learning. We will deduct the cube of the number of un-excused absences from the final Practical mark. Here are the components and their weights:

1. Notebook Mark 1 (0 Weights). After the first Practical the lab books will be collected and marked. However, this mark will not count towards your Practical mark. Instead it is intended to make our standards and requirements clear to you.

2. Notebook Mark 2 (6 Weights). After the last Practical before Test, a selection of Activities from Practical sessions completed so far will be chosen to be marked. The decision of which Activities will be marked will be chosen more-or-less randomly after the books have been collected. All Teams will have the same Activities marked.

3. Notebook Mark 3 (6 Weights). At the end of the term a selection of the Mechanics, Oscillations and Fluids Activities you have done since the Test will be chosen to be marked. The decision of which Activities will be marked will be chosen more or less randomly after the books have been collected. All Teams will have the same Activities marked.

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4. Measurement Project (3 Weights). Due on July 28, this is a 2-page report on a Measurement Project done by individual students outside of class time.

Computers and Networks The Practical server is: feynman.physics.utoronto.ca. You will access the server using your UTORid and password. You will have access to three folders on this server:

Your home directory. You have read and write privileges for this directory. Your team directory. All members of your team have read and write privileges here. public. This is an area of the server containing documents, computer programs, etc.

Everyone has read privileges for this directory. Note: you should never save work on the local PC. These discs will be ruthlessly purged on a regular basis. Remote Access You may access the server at: https://feynman.physics.utoronto.ca. You may upload and download files from your computer to the server. Printing There is a colour printer in the Practical Room. You may choose to print either in colour or black and white by choosing the appropriate printer in the print dialog. We charge for printing using your TCard. We charge:

10 cents per page for black and white printing. 15 cents per page of colour printing.

We do not (yet) have facilities in the building to add dollar values to your card. The locations of cash-to-card locations is at: http://content.library.utoronto.ca/finance-admin/photo/cash-to-card At present the nearest location is the Main Floor of the Earth Science building, just across Huron Street.

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PHY131 Practicals Manual Measurement Project

Measurement Project due July 28

Overview and Motivation If you took the pre-requisite for this course, PHY131, you completed the Error Analysis Assignment, which is still available for your review at: http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/ . We hope that as part of PHY131 and PHY132, you have been taught how to

• take careful measurements • report all measurements with a ± error • propagate errors when computing results based on measurements • compute the average and standard deviation of multiple measurements of the

same quantity • distinguish between accuracy and precision • report your findings carefully and convincingly

These are skills that will last you the rest of your life as you continue in any scientific, medical or other discipline in which measurements are important. Another important skill is writing. You should be able to write a clear, readable report in English that informs the reader of your findings and conclusions. To this end, I am assigning this Measurement Project, due July 28. Your report should be about 2 pages, type-written, and should be submitted both electronically and in paper format. Summer 2009 Topics I would like you to answer exactly ONE of the following five general questions:

1. What is the height of the Burton Tower, which is part of McLennan Physical Laboratories?

2. What is the volume of the water in the fountain pool in the courtyard between McLennan Physical Laboratories and Lash Miller Chemical Labs?

3. What is the average mass per unit length of the typical blade of grass from the courtyard between McLennan Physical Laboratories and Lash Miller Chemical Labs?

4. How fast do your own fingernails grow? [Please convert into some useful unit, such as mm/month.]

5. What is the average maximum temperature of the hot water coming out of the taps in the bathrooms in McLennan Physical Laboratories?

As part of your Motivation section, you should re-state the question so that it is more specific. Make sure that your final answer matches the question you are asking. [For example, if you are measuring the height of the Burton Tower, do you include the green astronomy domes on the roof, and where do you define zero height? ie, at the base of the tower where the doors are, or the street level on Huron? ] If you would like to answer a

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question that is very different from any of the above five, please first obtain permission from Jason Harlow, the Practicals Coordinator. Report Format Your report should include the following sections:

Title It should be clear from your title which of the 5 questions you answered. Author, Date Include your student number, Course Name, your Practical Section and

your Practical Group code. Collaborators List any friends who may have worked with you on taking the

measurements. Abstract This should be one or two sentences summarizing the main conclusions of

the report, including the final numerical result. Motivation What is the question you are trying to answer exactly? Procedure Please detail exactly what you did, what measuring devices you used, any

relevant environmental conditions, problems you encountered or innovations you may have devised to perform your measurements. You may wish to include a short table, summary or sample of your original measurements.

Analysis Describe any mathematical procedures you used to go from the raw original measurements to the final results.

Conclusions The Measurement Project will, in part, be marked on writing style and on the organization and presentation of the material. Good English structure, spelling and grammar are expected, and graphs and diagrams should be clearly labelled. Resources The technologists for PHY132 are Larry Avramidis, Lilian Leung, Phil Scolieri and Rob Smidrovskis. They all share an office in MP127. With their permission you may borrow metre sticks, stopwatches, measuring tape, Vernier callipers, thermometers, and the like from the Resource Centre in MP126. They can also make a digital scale available to you. If MP126 is not open you can knock on the door of MP127 during regular business hours M-F 9-12, 1-5. Due Date, Procedures for Turning in Report. The Measurement Project in electronic format is due to www.turnitin.com by 11:59 PM on Tuesday, July 28, 2009. It must be submitted in electronic format (Word, PDF and several other formats are acceptable) to www.turnitin.com by the deadline, and an identical paper copy must also be submitted to your demonstrator either during your 8th Practical session July 28. The paper copy may be turned in early if you wish, as can the electronic version. Your name, Student Number, Practical Section and Group code must appear clearly on the front of your Measurement Project.

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Late Measurement Projects will be penalized at the rate of 10% per day of lateness. The number of days of lateness will be the maximum of the electronic submission lateness, as based on the turnitin.com time-stamp, and the paper-copy lateness. A fractional number of days will always be rounded up to the nearest integer, and the penalty will be applied as a percentage of the unpenalized mark. Measurement Projects with an electronic or paper lateness of more than 10 days will receive a zero. To submit your assignment electronically you should follow these steps:

Log on to www.turnitin.com . If you don’t already have a user profile, set one up:

• Click Create a user profile. • Enter a valid utoronto.ca email address, password and your name. Please

enter the same name that is on your University of Toronto I.D. so we can easily tell who you are.

Enroll in this class From your turnitin homepage click the Enroll in a class button. For this class the Turnitin class I.D. is 2619965 and the Turnitin enrollment

password is sunshine . The name of the class should be “PHY132 Summer 2009”.

Submitting a paper. From your Turnitin homepage select this class Click on the Submit button and select File Upload from the pulldown menu. Enter a submission title for your paper, which should include your name. You

may use spaces in the title, but not commas or other special characters. Use the Browse button to select the file that you would like to submit. Click Submit.

NOTE: Turnitin automatically will generate a text-only version of your paper. This is what it uses to search for textual similarity with other documents in its database. This text-only version will NOT be used in the marking; please ignore it! If we wish to mark your electronic version, we will download the exact same file you uploaded, which will be complete with figures, tables, special characters, fonts, etc. If you prefer, you may choose to submit only a hard-copy of your project, but in this case you must also provide a photocopy of the relevant notes you took while performing your measurements with dates and times, with numbered references linking the text in your formal report to the original measurement notes. Please speak with the lab coordinator at least one week before the project deadline if you prefer to submit a hard-copy only. Length Limit The typed report should be approximately 2 pages long. The final version of the Measurement Project should contain no more than 800 words (including title, abstract, table and figure captions), and should take up no more than 5 letter-sized page sides total. Marks will be deducted if either of these length limits is exceeded. Note that www.turnitin.com sometimes overcounts the number of words,

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mostly depending on how you submitted your tables; in any case the turnitin.com word count should not exceed 1000 words. Poster Option The normal way to prepare the report is by using Microsoft Word with 12-pt font, into which you may insert figures, tables, etc. As an alternative, you may instead submit your Measurement Project as a poster. The poster should be 28” tall and 36” wide, and should not have any text on it smaller than 24-pt. There are several good programs for making posters, including Powerpoint, Macromedia Freehand, Adobe Illustrator, Adobe Photoshop and Adobe PageMaker. In any case, you should make a PDF of your poster and submit it to turnitin.com following the instructions above. You should NOT submit a poster in paper format. Instead, please attach the PDF file in an email to Jason Harlow by the electronic deadline. You will receive a confirmation that your report has been received. The very best posters submitted will be printed by the graphics department in Physics, and, with the author’s permission, posted in the hall on the first floor of the North Wing of McLennan. Posters should include all the necessary information about your measurements and analysis, but should also be eye-catching, colourful and succinct. Notes on “Originality” While your Procedure may include work you do with your friends (who should be listed as collaborators in your report), your Measurement Project should be primarily your individual work. You must perform the analysis and write the entire report yourself. For information on “how not to plagiarize”, please see http://www.utoronto.ca/writing/plagsep.html. The turnitin.com version will be treated as your official submission, and the marker may download your report from the turnitin.com web site. The marker will also have access to an “originality report”, which is a comparison of the text-portion of your report to millions of other documents, including all the online material for this course, all the other reports submitted to turnitin.com, and many documents which were available at some time on the world-wide-web. The originality report will probably not be used in the marking unless there is some evidence that an unusually large amount of your unquoted text is identical to some other source. If you do wish to quote a source, be careful to reference it and include the copied words in quotation marks, so it is clear to the reader that you did not write them. Students agree that by taking this course your measurement project may be subject to submission for textual similarity review to Turnitin.com for the detection of plagiarism. All submitted papers will be included as source documents in the Turnitin.com reference database solely for the purpose of detecting plagiarism of such papers. The terms that apply to the University’s use of the Turnitin.com service are described on the Turnitin.com web site.

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PHY131 Practicals Manual Waves Module

Waves Module 1 Student Guide

Concepts of this Module

• Traveling waves • Intensity • Reflection • Superposition • Standing Waves

Activity 1

A. Open the Java applet wave-on-a-string.jar which is at: Feynman:Public/Modules/Waves.

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• Set the Damping to 0 • Set the wave type to “Pulse” • Set the end to “No End” which will replace the vise on the right side with

an open window for the waves to go through. Click on the Pulse button that will appear. Imagine you are standing right beside the window as the pulse goes out of it, measuring the amplitude as a function of time as it goes by you. Sketch the amplitude as a function of time.

B. Click on the Rulers control in the upper-right corner of the simulation. The rulers that appear can be moved with the mouse. Estimate the speed, width and amplitude of the wave pulse. Add labeled tick marks on the axes of the sketch of Part A. This is a good time to experiment with different values of the Damping and tension. What happens as the Damping is increased? What happens as the tension in the string is decreased? You may wish to explore some of the other settings of the simulation too.

C. The triangular pulse of Parts A and B was symmetric. Here is a plot of an asymmetric triangular pulse traveling from left to right. At the moment shown the time t = 0. The wave is traveling with a speed of 0.5 m/s. Sketch the amplitude of the pulse at x = 0 as a function of time t as the pulse goes by. Include labeled tick marks on both the y and t axes.

D. Here is the same triangular pulse as Part B, but it is traveling from right to left at 0.5 m/s. At the moment shown the time t = 0. Sketch the amplitude of the pulse at x = 0 as a function of time t as the pulse goes by. Include labeled tick marks on both the y and t axes. Compare to the sketch from Part B.

E. Here is a sinusoidal wave pulse traveling from left to right at v = 0.5 m/s. At the moment shown t = 0. Sketch the amplitude of the pulse at x = 0 as a function of time t

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as the pulse goes by. Include labeled tick marks on both the y and t axes. What is the wavelength λ of the pulse? From your sketch what is the period T, frequency f, and angular frequency ω of the sinusoidal pulse? What is the relation between λ, f and v?

F. Here is a sinusoidal wave pulse traveling from right to left at v = 0.5 m/s. At the moment shown t = 0. Sketch the amplitude of the pulse at x = 0 as a function of time t as the pulse goes by. Include labeled tick marks on both the y and t axes. What is the wavelength λ of the pulse? From your sketch what is the period T, frequency f, and angular frequency ω of the sinusoidal pulse? What is the relation between λ, f and v?

G. Here is a sine wave traveling from left to right with v = 0.5 m/s. The wave extends to infinity in both directions along the x axis. At the moment shown the time t = 0. At the moment shown the amplitude as a function of position is:

)2sin(1.0)0,(λ

π xtx ==Ψ

In your own words, explain the factor 2π in the above equation. We can describe

the amplitude as the wave passes x = 0 either as )2sin(1.0),0(Txtx π==Ψ or as

)2sin(1.0),0(Txtx π−==Ψ . Which form is correct? Explain your own words.

Write down a form of ),( txΨ which is valid for all values of x and t. You may find the following Flash animation useful in visualizing this situation: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/TravelWaves/TravelWaves.html The above link is to a fixed size animation which works nicely if only one person is viewing it. If more than one person is viewing the animation, a version which can be resized is better. Here is a link to a resizable version of the same animation: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/TravelWaves/TravelWaves.swf

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Activity 2

Here is a link to a simple little Flash animation of a plane wave traveling through two different mediums: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/TwoMediums/TwoMediums.html Here is a link to a resizable version of the same animation, which is nicer if more than one person is trying to view it: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/TwoMediums/TwoMediums.swf Open one of the versions of the animation.

A. At what rate do the wave fronts from the left strike the medium in the centre? What is the period, frequency, and angular frequency of the wave to the left of the medium in the centre?

B. For the medium in the centre, at what rate do the wave fronts leave the left-hand side? Is this the same as your answer to Part A? Explain. Do the wave fronts strike the right side of the medium in the centre at this same rate? What is the period, frequency, and angular frequency of the wave while it is traveling through the medium in the centre?

C. How does the wavelength of the wave traveling from the left to the medium in the center compare to the wavelength of the wave while it is traveling through the medium in the centre? Show how you arrived at your answer.

D. The wave leaves the medium in the centre and travels off to the right. How do the period, frequency, angular frequency, and wavelength of the wave traveling to the right of the medium in the centre compares to the same quantities for the waves in the other regions?

Activity 3

In Activity 2 the waves strike the interface between the two mediums straight on, with zero angle of incidence. Here is a link to a Flash animation where the angle of incidence is not zero. http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html Here is a link to a resizable version: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.swf Open one of the versions of the animation.

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A. At what rate do the wave fronts from the left strike the medium in the centre? Is the rate the same regardless of what vertical position you are considering? What is the period, frequency, and angular frequency of the wave to the left of the medium in the centre?

B. For the medium in the centre, at what rate do the wave fronts leave the left-hand side? Is this the same as your answer to Part A? Explain. Do the wave fronts strike the right side of the medium in the centre at this same rate? What is the period, frequency, and angular frequency of the wave while it is traveling through the medium in the centre?

C. How does the wavelength of the wave traveling from the left to the medium in the center compare to the wavelength of the wave while it is traveling through the medium in the centre? Show how you arrived at your answer.

D. The wave leaves the medium in the centre and travels off to the right. How do the period, frequency, angular frequency, and wavelength of the wave traveling to the right of the medium in the centre compares to the same quantities for the waves in the other regions?

E. The figure to the right shows a portion of two wave fronts of the animation. What is the relation between θ1 and θ2? Notice that there are two right triangles in the figure with a common hypotenuse.

Activity 4

Here is a Flash animation of a molecular view of a sound wave traveling through the air: http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/long_wave.html The above link is to a fixed size animation which works nicely if only one person is viewing it. If more than one person is viewing the animation, a version which can be resized is better. Here is a link to a resizable version of the same animation: http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/long_wave.swf Open one of the versions of the animation.

A. The bottom shows the motion of the air molecules. You may wish to imagine that the molecules are connected to their nearest neighbors by springs, which are not shown. There is a wave of increasing and decreasing density of the molecules. Is the wave moving to the right or to the left? Explain. Is the wave longitudinal or transverse? Explain.

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B. Often instead of describing the wave as one of density we talk about a pressure wave. Does the higher density of molecules correspond to higher or lower pressure? Can you explain?

C. The top shows the displacement of the molecules from their equilibrium positions. It too is a wave, often called a displacement wave. Is the wave moving to the right or to the left? Explain. Is the wave longitudinal or transverse? Explain.

D. Use the step controls, pause the animation and position molecules 3 and 9 at their equilibrium position with molecule 6 at maximum displacement. The amplitude of the displacement wave is zero for molecules 3 and 9. Is the amplitude of the pressure wave at the position of molecule 3 also zero, or is it a maximum or a minimum? What about the pressure wave at the position of molecule 9?

E. Use the step controls to position molecules 3 and 9 at their equilibrium position with molecule 6 at minimum displacement. Is the amplitude of the pressure wave at the position of molecule 3 zero, or is it a maximum or a minimum? What about the pressure wave at the position of molecule 9?

F. From your results for Parts D and E, what is the phase angle between the pressure wave and the displacement wave?

Activity 5

A. Open the Java applet wave-on-a-string.jar which is at: Feynman:Public/Modules/Waves. Part A of Activity 1 shows a screen shot of the applet.

• Set the Damping to 0 • Set the wave type to “Oscillate” • Set the end to “No End” which will replace the vise on the right side

with an open window for the waves to go through. How does the amplitude of the wave change as it propagates down the string? Is this a one dimensional, two dimensional, or three dimensional wave? You may wish to look over Parts B and C before answering this question.

B. A two dimensional wave, such as a water wave, is propagating away from its source equally in all directions. Assume damping is negligible. How does the amplitude of the wave change with distance from the source?

C. A three dimensional wave, such as a sound wave, is propagating away from its source equally in all directions. Assume damping is negligible. How does the amplitude of the wave change with distance from the source?

D. What physical principle or conservation law gives the answers to Parts A – C? Explain

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Activity 6

A. Open the Java applet wave-on-a-string.jar which is at: Feynman:Public/Modules/Waves. Part A of Activity 1 shows a screen shot of the applet.

• Set the Damping to 0 • Set the wave type to “Pulse” • Leave the end in its default state of “Fixed End” which clamps the right

side of the string with a C-clamp.

Click on the Pulse button. What is the behavior of the wave pulse when it is reflected by a fixed end?

B. Change the end of the string to “Loose End” which terminates the right hand side of the string with a frictionless loop around a vertical rod. Click on Reset and then on Pulse. What is the behavior of the wave pulse when it is reflected by a free end?

C. Set the end of the string back to “Fixed End,” click on Reset and then on Pulse. Use the pause/play button and then the step one to step the wave pulse through a complete reflection at one end of the string. There is a point where the wave pulse nearly disappears. Where did the wave go? Where did the wave’s energy go? Explain what is happening.

D. Set the end of the string back to “Loose End,” click on Reset and then on Pulse. Click on the Rulers control in the upper-right corner of the simulation. The rulers that appear can be moved with the mouse. Measure the maximum amplitude of the wave pulse; you may already have done this measurement in Activity 1 Part B. Use the pause/play button and then the step one to step the wave pulse through a complete reflection at one end of the string. There was a point where the amplitude of the wave at the position of the free end was large. Use the ruler to estimate its amplitude. Explain your result.

E. Set the end of the string back to “Fixed End.” Set the Damping to 10. Set the wave type to “Oscillate” and click on Reset. You will see a “standing wave” on the right hand side of the string. Use the pause/play button and then the step one to step the wave pulse through a complete reflection at the right end of the string. There is a point where the wave pulse near the right hand side nearly disappears. Where did the wave go? Explain what is happening.

F. Set the end of the string back to “Loose End,” click on Reset. You will once again see a “standing wave” on the right hand side of the string. Is there a difference between this standing wave and the one you saw in Part E? Explain. Is there a point where the wave amplitude near the right hand side nearly disappears, as in Part E? Explain.

G. Set the end of the string back to “Fixed End.” Leave the wave type as “Oscillate.” Set the Damping to 0. Click on Reset. What happens? Explain.

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Although we have used the wave on a string applet in Activities 1, 5, and now here, there is still lots more Physics that you can learn from it. You are invited to explore further.

Activity 7

“Music is a hidden practice of the soul, that does not know it is doing mathematics.” --Leibniz

If Pythagoras had a guitar, it might have looked like this:

We assume that Pythagoras was a large man, so the length of the strings from the bridge to the nut is 1 m, as shown. The second string of six from the top is conventionally tuned to A two octaves below concert A. This is often written as A2, and has a frequency of 110 Hz. The notes of an A scale starting at A2 are: A2 – B2 – C1# - D1 - E1 – F#1 – G#1 – A1. Here are the frequencies of these notes in a Pythagorean tuning; also shown are the frequencies of the notes in an equally tempered tuning which is more common today.

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Note Pythagorean Tuning (Hz)

Equally Tempered (Hz)

A2 110.00 110.00 B2 123.75 123.47

C1# 139.22 138.59 D1 146.67 146.83 E1 165.00 164.81 F1# 185.63 185.00 G1# 208.83 207.65 A1 220.00 220.00

You will need to know that, as discussed in the textbook, the speed of a traveling wave on string with tension Ts is

μs

stringTv =

where μ is the string’s mass-to-length ratio

Lm

=μ

Note that the speed is independent of the frequency. To the right are shown the first four normal modes of a vibrating string. Here is a link to a simple Flash animation that shows the actual motion of the string for the first three normal modes: http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta2fix.html

A. When the second string from the top is playing the note A2 = 110 Hz the

frequency f of the first normal mode is also 110 Hz. What is the wavelength of the first normal mode? What is the speed of a traveling wave on the string? Explain why increasing the tension in the string increases the frequency of the note the string plays.

B. What is the wavelength of the second normal mode of the string? What is the frequency of the standing wave?

C. If you place your finger just to the right of the 12th fret the effective length of the string becomes 0.5 m, as shown in the figure. What is the wavelength of the first

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normal mode, and the frequency? What musical note is the string playing? How do your values compare to your result for Part B? Explain.

D. If you place your finger just to the right of the 7th fret the effective length of the string becomes 2/3 m, as shown in the figure. What is the wavelength of the first normal mode and the frequency? What musical note is the string playing? Explain.

E. Is there a pattern between the positions of the labeled frets in the figure of the guitar and the notes of the A scale in a Pythagorean tuning? What is the pattern? Are the lengths shown in the figure rational or irrational numbers? You may wish to know that if the guitar frets were set up to be equally tempered, except for the 12th fret the lengths would not be rational numbers.

F. As indicated in the table, the frequencies of the notes in a scale are slightly different in the Pythagorean tuning and the equally tempered tuning commonly used today. You may see if you can hear the difference by listening to a scale played with the Pythagorean tuning in the file Pythagorean.mid and a equally tempered tuning in EqualTempered.mid; both files are located at Feynman:public/Modules/Waves. You may also wish to explore further at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Temperament/Temperament.html

Activity 8

As investigated in Activity 4, we can think of a sound wave two different ways:

1. A pressure wave. The pressure oscillates around atmospheric pressure. 2. A displacement wave. The displacements of the air molecules oscillate around

their equilibrium positions. These two waves are 90 degrees out of phase: when one has a maximum or minimum the other is at zero amplitude.

You will want to know that microphones measure the pressure wave. You will also want to know that the speed of sound is:

19


)m/s(61.0331 2tvaccepted +=

where t is the temperature of the air in Celsius. In this Activity you will set up standing sound waves in a tube filled with air. A loudspeaker generates the sound wave. A rod inside the tube has a small microphone mounted on the end, so the sound wave inside the tube can be measured at different positions. The part of the tube with the loudspeaker is shown in the figure on the next page.

When the tube is closed at both ends, the possible displacement standing waves are the same as those for a standing waves on a string that is fixed at both ends: there is a node at each end of the tube. The figure to the right shows the first four possible standing waves. These are the same standing waves that for a string we called normal modes in Activity 7, and in fact this is the same figure that appears there!

A. What are the wavelengths of the shown standing waves? What is the wavelength of the m = 5 standing wave which is not shown? Generalise to a formula for the wavelengths for any value of m.

B. For the first two or three displacement standing waves, sketch the corresponding pressure standing wave.

20


Here is a link to a simple Flash animation that shows the displacement wave for the first three standing waves: http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta2fix.html

[DESCRIBE HOW TO SET UP THE APPARATUS]

Getting a Standing Wave in the Tube Have the tube closed at both ends. For some frequency between, say, 200 Hz and 2 kHz, adjust the frequency so that a standing wave exists in the tube. One way to adjust the frequency for a good standing wave is to place the microphone as close to the loudspeaker as possible. Now adjust so that the amplitude as measured by the microphone is a maximum; recall that this corresponds to the displacement of the air molecules from their equilibrium position being a minimum. A secondary adjustment can be made by placing the microphone at the position of a node in the pressure wave and making small adjustments of the frequency to make the measured amplitude as small as possible.

C. How much can you vary the frequency of the wave and not observe any difference in whether or not there is a good standing wave in the tube?

D. Put your ear close to the tube and note how loud the sound is. Adjust the frequency so that there is no longer a standing wave in the tube. How does the sound level compare to when there is a standing wave? Explain.

E. Get a good standing wave in the tube. Probe the standing wave with the microphone to determine the wavelength of the standing wave. What is your uncertainty in this determination? As you will discover in Part G, you should not just determine the wavelength from the number of nodes and the length of the tube. From your measurements of the frequency and wavelength calculate the speed of sound and its uncertainty. How does your value compare with the accepted value given above?

F. Repeat Parts C – E for a few more frequencies. G. For the lowest frequencies, the maximum amplitude as measured by the

microphone does not occur at the position of the loudspeaker, but a noticeable and measurable distance down the tube away from it. This is your first indication that the simple picture of these waves as described above is not quite complete. Choose one of the low frequency standing waves that you have discovered, and determine the value of this distance.

When one end of the tube is open to the air, the standing waves that are possible are the same as those for a vibrating string with one loose end. Here are some of these standing waves:

21


These standing waves occur because part of the incident sound wave is reflected from the open end of the tube. However, the effective reflection point of the wave is not the exact position of the open end of the tube but is slightly beyond it, and so the effective length of the tube is greater than its real length:

LLL realeffective Δ+= where:

DL 3.0≈Δ and D is the diameter of the tube. Sometimes effectiveL is called the acoustic length. Here is a link to a simple animation that shows the first three standing waves: http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta1fix.html

H. Set up one or more standing waves in the tube with one end open and determine the effective length of the tube. How well do you measurements agree with the value given above?

I. If someone designs a pipe organ without being aware of the acoustic length, what will be the consequences?

Activity 9

If the apparatus of Activity 8 were perfect, then when the tube is closed on both ends we would not hear any sound outside the tube. Similarly, if the air inside the tube were perfect, all molecule-molecule collisions would be perfectly elastic; this means that as a sound wave travels through the air none of its energy would be converted to heat energy of the air. However, neither the apparatus nor the air is perfect, The Quality Factor Q measures the degree of “perfection” of the system. Say we have a standing wave when the frequency is 0f . For frequencies close to the "resonant frequency" 0f the amplitude A of the sound wave at the position where there was an maximum in the pressure wave is given by:

22


⎟⎟⎠

⎞⎜⎜⎝

⎛ff

- ffQ + 1

1 A = A(f)o

o

22

o

Note in the above that the amplitude A(f) is equal to 0A when the frequency f is equal to the resonant freqency . The figure to the right shows A(f) for 0A equal to 1, Q equal 2, and for a resonant frequency of 50 Hz. Note that we have indicated the width of the curve where the maximum amplitude is

2/1 times the maximum amplitude 0A . A nearly trivial amount of algebra shows that the amplitude A is 2/1 times the maximum amplitude 0A for positive frequencies when the frequency is:

)141(2

20 ±+= QQf

f

Thus, if the width of the curve is fΔ , then Q is:

f

fQ

Δ= 0

A. For a given resonant frequency 0f how does the width of the curve of amplitude

versus frequency depend on the Quality Factor Q? B. When the Quality Factor Q is zero, the maximum amplitude 0A is zero. When Q

is infinite so is the maximum amplitude. Explain. C. Close the tube at both ends and adjust for a standing wave in the range of 200 Hz

- 1 kHz. Place the microphone at a maximum in the pressure wave and take data for the amplitude as a function of frequency for frequencies close to the resonant frequency. Calculate the Quality Factor of the tube.

This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in the Fall of 2008.

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Last revision: November 29, 2008. The Java applet used in Activities 1, 5 and 6 was written by the Physics Education Technology (PhET) group at the University of Colorado, http://phet.colorado.edu/index.php. Retrieved November 9, 2008. The figure of normal modes of a vibrating string in Activity 7 is slightly modified from Figure 21.22 of Randall D. Knight, Physics for Scientists and Engineers, 2nd edition (Pearson Addison-Wesley, 2008), pg. 640. The same figure is used in Activity 8. The Pythagorean and equally tempered scales used in Activity 6 are from Wikipedia, http://en.wikipedia.org/wiki/Pythagorean_tuning. Retrieved November 15, 2008. Activities 8 and 9 are based on a Student Guide written by David M. Harrison in October 1999 and revised in June 2001.

24

PHY131 Practicals Manual Ray Optics Module

Ray Optics Module Student Guide


• Ray Tracing. • Reflection and Image Formation in Plane Mirrors. • Refraction and Total Internal Reflection. • Thin Lenses: Magnification and the Thin Lens Equation.

Activity 1

A. Imagine a room that contains a light source, L, a plant P, and an observer with an eye, E, as shown. Sketch the situation in your notebook and some rays that will allow the observer to see the plant. If light is reflected, indicate if the reflection is diffuse or specular.

B. Imagine a room that

contains a light source, L, a plant P, an observer with an eye, E, and a mirror, M, as shown. Sketch the situation in your notebook and some rays that will allow the observer to see the plant in the mirror. If light is reflected, indicate if the reflection is diffuse or specular.

25

PHY131 Practicals Manual Ray OpticsModule

Activity 2

Here is a picture of Jason Harlow in front of a chalk-board, reversed several times to produce 4 possibilities. One image is the original, one is reversed left and right, one is reversed up and down, and the other is reversed both left and right and up and down.

1

2

3

4

A. Can you tell which image of Harlow is the original (1, 2, 3 or 4)? Explain your reasoning. [Note: this question is meant for fun and may end up being more about fashion trends than physics – but it is meant to get you thinking about image reversal.]

B. “Why do mirrors reverse left and right and not up and down?” Alice and Bob stand

in front of a large mirror in a dance studio, looking at themselves. Alice wonders

26


if their images are reversed “left and right” or “up and down”. She looks at Bob’s image in the mirror and memorizes what he looks like. Then she asks Bob to turn and face her, so she can compare the image to what Bob looks like in real life. Bob takes a couple of steps forward, turns around and faces Alice. Alice notes that, compared his the mirror image, Bob appears reversed left and right! Alice concludes: Mirrors reverse left and right, not up and down. Is this true? Can you see any flaws in Alice’s reasoning?

C. You are supplied with a small sign on which is printed “Charles Dodgson”,

which was Lewis Carroll’s real name. Hold the sign up to the supplied mirror with the writing facing the mirror: you will see the text in a mirror image. Now hold the sign up to a light and look at the writing on it from behind. Is there any difference between the appearance of the writing as seen in the mirror and the writing as seen from behind? Now curve the sign so the edges are closest to you and centre, about where the D is, is furthest away from you. Look at the sign in the mirror. In the mirror image are the edges curving towards you or away from you? In Part B, you may have concluded that mirrors reverse left and right. Do you want to change that conclusion? What do mirrors really reverse?

Activity 3

This activity uses the “Ray Box” feature of the PASCO Basic Optics Light Source. Place the light source flat on the table or on your notebook so it is sitting on its four little legs and plug it in. There is a wheel to select one, three or five parallel rays projected onto the table. If you place it on your open notebook the rays will be easier to see and you can trace them with a pen or pencil. You also should have a transparent glass trapezoidal prism, a small protractor and a ruler.

Trapezoidal prism:

A. Select the 1-ray and shine it on an open page of your notebook. Place the trapezoidal prism in the beam. You should see that part of the ray is refracted through the trapezoidal prism, but there also is a reflected ray. Adjust the angle of incidence. How does the brightness of the reflected ray vary with the angle of incidence? [Note, the angle of incidence is defined as the angle between the incident ray and the normal from the surface which emerges at the point where the ray touches the surface.]

B. Choose an angle of incidence, and carefully sketch and label the incident ray, the reflecting surface of the trapezoidal prism, and the reflected ray. Use the ruler and protractor to sketch and label the normal to the surface at the point where the

27


ray reflects. Measure the angle of incidence and angle of reflection. Is the Law of Reflection obeyed to within your errors? Repeat for twice for a total of three different incident angles. What is the largest source of error in measuring these angles?

C. Select the 1-ray and shine it on an open page of your notebook. Place the

trapezoidal prism in the path of the ray so that: • the angle of incidence is at least 45°, and • the ray emerges from the other side of the trapezoidal prism. The side

from which the ray emerges should be parallel to the side into which the beam enters. You should note that the emerging ray is parallel to the incident ray.

Sketch and label the incident ray and the surface of the trapezoidal prism through which you are refracting, the emerging ray, and the surface of the trapezoidal prism from which the ray emerges. Remove the trapezoidal prism, use the ruler to clearly mark all three parts of the ray, including the ray when it is inside the trapezoidal prism. Note that if you sketched the sides of the trapezoidal prism by running a pen along the surface, the line you drew will be about 1 mm in front of the actual glass surface; you should correct for this. Use the ruler and protractor to sketch and label the normals to the surface where the ray enters and exits the trapezoidal prism. Measure the angle of incidence and the angle of refraction for the ray when it first enters the trapezoidal prism. Use Snell’s Law to determine the index of refraction of the glass.

Activity 4

Here is the URL of a Flash animation of a ray of light traveling between air and glass. http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Optics/Refraction/Refraction.html Open the animation. There are two scenes you can toggle through: the first is of a ray traveling from air into glass, and the second is of a ray traveling from within some glass out into air. You may set the index of refraction of the glass within the range n = 1.25 to n = 1.75.

A. In the Air to Glass scene, set nglass = 1.5, and explore various angles of incidence. Record what happens for angles of incidence θ = 0°, 5°, 40°, 50°, 85° and 90°. Note the strengths of the reflected and refracted rays, if they exist, and the angle of refraction.

B. In the Air to Glass scene, set the angle of incidence to θ = 45°, and explore

various indices of refraction of the glass. Record what happens at the minimum nglass, and what happens as you increase nglass.

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C. In the Glass to Air scene, set nglass = 1.5, and explore various angles of incidence. Record what happens for angles of incidence θ = 0°, 5°, 40°, 50°, 85° and 90°. Note the strengths of the reflected and refracted rays, if they exist, and the angle of refraction.

D. In the Glass to Air scene, set the angle of incidence to θ = 45°, and explore

various indices of refraction of the glass. Record what happens at the minimum nglass, and what happens as you increase nglass.

Activity 5

This activity uses a Pyrex rod and two transparent beakers, one filled with water and the other with Wesson Oil.

A. Pyrex has an index of refraction of 1.47, water has an index of refraction of 1.33, and Wesson vegetable oil has an index of refraction of 1.47. All three materials are transparent. Predict what will happen if you dip the Pyrex rod in the water. Predict what will happen if you dip the Pyrex rod in the oil.

B. Test your predictions from Part A. Describe and explain your observations.

Activity 6

This activity uses the “Ray Box” feature of the PASCO Basic Optics Light Source. Place the light source flat on the table or on your notebook so it is sitting on its four little legs and plug it in. There is a wheel to select one, three or five parallel rays projected onto the table. If you place it on your open notebook the rays will be easier to see and you can trace them with a pen or pencil. You also should have a flat glass convex lens, a flat glass concave lens, and a ruler.

Convex Lens:

Concave Lens:

A. Select the 5-rays and shine them on an open page of your notebook. Take the

convex lens and focus the rays, so that the focal point is on your page. Sketch the five rays and the exterior shape and position of the lens. Label the focal point. Measure the focal length of the lens, which is the distance between the centre of the lense and the focal point for initially parallel rays.

B. Select the 5-rays and shine them on an open page of your notebook. Take the

concave lens and de-focus the rays. Leave enough room on the page so that you

29


will be able to sketch the rays backwards to the virtual focal point from which they appear to be emerging. Sketch the five rays and the exterior shape and position of the lens. Remove the lens and use a ruler to trace the rays backward to the spot from where they all seem to be emerging. Label the virtual focal point. Measure the focal length of the lens, which is related to the distance between the centre of the lens and the virtual focal point for initially parallel rays. Is the focal length for this lens negative or positive?

C. Switch the wheel to the red, green and blue thick beams. Using the lenses and

these coloured beams, can you create white light?

Activity 7

This activity uses optical components clipped to the 2.2m aluminum track. It is easy to slide these components along the length of the track, and to measure their position using the ruler on the track. Set up Viewing Screen at 50 cm. This means the front surface of the white screen should be above the 50 cm mark on the track, and facing down the length of the track where you will be placing other components. Set up the board with 5 mm hole in it at 100 cm and the light source with the single open hole at 150 cm, so the light shines through the hole onto the screen.

A. What is the size of the image of the hole on the viewing screen? Predict will the image of the hole get bigger or smaller when you move the light toward the hole, leaving the board and viewing screen fixed. Observe and record. Do your observations match your predictions?

B. Place the light source at 150 cm again, 50 cm away from the board with the hole.

Leaving the viewing screen and board fixed, predict what will happen to the image of the hole if you detach the Light Source from its holder and move it in a direction perpendicular to the direction to the track. In particular, if you move the light source sideways or up and down a certain distance x, how far will the image of the hole move on the viewing screen, and in what direction? Observe and record. Do your observations match your predictions?

C. Repeat the procedure of Part B with the light source placed at 200 cm, 1 m away

from the board with the hole.

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Activity 8

For this activity you are provided with two lenses which look almost identical. They are both convex, and therefore focusing or “converging” lenses. They both have diameters of 50 mm, but one is labeled “f/4”, and the other is labeled “f/2”, meaning that their focal ratios are 4 and 2. For a thin lens:

fss111

=′

+

where f is the focal length, s is the distance between the object and the lens, and s' is the distance between the image and the lens. By measuring s and s' the focal length can be determined. In this activity, you will estimate the focal length of the lens by making a single measurement of s', with s ≈ ∞. You will require two convex lenses, a screen or something white to project the image on, and a distant light source, such as a light bulb set up across the room.

A. Hold the lens in one hand and the screen or a white piece of paper in the other. Focus the image of a distant bright object (such as a window or light bulb across the room) on the screen. When the image is in focus, have a partner measure the distance from the lens to the screen. This is the image distance, s'. If you assume s = ∞, what is the focal length, f ?

B. Make a rough estimate of the actual object distance for your measurement in part

A. What percentage error did you introduce to your determination of f by assuming that s was infinity? By using the method of part A, do you think you will slightly overestimate or underestimate f ? There are two sources of error here, one is the random error introduced when you measure the image distance, and the other is the systematic error introduced by assuming that the object was at infinity. In this case, which is larger, the random error or the systematic error?

C. Repeat the procedure of part A for a second convex lens with a different focal

length.

Activity 9

An object and a viewing screen are held at a fixed distance d, and a focusing lens with positive focal length f is placed part-way between them. (This is the situation you will be

31


setting up in Activity 10.) In order to form a focused image, the sum of the object and image distances must be equal to d: s + s' = d.

A. Knowing the distance between the object and image d, and the focal length of the lens f, solve for the image distance of a focused image, s' in term of f and d. You will have to eliminate the variable s. [You should end up with a quadratic equation.]

B. Identify the discriminant of the quadratic equation for Part A. If the discriminant

is negative, then the solution for s' will have an imaginary component. Physically, this means that a focus is impossible and the image will always be blurry. For what condition on d will a focused image be impossible?

Magnification, m, is the ratio of the image size to the object size. By definition, |m| = h'/h. If the image is inverted, m is negative. For an image formed by a thin lens:

ssm′

−=

C. Consider a situation where d = 1.0 m, and f = 0.2 m. What are the two solutions

of s' ? What is the magnification, m, for the two solutions?

Activity 10

This activity uses optical components clipped to the 2.2m aluminum track. It is easy to slide these components along the length of the track, and to measure their position using the ruler on the track. Set up Viewing Screen at 50 cm. This means the front surface of the white screen should be above the 50 cm mark on the track, and facing down the length of the track where you will be placing other components. Place the f/4 convex lenses in the Adjustable Lens Holder and set up the lens holder at 100 cm and the light source with the illuminated crossed arrows pattern at 150 cm, so the pattern is facing toward the lens. For this set up, the distance between the source and the screen, d, is 1 m.

A. Starting with the f/4 lens close to the screen, slide the lens away from the screen to a position where a clear image of the crossed-arrow object is formed on the screen. Measure the image distance s' and the object distance s. Also measure the object size h and the image size h'. The object size is the distance between two pattern features on the crossed-arrow object, and the image size is the corresponding distance between these features in the image.

B. From measurements of s and s' you can predict the magnitude of the

magnification, m. For the two different images you focused for Part A, with d =

32


100 cm, how does the predicted |mpred| = s / s' compare with the measured magnification |mmeas| = h' / h?

C. Is it possible to find a focus for the f/4 lens when d = 50 cm?

D. For the f/4 lens, repeat measurements of s and s' for d = 90 cm and 110 cm, and

fill them in along with their inverse into a table in your lab notebook. Plot 1/s versus 1/ s' and find the best fit line (linear fit). This will give a straight line with the y-intercept (A0) equal to 1/f. What is the value of f for this lens?

E. Replace the f/4 lens with the f/2 lens. Make measurements of s and s' for d = 50

cm, 60 cm and 70 cm, recording the inverses as you go. Plot 1/s versus 1/ s' and find the best fit line (linear fit). This will give a straight line with the y-intercept (A0) equal to 1/f. What is the value of f for this lens?

This Student Guide was written by Jason B. Harlow, Dept. of Physics, Univ. of Toronto, in the Fall of 2008. Part C of Activity 2 is by David M. Harrison, Dept. of Physics, Univ. of Toronto. Last revision: March 14, 2009.

33

PHY131 Practicals Manual Electricity and Magnetism Module 1

Electricity and Magnetism Module 1 Student Guide


• Electric Charge • Coulomb’s Law • Addition of Electrostatic Forces

The Activities

Background for Activities 1 - 3 Here are four hypotheses about electric charge:

Hypothesis 1: The interaction between objects that have been rubbed or separated is due to a property of matter we call charge. There are two types of electric charge. Hypothesis 2: The strength of the interaction between electric charges increases as the distance between the charges decreases. Hypothesis 3: The strength of the interaction between electric charges increases as the quantity of charge increases and decreases as the quantity of charge decreases. Hypothesis 4: Excess charge moves readily on certain materials, known as conductors, and not on others, known as insulators. In general, metals are good conductors while glass, plastic, and rubber tend to be insulators.

34


Activity 1

A. Press a length of sticky tape 10 – 20 cm long firmly on the table top or other unpainted surface, with a few cm hanging over the edge. Form a non-sticky handle by looping the tape hanging over the edge onto itself. Do this for a second length of sticky tape. Peel one of the tapes off the table and hang it from the edge of the cupboard. Peel the second tape off the table and holding its handle bring it near the first tape. Try to keep your hand holding the second tape far away from the tapes that are hanging down. What happens? How does the distance between the tapes affect the interaction between them?

B. Place two more strips of sticky tape on the surface as in Part A. Using a pencil or ball point pen, but not a rollerball pen, mark the tapes with B for bottom. Press another strip of tape on top of each of the B strips; label these strips T for top. Pull one pair of strips off the surface, separate them, and hang them from the edge of the cupboard at least 50 cm away from each other. Pull the second pair of strips off the surface and separate them. Describe the interaction between two top strips when they are brought toward one another. How does the strength of the interaction depend on the distance between the tapes? Caution: if the two tapes come into contact with each other the charges on them may change. In the lab notebook describe the interaction between two bottom strips when they are brought toward each other. How does the strength of the interaction depend on the distance between the tapes? Describe the interaction between a top strip and a bottom strip when they are brought toward each other. How does the strength of the interaction depend on the distance between the tapes?

C. Rub the supplied plastic rod with the fur, hold the rod horizontally and bring the it near but not touching the hanging bottom and top strips. Describe what you observe. Caution: if the tape touches the rod the charge on the tape can change. Again being careful not to touch the hanging strips, bring the fur near them. What do you observe? Rub the supplied glass rod with the polyester cloth, hold the rod horizontally and bring it near but not touching the hanging bottom and top strips. Describe what you observe.

D. Do your observations in Parts A – C support the hypotheses? Which hypotheses are supported and which are not by your observations? Please explain in the lab notebook using complete sentences. You should not just state results that you may have learned about in class or from the textbook. Rather we wish you to devise a sound and logical set of reasons based on your observations.

E. Following Benjamin Franklin, we arbitrarily call the charge on the glass rod after being rubbed with polyester positive and the charge on the plastic rod after being rubbed by fur negative. For the sticky tape, what type of charge is on the top strip? What type of charge is on the bottom strip?

35


What type of charge is on the fur after rubbing the plastic rod? Devise and carry out an experiment to test your answer. Describe the experiment and the result in the lab notebook. Predict what type of charge is on the polyester after rubbing the glass rod? Devise and carry out an experiment to test your prediction. Describe the experiment and the result in the lab notebook.

Please remove all the sticky tapes from the table top when you have completed this Activity.

Activity 2

Blow up the two balloons, tie them off, and tie each to the end of one of the supplied strings. Hang each balloon from the horizontal rod which is attached to the vertical rod that is clamped to the table. You will want each balloon to be as far as possible from the vertical rod, the edge of the tabletop, and any other objects. At the end of today’s Practical you may keep the balloons. Below you will also be using a small white ball of pith1, which is hanging by a silk thread onto a small stand. You will also be supplied a dark gray ball hanging by a thread from a stand; these are pith balls that have been coated in Aluminum.

A. Rub the glass rod with the polyester, and slowly bring the rod near the Aluminum coated pith ball, letting the two touch. Describe what happens. Can you explain your observations? Do your observations confirm or reject any of the hypotheses described at the beginning of this document?

B. What is the sign of the electric charge on the coated ball? Confirm your answer using the glass rod after rubbing with polyester and the plastic rod after rubbing with fur, being careful not to let the rods touch the ball.

C. Remove the electric charge on the Aluminum coated ball by touching it with the metal plate. Repeat Part A using the plastic rod after rubbing it with fur.

D. Now what is the sign of the electric charge on the ball? Confirm your answer using both the plastic rod after rubbing it with fur and the glass rod after rubbing it with the polyester.

E. Rub the glass rod with the polyester, and slowly bring the rod near the uncoated pith ball, letting the two touch. Describe what happens. Can you explain your observations? Do your observations confirm or reject any of the hypotheses described at the beginning of this document?

F. What is the sign of the electric charge on the uncoated ball? Confirm your answer using both the plastic rod after rubbing it with polyester and the glass rod after rubbing it with the polyester. Are your results the same as for Part B? Explain.

1 Pith is an insulator found in vascular plants.

36


G. Rub the two balloons with the fur. Bring the balloons closer together. Describe what happens. Do your observations confirm or reject any of the hypotheses described at the beginning of this document?

H. Determine whether the charges on the balloons are positive or negative. I. As you showed in Activity 1, when you rub the fur with the plastic rod it becomes

charged. Does the charging of the balloons depend on whether or not the fur is charged? Explain.

Activity 3

Coulomb devised a clever trick for determining how the interaction between charges depends on the quantity of charge without knowing the actual amount of charge. Say one of the Aluminum coated balls has a charge Q, and the other Aluminum coated ball is uncharged. If the two balls touch, they will quickly exchange charge until both have a total charge Q/2 on them. If one of the balls is then discharged by touching it to a large piece of metal, the procedure can be repeated to get Q/4 on both balls.

A. As you saw in Activity 2 Part G, if two objects have the same type of charge when they are brought near each other they repel each other. How is the force that they exert on each other related to the angle that their supporting strings make with the vertical? Just a qualitative answer is sufficient, although you may derive the formula if you wish.

B. Using Coulomb’s trick, charge the two Aluminum coated balls with a charge Q/2. Bring them close together. Describe what happens. Do your observations confirm or reject any of the hypotheses described at the beginning of this document?

C. Now put a charge Q/4 on the two Aluminum coated balls and bring them close together. Describe what happens. When the two charges are the same distance away from each other as in Part B, how does the force the charges exert on each other compare to your observations for Part B? Do your observations confirm or reject any of the hypotheses described at the beginning of this document?

D. Put a charge Q on one of the Aluminum coated balls and discharge the other Aluminum coated ball. Bring the two balls close to each other. Describe what happens. Can you explain your observation?

Activity 4

In the late eighteenth century Coulomb used a torsion balance and a great deal of patience of discover how the electric force between small spherical charged objects depends on the distance between the objects. A modern implementation of his apparatus is shown on the next page; using it also requires considerable patience.

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It is also possible to do a similar determination using the charged balls that you may have used in Activity 3, and these experiments have also been done. However, in practice this method is even more difficult than Coulomb’s. An animation which side-steps these difficulties by simulating the experiment is available at: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/EM/Coulomb/Coulomb.html The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/EM/Coulomb/Coulomb.swf This version will only work if your browser is configured to display Flash animations directly without an html wrapper.

A. Open the animation and explore how it works. B. Move the right hand charge with the slider to some distance between the support

points of the strings, move it to a new position, and then return it to the same original distance. You will notice that the measured angle the strings make with the vertical usually has a slightly different value for the same distance. Under what circumstances would a real apparatus exhibit this behavior?

C. If you had the patience of Coulomb and repeated the process of Part B a large number of times and made a histogram of the measured angles, what would you predict the shape to be? How would you characterize the width of the shape?

D. Set the right hand charge to any position that you like and record the distance and the angle. Calculate the value of the electric force F exerted on the left hand charge by the right hand charge. What is the direction of that force?

E. What is the error in this experimental determination of the value of the force? You may find one or more of the following error relations useful:

)(cos)tan(

)sin()cos(

)cos()sin(

2 θθθ

θθθ

θθθ

Δ=Δ

Δ=Δ

Δ=Δ

Note: in the above equations the error in the angle must be expressed in radians. F. For this same position of the left hand charge calculate the distance r between the

centers of the two 1 gram masses. G. Calculate the error in this measurement of r.

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H. We took 10 datapoints from the animation, calculated r and F and their errors, and placed the result in a dataset named CoulombDistForce.2 A copy is located in the following area: public:Modules/E&M/Module01/Data Explore this data with the ViewDataset program which is on the desktop of your computer and/or a spreadsheet program such as Excel. As always, the datasets are tab separated, with the first row a title, the second row the names of the variables, and the rest of the rows the actual data. How could you use a collection of force-distance, F versus r, data for different values of the distance to determine how the force depends on the distance? How many different ways can you and your partners think of? Which do you think might be best?

I. Here is a method which you may have thought of in Part I. Imagine that the force F varies in an unknown way with the distance r:

nrcF = (1)

You wish to determine n from data of F versus r. A good way to do this is to fit the data to Eqn. 1. However, n does not have a linear relation to F and r, so a non-linear fitter has to be used. We have non-linear fitters, but using them is not as simple as the PolynomialFit program which you may have used in other Modules. But, if we take the logarithm of both sides of Eqn. A1, we get: )ln()ln()()ln( crnF +−= (2) Recall that the generic equation of a straight line is: bmxy += So if we take the logarithms of F and r and fit this data to a straight line: Slope = (-n) Intercept = ln(c) Use this method using PolynomialFit which is on the desktop of your computer. The following file, located in the same directory as the one you looked at in Part H, contains data of the logarithm of the force F versus the logarithm of the distance r: CoulombLnDistLnForce

2Not having Coulomb-like patience, I did not do the error calculations by hand. Instead I used some error propagation routines with Mathematica software.

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J. There is another file which you may wish to use to explore the relation between the force and the distance, located in the same directory as the other two data files. CoulbDist2Force: F versus r2

Activity 5

We commonly say that there are two types of electric charges, positive and negative. Imagine two charges Q and q are separated by a distance r as shown. We construct a unit vector r pointing from Q to q. Then the force exerted on q by Q is given by Coulomb’s Law:

rrqQcF qon ˆ

41

20πε

=r

Here c is either +1 or -1. Which value of c correctly expresses the direction of the force exerted on q by Q for all four combinations of the two charges being positive or negative? You may find it useful to build a table of the four possible sign combinations and the direction of the force on q for each combination.

Activity 6

There is another model of electricity, due to Benjamin Franklin and William Watson in the 18th century, called the one fluid theory. In this model all objects contain some amount Q or q of electric “fluid”. Portions of the fluid repel each other, and portions of fluid are attracted by matter that does not have fluid. There is a “natural” quantity of this electric fluid, Q0 or q0. When an object has more fluid than this natural amount we say that it has a positive amount of fluid, and when it has less fluid than this we say that it has a negative amount.

A. Object 1 has a natural quantity of electric fluid Q0 and has an amount Q of fluid. Object 2 has a natural quantity of electric fluid q0 and contains q fluid. The two objects are separated by a distance r. Write down a form of Coulomb’s Law for the electric force that is exerted on object 2 by object 1.

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B. Imagine that instead of thinking about an electric fluid we think about the total number of electrons contained in some object. Is there a “natural” quantity of electrons in the object? If so, what might it be related to?

C. We now have two different models of electricity, the usual one of positive and negative charges and the one fluid one of Franklin and Watson. Both seem to explain all the data. What are all the differences you can think of between these two models?

Activity 7

In the figure three charged particles lie on a straight line and are separated by distances d. Charges q1 and q2 are held at fixed positions. Charge q3 is free to move but happens to be in equilibrium (i.e. no net electrostatic force acts on it). Charge q2 has the value Q. What value must the charge q1 have?

Activity 8

In Case A a point charge +q is a distance s from the center of a small ball with charge +Q. In Case B the point charge +q is a distance s from the center of a rod which is uniformly charged with a total charge +Q. Consider the following student dialog:

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Student 1: “The charged rod and the charged ball have the same charge +Q and are the same distance from the point charge +q. So the force on +q will be the same in both cases.” Student 2: “No, in Case B there are charges spread all over the rod. The charge directly below the point charge will exert the same force on +q as the ball in Case A. The rest of the charge on the rod will make the force in Case B bigger.” Neither student is correct. What are the errors made by each student? What is a correct description of how the forces compare? Explain.

Activity 9

A. A point charge +Q is located a distance R away from three identical point charges, each of charge +q/3, equally distributed along a semicircular arc of radius R as shown. What is the total force, magnitude and direction, exerted on +Q?

B. A point charge +Q is located a distance R away from five identical point charges, each of charge +q/5, equally distributed along a semicircular arc of radius R as shown. What is the total force, magnitude and direction, exerted on +Q?

C. A point charge +Q is located a distance R away from a semicircular arc that is uniformly charged with a total charge of +q as shown. The charge per arc length λ along the semicircle is:

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Rq

πλ +

=

What is the total force, magnitude and direction, exerted on +Q?

Activity 10

We commonly call forces such as you exert with your hand on a door to close it a contact force. But physicists say that there are exactly four types of interactions that can occur:

1. Gravitational 2. Electromagnetic 3. Weak 4. Strong or Nuclear

Do contact forces fit into this classification? If yes, how? If no, should they be added to this list as a fifth type of interaction? This Guide was written in October 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. Activities 1- 3 are based on Priscilla W. Laws et al., Workshop Activity Guide, Module 3, Unit 19, (John Wiley, 2004), pg. 531-533. Activity 7 is from Edward F. Redish, Rachel E. Scherr and Jonathan Tuminaro, “Reverse-Engineering the Solution of a ‘Simple’ Physics Problem”, The Physics Teacher 44, 293 (2006). Activity 8 is from Lillian C. McDermott et al. Tutorials in Introductory Physics (Prentice Hall, 2002), pg 75. Last revision: January 19, 2008.


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Concepts of this Module • Introducing current and voltage • Simple circuits • Circuit diagrams

Background When water flows through a garden hose, we can characterize the rate of flow as the volume of water passing any cross section of the hose per time. Units for this flow could be m3/s. Similarly, for a conducting wire electric charge can flow down the wire. We call the rate of flow of electric charge the current, which is the charge Q passing a cross section of the wire per time t. In SI units this is C/s. 1 C/s is also called an ampere, A. Conventionally the current is given the symbol I or i, so the definition of current is:

tQI

ΔΔ

≡

In order for water to flow in a hose a source of pressure is required. Similarly, for a current to flow in a wire a source of voltage is required. Common voltage sources are batteries, electric generators, and power supplies. In this Module we will be using a battery.

Record Keeping The keeping of good records in the laboratory notebook is an important skill for any experimental work. In this Module, some of the data you take may be used in later Modules. So be particularly careful to insure that the notebook contains a complete record of all the Activities you perform.

The Activities Note: the battery you will be using in the Activities is filled with acid. Do not lay it on its side or turn it upside down.

Activity 1

Mounted on a plastic frame is a light bulb and two banana sockets. On the bottom of the light bulb are two metal contacts which are connected to wires. The


44

other ends of the wires are connected to the banana sockets, which are a convenient way to attach wires with a corresponding plug. The figure on the previous page shows the bulb, wires and sockets. The figure to the right traces the conductors from the banana sockets through the light bulb. If you are viewing this in color, the conductors are in red.

A. Examine the mounted light bulb and identify the parts that are indicated in the above figures. Connect a wire from each terminal of the battery to each of the banana sockets. The light bulb should light. It is good practice to use a red wire to connect to the red terminal of the battery marked +, and a black wire to connect to the black terminal of the battery marked -.

B. Here are four possible models for how the current flows in the wires when the light bulb is lit:

Which case is most correct? Why?

C. You are supplied with a clamp meter, which measures the current in a wire that goes between the jaws of the clamp. Appendix 1 describes how to use this meter. Use the meter to measure the current in one place along one of the wires. As you slightly move the position of the clamp the measured current will change a bit. Quantify this by guessing the error ΔI to one significant figure.3

D. Use the clamp meter to check your prediction of Part B. Were you correct?

3 Although one can carefully repeat the measurements of the current and calculate the standard deviation to get a value for ΔI, that will not be necessary here. This is a general principle: do things the simple way first. If you later discover that you need a more careful determination you can always go back and do so.


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Please disconnect all the wires and turn off the meter when you are done with this Activity.

Activity 2

Instead of drawing a picture of an electric circuit, we can schematically represent it with a circuit diagram. Here are a few elements of circuit diagrams.

Wire

Wires that are joined

Wires that are not joined

Light Bulb

Battery

For the Battery, the positive terminal is on the right and the negative terminal is on the left. Here is a mnemonic for remembering this: a + symbol has more line in it than –, and the longer line of the battery is the + terminal. Draw a circuit diagram of the circuit of Activity 1.

Activity 3

In Activity 1 the light bulb had two conducting contacts on the bottom. Most light bulbs only have a single contact on the bottom, and use the conducting side of the base for the other contact. Using the supplied unmounted flashlight light bulb, the battery, and only one wire can you make the light bulb light? You may not cut the wire.


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Activity 4

A. A knife switch is just a length of conductor on a pivot and a contact for the conductor. Connect the knife switch as shown between one terminal of the battery and one terminal of the mounted light bulb as shown. When the “knife” is closed what happens? Explain why the circuit behaves like this. You will want to notice how bright the light bulb is when the switch is closed.

You are supplied with a common light switch which is mounted with banana sockets for easy connection. A photograph of the insides of such a switch appears to the right, and a simple Flash animation of it is at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/LightSwitch/LightSwitch.html. The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/LightSwitch/LightSwitch.swf This version will only work if your browser is configured to display Flash animations directly without an html wrapper.

Explain how this switch works. B. Place the light switch in the circuit in place of the knife switch that you used in

Part A. Close and open the switch. Is its effect on the circuit and the light bulb any different than the knife switch?


47

C. The symbol for a switch, either the knife switch or the light switch, is shown to the right. So here is the circuit diagram for the circuits of Parts A and B. In the circuit, we say the switch is wired in series with the battery and the light bulb. Note that although the components are laid out differently from the figure in Part A, the two representations are completely equivalent. Here is a circuit in which we say the two switches are wired parallel to each other. If both switches are closed at the same time predict how the brightness of the light bulb will compare to when only one switch is closed. Explain your prediction. Wire the circuit and check your prediction. You will find it convenient to note that the banana plugs on the ends of the wires have an extra hole into which another banana plug may be inserted.

Please disconnect all wires from all the circuit elements when you have completed this Activity.

Activity 5

Meters that measure currents are called ammeters. Conventional ammeters, as opposed to the clamp meter you used in Activity 1, must be inserted in series into the circuit, just as the single switch was inserted into the circuit in Parts A and B of Activity 4. The circuit diagram symbol for a conventional ammeter is shown to the right. Here is the circuit diagram for using a conventional ammeter to measure the current in a wire of the same setup you investigated in Activity 1.


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You are supplied with multimeters which can be used as conventional ammeters. Details on how to do this are in Appendix 2.A. Wire the circuit and measure the current in the wire. Check your measurement using the clamp meter. Do they give the same results for the magnitude and direction of the current? How do the values compare to the results of Part D of Activity 1? Please disconnect all wires from all the circuit elements and turn off the meters when you have completed this Activity.

Activity 6

Voltmeters measure voltages of, say, batteries. The circuit symbol for a voltmeter is shown to the right. Voltmeters are typically wired in parallel. So the circuit diagram that measures the voltage of the battery while the light bulb is being lit is shown to the right. Instructions on how to use a multimeter as a voltmeter are given in Appendix 2.B. Use a multimeter to measure the voltage of the battery. The rated voltage is written on the front of the battery. How do the two values compare? Disconnect the battery from the circuit and use the voltmeter to measure its voltage. How does it compare with the voltage when it was in the circuit? Please disconnect all wires from all the circuit elements and turn off the meter when you have completed this Activity.

Activity 7

Rewire the circuit that lights the light bulb with the ammeter in the circuit again. In the circuit diagram to the right we have indicated a number of points in the circuit. Use the voltmeter to measure the voltage difference between 1 and 2, 2 and 3, 1 and 3, 4 and 5, etc. If the meter reads a very small voltage difference between two points, you should decrease the scale of the reading by rotating the


49

upper knob: when the scale is too small the meter will read -1; in this case increase the scale of the reading. Do you see a pattern? What is the voltage “drop” across the light bulb? What about across the ammeter? One of the wires? Summarise your findings. Can you explain them? Why did we use the word “drop” above? Please disconnect all wires from all the circuit elements and turn off the meters when you have completed this Activity.

Appendix 1 – The Clamp Meter A clamp meter measures the current in a wire that passes through the jaws of the circular clamp. For now we will treat how the meter does this as “magic”; in a later Module we will return to investigate how it works. The jaws may be separated by pressing on the Clamp Opening Handle. When the current is flowing in the direction shown, the reading will be positive; if the current is flowing in the opposite direction to that shown the reading will be negative. There is a small arrow on the inside of the jaws of the clamp indicating the current direction shown in the figure. Here is a close-up of the controls of the meter. The Function Select knob has three positions:

1. Off 2. 400A 3. 40A

We will be using the 40A function. After turning the meter on it must be zeroed.

1. Place the meter close to the part of the wire whose current will be measured and orient the meter as it will be when it is clamped around the wire.

2. Press ZERO: The display will read ZERO. 3. Press on the Clamp Opening Handle to separate the jaws of the clamp, place the

clamp around the wire, and release the handle. The meter will now read the current in the wire in amperes.


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To measure the current at a different location or with the meter at a different orientation:

1. Move the meter close to where it will do the new measurement, oriented as it will be when clamped around the wire.

2. Press on ZERO; the display will no longer read ZERO. 3. Press the ZERO button again: the display will read ZERO. 4. Clamp the meter around the wire and read the current on the display.

If it is difficult to see the display because of the orientation of the meter:

1. Press on the HOLD button. This will cause the reading to be held and the display will read HOLD.

2. Remove the meter and read the current on the display. 3. Press on the HOLD button to return the meter to normal operation. The display

will no longer read HOLD.

Appendix 2 – The Multimeter This module uses multimeters, which are devices capable of a number of different electrical measurements. With the flexibility of this instrument comes a price: at first glance there is a bewildering array of controls and inputs. This Appendix will guide you through this complexity to learn how to use the meter to measure currents and voltages. Just as for the clamp meter, for now we will treat how the instrument actually works as “magic”.

2.A – Measuring Currents The figure shows the multimeter configured to measure currents. Not visible in the photograph is the On/Off Switch, which is on the right side of the meter. The upper knob controls the scale of the readings, and should be set to the shown 2000mA position. The lower knob selects the type of measurement that the meter will do, and should be set to the shown DCA position. DCA stands for DC Amps. The wires are connected to the terminals as shown, which are labeled mA and COM. COM stands for Common. Note that the meter reads current in milliamperes, mA, while the clamp meter reads in amperes.


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When the direction of the current in the wires is as shown, the meter will read a positive current. If the current is going in the opposite direction to that shown the meter will show a negative value.

2.B – Measuring Voltages Here is the meter configured to read voltages. Not visible in the photograph is the On/Off Switch, which is on the right side of the meter. The upper knob controls the scale of the readings, and should be set to the shown 20V position. The lower knob selects the type of measurement that the meter will do, and should be set to the shown DCV position. DCV stands for DC Volts. The wires are connected to the terminals as shown, which are labeled COM and V-Ω. COM stands for Common, and the V stands for Volts. If the wire connected to the V-Ω terminal of the meter is connected to the + terminal of the battery the meter will read a positive voltage; if it is connected to the – terminal the reading will be negative. If you are viewing this in color, this is the red wire in the photograph. This Guide Sheet was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in November 2007. Activity 1 draws on material from Priscilla W. Laws et al., Workshop Activity Guide, Module 4, Unit 22.6, (John Wiley, 2004), pg. 604. The photograph of the interior of a light switch is by Scott Erhardt, and placed in the public domain at http://en.wikipedia.org/wiki/Image:Light_switch_inside_explained.jpg. Last revision: January 25, 2008.


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Electric Fields - An Introduction A field is a function, f(x,y,z), that assigns a value to every point in space (or some region of space). Charged particles alter the space around themselves to create an electric field, which, in turn, determines the electric force that will be exerted on a positive test charge placed at each point. The electric field is a vector field, which means that a vector (with magnitude and direction) is assigned to every point in space. For positive test charge q, it is defined as:

q)z,y,x(F)z,y,x(E q on

rr≡ .

Therefore, the electric field is the electric force per unit charge and is equal to the electric force acted on by a particle with a charge of 1 C. Electric fields can be represented using electric field lines, which have the following properties: • The tangent to a field line at any point is in the direction of the electric field at that

point. • The field lines are closer together where the electric field strength is larger. • Every field line starts on a positive charge and ends on a negative charge. • Field lines cannot cross. The electric potential is defined as:

qUV ssourceq+≡ where Uq+sources is the electric potential energy. It describes the “ability” of the source charges to interact with any charge q and is present in space whether or not charge q is there to experience it. It is a property of the source charges, and is independent of test charge q. The electric potential and the electric field are related by the following pair of equations:

∫−=Δf

i

s

ssdsEV

dsdVEs −=

where ΔV is the potential difference, and s is the position along the line from the initial position si to final position sf. The negative sign indicates that the electric field lines always point in the direction towards decreasing electric potential, so a positive charge loses potential energy and gains kinetic energy as it accelerates along the direction of the electric field lines. The electric potential can be represented in four ways, using a potential graph, an elevation graph, equipotential surfaces, and contour maps. Equipotential surfaces are frequently used. When moving a charged particle between any two points on an equipotential surface, the following conditions hold:


53

• The direction of the electric field at any point is perpendicular to the tangent lines to the equipotential surface.

• The electric potential energy is constant (ΔU = 0). • The electric potential is constant (ΔV = ΔU/q = 0). • No work is done on moving the charged particle (ΔV = 0, so W = F ds = q E ds = -q

dV = 0). It is also worth noting that the tangent lines to equipotentials are perpendicular to electric field vectors. In this module, we will explore the concept of the electric field and the usefulness of equipotential surfaces.

The Activities Unless otherwise instructed, answer all questions in your lab notebook, including your own sketches where appropriate. For some activities, which will be indicated below, you will need to add sketches to diagrams that are provided at the end of this Student Guide. When completed, these should be stapled, taped, or glued into your lab notebook at the appropriate place.

Activity 1

The electric field is an example of a vector field. You can think of a vector field as a region of space filled with an infinite number of arrows, with each arrow’s length proportional to the value of the vector quantity at that point in space, and the direction of the arrow showing the direction of the vector. A wind-velocity field is an example of a vector field. Such a field can be represented in several ways. One way to describe the wind-velocity field is to assign to each of the infinite number of points in space a pair of numbers: a wind speed, and a wind direction. Another way to describe the wind-velocity field is the graphical method: drawing an arrow whose length is scaled to give the speed of the wind, and whose direction points in the direction of the wind. Naturally, with the graphical method, one would just draw a representative number of arrows, not one for each one of the infinite points in space. The two figures below show two different representations (weather maps) used by meteorologists to indicate wind speed and direction across an area (taken from http://www.intellicast.com/Global/Wind/ and http://squall.sfsu.edu/). Note: The knot is a unit of speed sometimes used in meteorology, which has its origins in navigation, as the speed of a ship used to be measured with the help of a knotted rope. 1 knot = 1 nautical mile per hour = 1.151 statute mile per hour (mph) = 1.852 km per hour. A. How are direction and magnitude indicated on these two maps?


54

B. In what ways is the first map better? In what ways is the second map better? C. Can you think of any other ways to provide information on wind vectors? D. Can you think of any other examples of vector fields?


55

Activity 2

A. The electric field of a point charge (not shown) is given below at one point in space. Can you determine whether the charge is positive or negative? Explain why or why not.

900 N/C ⎯→

| | | | | | B. In the figure below, the electric field of a point charge is shown at two positions in

space. Now can you determine whether the charge is positive or negative? Explain why or why not.

900 N/C 400 N/C ⎯→ →

| | | | | | C. Can you determine the exact location of the charge? If so, draw it on the figure

shown in Part B. If not, explain why not.

Activity 3

At each of the dots in the four figures shown below, use a black pen or pencil to draw and label the electric fields 1E

r and 2E

r due to the two point charges. Sketch these figures and

draw the electric fields in your lab notebook. Make sure that the relative lengths of your vectors indicate the strength of each electric field. Then use a red pen or pencil to draw and label the net electric field netE

r at each dot.

(b)

q1 q2

(a)

(c) (d)

q1 q2

q1 q2 q1 q2


56

Activity 4 The graphs below of electric potential versus position are for regions

in which there may be electric fields. On each of these graphs, draw a second line that is consistent with each of the following modifications. A. The direction of the electric field in Region A is reversed, while its magnitude is

unchanged and the potential at x = 0 cm remains the same. B. The electric field in Region B remains the same, but the potential at x = 2 cm

increases to 10 Volts. C. The electric field in Region C remains the same, but the potential is cut in half at x =

2 cm. D. The magnitude of the electric field in Region D is increased keeping the same

direction, and the potential at x = 2 cm remains the same. E. The direction of the electric field in Region E is reversed and its magnitude increases,

but the potential at x = 4 cm remains the same. F. The electric field in Region F remains the same, but the potential at x = 2 cm is 20

Volts. Note: An additional copy of these graphs is provided at the end of this Student Guide. Draw your lines on that pull-out copy and attach it to your lab notebook at the appropriate place.


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Activity 5

Two very large, parallel insulating sheets have the charge densities, η, shown in the figure to the right. (Only a small portion near the centre of the sheets is shown.) For each change listed in the table below, state whether the magnitude of the electric field at point P just to the left of the left sheet increases, decreases, or remains the same. Provide your answers in your lab notebook. You may find it useful to tabulate your answers in the format shown. Change Increases Decreases Same 1. Sheet B is moved to the left. 2. Sheet A is moved to the right. 3. Sheets A and B exchange positions. 4. The area of both sheets is doubled while charge density η remains constant.

5. The area of both sheets is doubled while the total charge on each remains constant (η changes).

6. The charge density of sheet A is changed to -3μC/m2. 7. The charge density of sheet B is decreased. 8. The sign of the charge on sheet B is changed. 9. A positive point charge is placed at Point R. 10. Point P is moved a small distance to the left.

Now, for each change listed in the table below, state whether the magnitude of the electric field at point R midway between the two sheets increases, decreases, or remains the same. Note that when one of the sheets is removed, point R remains fixed at its original location. Change Increases Decreases Same 11. Sheet B is moved to the left, but is still to the right of point R.

12. Sheet A is moved to the right, but is still to the left of point R.

13. Sheets A and B exchange positions. 14. The area of both sheets is doubled while the total charge on each remains constant.

15. The charge density of sheet A is changed to -3μC/m2. 16. The charge density of sheet B is changed to -


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12μC/m2. 17. The sign of the charge on sheet B is changed. 18. A positive point charge is placed at Point P. 19. Point R is moved closer to sheet A.

Activity 6

The graph in the figure below shows the electric force in the x-direction acting on a proton at different times. Four students are discussing inferences that might be drawn from this graph: Student 1: “The electric fields at C and E are zero because the slope of the line is zero.” Student 2: “No, the electric potentials at C and E are zero because the slope of the line is zero.” Student 3: “I think the electric field is zero at F because the electric force is zero.” Student 4: “No, you are all wrong. Force and electric field vary as 1/r2, so we should be looking at curved lines, not straight lines.” Which, if any, of these students is correct? Explain your answer.

Activity 7

You have been assigned the task of determining the magnitude and direction of the electric field at a point in space. Devise an experiment to determine the electric field. (Note: you will not actually perform this experiment.) List any objects that you will use, any measurements that you will make, and any calculations that you will need to perform. Describe briefly how you would analyze the results of your experiment. Make sure that your measurements do not disturb the charges that are creating the field.


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Activity 8 The dashed lines in the figure below show equipotentials in a region

in which there is an electric field. A positive point charge is placed at each of the seven labelled points in turn. On the diagram, draw vectors to show the direction of the force on the charge at each of the labelled points. Briefly explain your reasoning. Note: An additional copy of this figure is provided at the end of this Student Guide. Draw your vectors on that pull-out copy and attach it to your lab notebook at the appropriate place.

Activity 9 - Electric Field Mapping

In this experiment, charges will be placed on conducting surfaces and the electric field set up by these charges will be mapped. In Module 2 you used a battery as a source of voltage. Here you will be using a DC power supply, which has exactly the same functionality. Work Done When a Charge Is Moved In an Electric Field


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Recall that work = force × displacement, where "force" is the component of the force in the direction of the displacement. Suppose a positive charge q = 5 x 10-6 C is placed at point A in Figure 1. The electric field strength E is assumed to be constant, E = 7 N/C, in the direction indicated by the arrows. The unseen charges which set up the electric field are responsible for the push on the charge q, and the net force on the charge is F = qE = 35 x 10-6 N. Assuming the distance through which the charge q moves from A to B is d = 2 m, the work done by the electric field is W = Fd = 70 x 10-6 Joules.

Fig.1

The situation is quite different in Figure 2. There, the path along which the charge moves is perpendicular to the electric field. This means that the force exerted on q by the charges which set up the electric field is perpendicular to the displacement of the charge, so the component of the electric force along the direction of the displacement is zero.

Fig.2

Thus, the electric field does no work on the charge q as it moves from point A to point B. Since no force resists the motion along the path, we are free to imagine that virtually any small force whatever could have moved the charge q along this path; the cause of this force is irrelevant. The point here is that the electric field does no work on any charge moved along a path that is everywhere perpendicular to the electric field. Part A Along which paths A, B, C, and D, shown in the figure below, is work done by the electric field (indicated by the arrows) on a positive charge moved along the path, and along which paths is zero work done?


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Using a Multimeter to Find Paths of Zero Work A multimeter in the DC-V (direct current, voltage) mode can be used to map paths of zero work in an electric field. It's done this way: the negative lead from the black jack is connected to some object whose charge state is constant; this is often the earth ("ground"), which is so large that its surface charge may be regarded as constant. Other objects can be treated as if they were ground, and with the negative lead connected to them. The lead from the red jack (the positive jack) of the multimeter is then used as a probe to pick a point--any point--and measure its voltage; then the probe is moved over the region to find points in space which provide the same voltage reading. Paths made up of points which are at the same voltage are paths along which zero work would be done if any charge is moved along this path. Voltage is sometimes referred to as "potential", so the lines or curves of equal voltage are called "equipotential paths", or "equipotential curves", or "equipotential lines". If one can map the equipotential curves of an electric field, one then has only to draw perpendicular curves or lines which intersect the equipotential curves at right angles in order to specify the electric field. How is the Direction of Electric Fields Determined?


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The direction of the electric field lines may be shown by arrows on the field lines; they point toward regions of lower potential, or voltage. Note: Additional copies of the figures in Parts B and C are provided at the end of this Student Guide. Draw your vectors on that pull-out copy and attach it to your lab notebook at the appropriate place. Part B Sketch several representative lines representing the electric field pattern for the three equipotential patterns shown below. Draw arrows on the lines to show the direction of the electric field.

Part C Sketch the equipotential patterns corresponding to each of the electric field patterns shown below. Label three of the equipotential curves in each diagram "20 V", "10 V", and "5 V"; these are completely arbitrary. Remember, the electric field lines point toward regions of lower potential.

Experimental Procedure


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You are going to map the electric field patterns set up by pairs of objects that are oppositely charged. One object will be negatively charged, and will be referred to as "ground"; the other object will be positively charged. The pairs of objects are: (i) parallel lines, and (ii) a pair of points. You will be provided with two sheets of black conductive paper that are capable of carrying small current and have painted with conducting silver paint. Since the charged paths will actually be conductive ink electrodes, they will be referred to as electrodes. Diagrams of these traces are included in this handout.

1. Choose one of the sheets and place it on your corkboard. NOTE: The traces for parallel lines have a large silver pad at the midpoint of each line. Use these pads as electrical contacts for the parallel electrodes.

2. Select the red wire with a banana plug on one end, and a ring terminal on the other. Insert the banana plug into the red (positive) jack of the power supply, and place the ring terminal on top of one of the silver pads on the conductive paper. Ensure the flat side of the ring terminal is the side touching the paper, and press a conductive push pin through the ring terminal into the corkboard. Make sure that good metal-to-metal contact is made between the ring terminal and the silver pad.

3. Select the black wire with banana plug and ring terminal, this time connecting the

black (negative) power supply jack to the other electrode with another push pin.

4. On the diagram in this handout (not on the black paper), label the corresponding electrode "20 V", and the other one "0 V"; it doesn't matter which one is positive, and which is negative.

5. Select the wires (one red, one black) with a banana plug on one end and a pointed

metal rod on the other. These are the probes for the multimeter. Connect the red banana plug to the V-ohm jack and the black banana plug to the COM (common) jack on the multimeter.

6. Switch on your multimeter and select DC-V to enable measurements of voltage

(potential). If the display shows "batt", replace the battery of the multimeter.

7. Plug in the power supply, turn it on, and set its voltage to about 20 Volts. You can check the voltage using the multimeter. To prevent the highest voltages to be out of range for your multimeter, use the 200 V setting.

8. The power supply acts just as a DC battery would, so the objects on the corkboard

are now charged and have an electric field in the region between them.

9. When measuring potential of a point on the conductive sheet using the multimeter, bring and hold the black probe in contact with the push pin connected to the negative (-) jack of the power supply, bring the red probe in contact with the point of interest, and observe the reading on the multimeter.


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When the circuit is complete, it should look something like the photo below. When the circuit is complete, it should look like the one below.

10. Gently scrape the banana plug over the paper between the two objects until you find a point at which the potential (voltage) is about 5 Volts; don't make any pencil or pen marks on the black paper. Next, find the path from this point along which the potential stays constant at 5 Volts; follow this path wherever it leads, even if the path curves around behind the edges of the objects. This path traces a curve or line called the "5-Volt equipotential line". You may find it easier to plot a series of points rather than tracing a continuous line. For example, you could work across each row of the grid to find the location of the 5-Volt potential, mark these points, and then connect them to obtain the 5-Volt equipotential line.

11. After you have located the 5-Volt equipotential line, sketch its approximate shape

and location on the diagram below; it is not necessary to be exact. Label this line "5 V".

12. Repeat Steps 9 and 10 to find the 10-V amd 15-V equipotential lines.

Keep in mind, as you plot the equipotential lines, that the electric field is strongest in those regions where the equipotential lines are most closely spaced. This is because the


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electric field E is related to the gradient of the potential field: E = - ΔV/Δr, where ΔV is the change in the potential which occurs over a change in location Δr.

13. The electric field pattern consists of lines or curves which cross the equipotential curves at right angles. Sketch a representative number (15-20) of these electric field lines, and place arrows on a few of them to indicate the direction along which a positive test charge would move if one were placed at that point. Make sure that an electric field curve passes through each of the six small circular points shown on each pattern.

14. In each of the patterns shown, there are tiny circles through which one of the many electric field curves pass. Using the procedure described below, calculate an approximate electric field strength at each one of these points.

Magnitude of electric field strength = |ΔV/Δx|. (It's |dV/dx|, for those who know calculus.) This calculation is illustrated in the figure at the right for Point Q. Locate two points 0.5-cm on either side of Point Q along the electric field line passing through it. Estimate the potential at Points P and R, and subtract the smaller from the larger to get ΔV.

Note: the distances on your plots on the next pages probably won't be the same as on the actual configuration. This doesn't matter; what's important is to record the relative values of the electric field strengths; these will be the same irrespective of scale. Divide this potential difference by Δ x, in centimeters (no need to convert to meters). Label each of the six points in each pattern with these numbers. Please repeat all steps with both patterns.


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Question The more closely spaced the electric field lines are in any region, the more intense is the electric field. Looking at your four field plots, can you make a generalized statement about how the electric field strength near a conductor depends on the shape of the conductor? This Guide was written in October-November 2007 by Kimberly Strong, Dept. of Physics, Univ. of Toronto. Activity 1 is based on ILD 10, Representation as Communication: Fields, University of Maryland Physics Education Research Group (Spring 2003) and The Electric Field Mapping experiment, by Joseph Alward and Jason Harlow, Introductory Laboratories, Physics Department, University of the Pacific (2004). Activities 2, 3, and 4 are taken from Randall D. Knight, Student Workbook (Pearson, 2004). Activities 5, 6, 7, and 8 are taken from Curtis J. Hieggelke et al., E&M Tipers: Electricity and Magnetism Tasks (Pearson Prentice Hall, 2006). Activity 9 is based on The Electric Field Mapping experiment, by Joseph Alward and Jason Harlow, Introductory Laboratories, Physics Department, University of the Pacific (2004). Activity 9 was modified by Jason B. Harlow, Dept. of Physics, Univ. of Toronto. Last revision: March 19, 2009.


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Diagram for Activity 4


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Diagram for Activity 8


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Diagrams for Activity 9, Parts B and C

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In Module 2, each time you finished with a circuit you were asked to disconnect all wires, so that the next circuit you investigated starts with a “blank slate”. This is always a good idea and we recommend that you continue that practice in this Module.

Concepts of This Module

• Resistance • Ohm’s Law • Series and Parallel Circuits • Electric Power

Activity 1

Consider the circuit shown to the right using three of the supplied light bulbs labeled 6V 6W. The label means the bulb is rated for 6 watts at 6 volts

A. Without doing any calculations, predict what will happen to the brightness/dimness of Bulb 1 when the switch is closed. Explain your prediction without equations.

B. Wire the circuit and check your prediction. Was your prediction correct? If not, describe what happened.

C. How does the brightness of Bulb 2 change when the switch is opened and closed? D. With the switch closed, how does the brightness of Bulbs 2 and 3 compare?

We will return to this circuit in Activity 10.

Activity 2

In Module 2 you used a 6V 6W light bulb and measured the voltage drop ∆V across it and the current I flowing through it in a circuit such as shown to the right. Repeat those measurements for each of the three supplied 6V 6W light bulbs. Are all the values exactly the same? Quantify the spread in values by assigning an uncertainty to the voltage drops and currents.


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Activity 3

A. The light bulb labeled 6V 1W is rated for 1 watt at 6 volts. Measure the current I and the voltage drop ∆V across it in the shown circuit. Compare the brightness and the values for the voltage and current to those of the 6V 6W light bulb.

B. Can you combine the numbers from Part A and Activity 2 to give a single formula that gives the rated wattage for each bulb? What is the unit of the quantities that are combined in the formula to give the wattage?

Activity 4

In Electricity and Magnetism Module 2, an analogy of current flowing in a wire was made to water flowing in a garden hose. A Flash animation of this analogy is available at http://faraday.physics.utoronto.ca/IYearLab/Intros/DCI/Flash/WaterAnalogy.html. We shall be extending that analogy in the next Activity. Open the animation and explore the two possible values of the Voltage/Pressure that are available.

A. In the animation of the electric circuit, the movement of the negatively charged electrons in the conducting wire is shown. Is this in the same direction as the conventional current that you explored in Module 2?

B. As discussed in Module 1, in the 1700’s Benjamin Franklin and William Watson arbitrarily called the charge on a glass rod after being rubbed with silk positive and the charge on rubber after being rubbed with fur negative. If they had made the opposite choice, how would your answer to Part A change?

Activity 5

For a garden hose a pressure difference ∆p generates the flow of the water. We shall give the symbol w to the volume of water per time passing a cross-section of the wire in m3 / s. The hose has a resistance R to the flow of the water and we can define the resistance as:

wpR Δ

≡ (1)

This resistance is approximately constant for a given hose. Similarly, a voltage difference ∆V causes the electric current I to flow in the wire, and the wire has a resistance R to the flow:

IVR Δ

≡ (2)


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Just as for the hose, for a given conductor the resistance is usually approximately constant. Equation 2 is called Ohm’s Law. The unit of resistance is volts / ampere, which is called an ohm Ω. (Ω is the Greek letter omega.) The circuit diagram symbol for a resistance is:

A. A hose of length L and area A is shown. How would you expect the resistance of the hose to the flow of water to depend on its length? How might the resistance depend on its area?

B. A wire of length L and area A is shown. How would you expect the resistance of the wire to the flow of electric charge to depend on its length? How might the resistance depend on its area?

C. A perfect ammeter and a perfect wire both have zero resistance. In Module 2 Activity 7 you measured a number of voltage differences for the circuit shown to the right. From that data, what is the actual resistance of the ammeter? Of the wires? Calculate the resistance of the 6V 6W and the 6V 1W light bulbs; you may already have the data for this calculation. For the light bulb, you will need to know that as it heats up its, resistance changes. Thus when doing measurements of a light bulb in a circuit, be sure to give it a few seconds to reach equilibrium. In terms of the resistances of the wires, ammeter, and light bulb we can represent the circuit as shown: in this representation the lines connecting the circuit elements have zero resistance.


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Activity 6

You are supplied with two short wires which we will assume are perfect and have zero resistance. The plastic cylinder of the banana sockets of the mount for the light bulbs can be unscrewed so that a wire can be inserted into a hole in the conductor. Use the two wires to connect two of the 6V 6W light bulbs together in parallel and place the combination in a circuit with an ammeter and the battery. On the left is a circuit diagram, and the diagram on the right represents the components as resistors.

A. If the resistance of each light bulb is R, predict the effective resistance Reff of the two bulbs together in the circuit. You may find Activity 5 Part B and a result from Activity 5 Part C useful in making your prediction.

B. Measure the effective resistance. Was your prediction correct? C. Why are we justified in ignoring the resistance of the two short

wires? You may find Activity 5 Part B useful here too.

D. If two different resistors R1 and R2 are wired in a circuit in parallel, what is the effective resistance Reff of the two? You may need to do a derivation to answer this question.


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Activity 7

Use one of the short wires to connect two of the 6V 6W light bulbs in series, and place the combination in a circuit with an ammeter and the battery, as shown.

A. How does the brightness of the light bulbs compare to their brightness in the parallel circuit of Activity 6?

B. The brightness of the bulb is a function of the temperature of filament wire. Therefore, the resistance of each light bulb is different than the value you have been using. Measure the resistance of each individual light bulb when they are in the above circuit.

C. If the resistance of each light bulb is R’, predict the effective resistance R’eff of the two bulbs together in the circuit. You may find Activity 5 Part B useful in making your prediction.

D. Check your prediction by measuring the resistance of the combination. E. If two different resistors R1 and R2 are wired in a

circuit in series, what is the effective resistance Reff of the two resistors? You may need to do a derivation to answer this question.

Activity 8 The diagram to the right has 6 different resistors that are connected by perfect wires. Describe how you might calculate the total effective resistance between points A and B. You may find it helpful to re-draw the diagram with the resistors and wires laid out in a more standard


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fashion. You will not need to write down any formulas to do this Activity, but just describe how to do the calculation.

Activity 9

A. In Activity 3 Part B you devised a formula for the wattage of the light bulbs. Modify the formula so that it involves only the voltage drop across the light bulb ∆V and its resistance R.

B. Modify the formula so that it involves only the current flowing through the light bulb I, and its resistance R.

Activity 10 Assume that the change in resistance of a light bulb when it has a different brightness is negligible. Explain all the results for the brightness of the bulbs in Activity 1 using reasoning and/or expressions you have derived or learned in class.

Activity 11

A. If a 6V 6W light bulb and a 6V 1W one are wired in series, as

shown, predict which light bulb will be brighter. B. Wire the circuit and check your prediction. Were you correct?

Explain.

Activity 12

So far we have been treating the battery as perfect: it delivers a constant voltage in all circuits. Real batteries have a non-zero resistance, and we can represent the internal resistance r of the battery as shown. Now the battery symbol represents a perfect battery in series with the internal resistance of the real battery. Call the voltage of the perfect battery ε.4

4 ε is sometimes called the emf.


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Describe how the voltage delivered by a real battery ∆Vreal varies with the current I being drawn from it.

Activity 13

The earth contains minerals and moisture, which means that it is a usually a good conductor of electricity. Therefore, it is possible to wire a light bulb using the earth as the return path for the current: in the figure two metal rods have been stuck into the ground and connected to the circuit as shown. The symbol for a connection to ground is: Therefore we can represent the above circuit as shown to the right.

A. The resistance of your arm from your palm to your shoulder is about 150 Ω. Estimate the resistance of your body from your palm to the soles of your feet.


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B. An electric fence consists of a single wire connected to one

terminal of a voltage source; the other terminal of the voltage source is connected to ground. If the voltage ∆V is 120 volts and you touch the fence with your hand while standing on the ground in your bare feet, what is the number of watts your body will absorb while in contact with the wire?

This Guide was written in November 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. The cartoon of the person touching an electric fence is from the Glasgow Digital Library: http://gdl.cdlr.strath.ac.uk/hewwat/hewwat0206.htm Last revision: January 30, 2008

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Concepts of this Module In this module, we will explore concept involving capacitors, including:

• The electric field inside a capacitor • Potential and kinetic energy of charged particles inside a capacitor • Charging and discharging capacitors • RC circuits

The Activities

Activity 1 Positive and negative test charges are placed inside a parallel plate capacitor as shown. These test charges interact only with the capacitor. Their presence does not alter the field of the capacitor, nor do they interact with each other. G. An enlarged version of the diagram shown to the right is

provided at the end of this module. Tear off the page and staple it into your lab notebook. Use a black pen or pencil to draw the electric field vectors due to the capacitor inside the capacitor on that enlarged version.

H. Use a red pen or pencil (if available) to draw the forces acting on the two charges.

I. Pick a point of your choosing for the zero of potential energy. Label it “U = 0” on the diagram.

J. Is the potential energy of the positive point charge positive, negative, or zero? Explain. K. In which direction (right, left, up, or down) does the potential energy of the positive charge

decrease? Explain. L. In which direction will the positive charge move if it is released from rest? Use the concept

of energy to explain your answer. M. Does your answer to part F agree with the force vector that you drew in part B? N. Repeat steps D to G for the negative point charge.

++++++

- - - - - -

+

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Activity 2

The figure to the right shows the potential energy of a positively charged particle in a region of space. A. What arrangement of source charges is responsible for this

potential energy? Draw the source in your lab notebook using the axis given below.

B. With what kinetic energy should the charged particle be launched from x = 0 mm to have a

turning point at x = 3 mm? Explain. C. How much kinetic energy does the charged particle of part B have as it passes x = 2 mm?

Activity 3 - Measuring Capacitance

The unit of capacitance is the Farad, F, named after Michael Faraday. One Farad is equal to one Coulomb/Volt. As you will demonstrate shortly, one Farad is a very large capacitance for a conventional capacitor (see discussion in Appendix). Thus actual capacitances are often expressed in smaller units with alternate notation: microfarad: 10-6 = 1 μF nanofarad: 10-9 = 1 nF picofarad: 10-12 = 1 pF Typically, there are several types of capacitors used in electronic circuits, including disk capacitors, foil capacitors, electrolytic capacitors, and so on. You might want to examine some typical capacitors. To do this, you’ll need:

• 4 capacitors (assorted collection) To complete this activity, you will need to construct a parallel plate capacitor and use a multimeter to measure capacitance. You will use the following items:

• 2 pieces of thin aluminum sheet or aluminum foil, 15 cm × 15 cm • 1 textbook • 1 multimeter with capacitance-measuring capability • 2 insulated wires • 1 ruler • 1 Vernier caliper

You can make a parallel plate capacitor out of two rectangular sheets of aluminum foil separated by pieces of paper. A textbook works well as the separator for the foil since you can slip the two

0 1 2 3 4 x (mm)


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foil sheets between any number of sheets of paper and weight the book down with something heavy and non-conducting like another massive textbook. You can then use your digital multimeter in its capacitance mode for the measurements. Note: Insert the wires into the capacitance slots of your multimeter as “probes”. When you measure the capacitance of your “parallel plates”, be sure that the aluminum sheets arranged carefully so they don’t touch each other and “short out”. A. Devise a way to measure how the capacitance depends on the foil area and on the separation

between the foil sheets. (i) First, hold the area constant and do a series of measurements while varying the separation. (ii) Then hold the separation constant and do a series of measurements while varying the area. In both cases, be sure to record the dimensions of the foil so you can calculate its area, and record the distance between the foil sheets. Take at least five data points in each case. Describe your methods and then create a data table with proper units and display a graph of the results for each case.

B. Can straight lines be drawn through the data in your graphs? If not, you should make a guess

at the functional relationship that your data set represents and create a model that matches the data. Draw a corresponding graph and compare it with your graph in part A for each case. Be sure to label your graph axes properly. Can you explain your results based on physical reasoning?

C. Use the ohmmeter to measure the resistance of a page in your textbook. What is its

resistance? Can current flow through the pages of your book?

Activity 4 A parallel plate capacitor with plate separation d is connected to a battery that has potential difference ΔVbat, as shown. Without breaking any of the connections, insulating handles are use to increase the plate separation to 2d. A. Does the potential difference ΔVC across the

capacitor change as the separation increases? If so, by what factor? If not, why not?

B. Does the capacitance C change? If so, by what factor? If not, why not?

C. Does the capacitor charge Q change? If so, by what factor? If not, why not? D. As the plates are being pulled apart, does current flow clockwise, counterclockwise, or not at

all? Explain.

+

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Activity 5 Light bulbs can be used to indicate current flow in a circuit. The brightness of a bulb increases with increasing current passing through it. The figure shows a battery, a switch, two light bulbs, and a capacitor that is initially uncharged. A. Immediately after the switch is closed, are

either or both bulbs glowing? Explain. B. If both bulbs are glowing, which is brighter?

Or are they equally bright? Explain. C. For any bulb (A or B or both) that lights up

immediately after the switch is closed, does its brightness increase with time, decrease with time, or remain unchanged? Explain.

Activity 6 - Capacitors, Batteries, and Light bulbs

Now let’s do an experiment using capacitors, batteries, and light bulbs to see what happens to the current flowing through a resistor (the bulb) when a capacitor is charged by a battery and then discharged. You are provided with two light bulbs (6V 1W and 6V 3W), two capacitors (0.47 F and 1 F), and a 6 V battery. The capacitors are built using nanotechnology and are called supercapacitors; further information appears in the Appendix to this Guide. A. Connect the 6V 1W light bulb (the elongated one) in series with the 0.47 F capacitor, a

switch, and the 6 V battery. Draw a circuit diagram of your setup. Describe what happens when you close the switch.

B. Now, can you make the bulb light up again without the battery in the circuit? Mess around

and see what happens. Describe your observations and draw a circuit diagram showing the setup when the bulb lights up without a battery.

C. Repeat parts A and B with a voltmeter across the capacitor. Draw a rough sketch of the

voltage across the capacitor as a function of time for both cases. D. Draw a sketch of the approximate brightness of the bulb as a function of time when it is

placed across a charged capacitor without the battery present. Let t = 0 when the bulb is first placed in the circuit with the charged capacitor. Note: You can examine the change in the current is by wiring an ammeter in series with the bulb.

E. Explain what is happening. Is there any evidence that charge is flowing between the plates

of the capacitor as it is charged by the battery with the resistor (the bulb) in the circuit, or as

+

_


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it discharges through the resistor? Is there any evidence that charge is not flowing through the capacitor? Hints: (1) You may want to repeat the observations described in parts A and B several times; placing the voltmeter across the capacitor or placing an ammeter in series with the capacitor and bulb in the two circuits you have devised might aid you in your observations. (2) Theoretically, how should the voltage across the capacitor be related to the amount of charge on each of its conductors at any given point in time?

F. What happens when more capacitance is put in the circuit? What happens when more

resistance is put in the circuit? You can use the 6V 3W light bulb (the rounded one) in the circuit to get more resistance. Hint: Be careful how you wire the extra capacitance and resistance in the circuit. Does more capacitance result when capacitors are wired in parallel or in series? How should you wire resistors to get more resistance?

Activity 7

The charge on the capacitor shown in the figure is zero when the switch closes at t = 0 s. A. What will be the current in the circuit after

the switch has been closed for a long time? Explain.

B. Immediately after the switch closes, before the capacitor has had time to charge, the potential difference across the capacitor is zero. What must be the potential difference across the resistor in order to satisfy Kirchoff’s Loop Law? Explain.

C. Based on your answer to part B, what is the current in the circuit immediately after the switch closes?

D. Sketch a graph of current versus time, starting from just before t = 0 s, and continuing until the switch has been closed a long time. There are no numerical values for the horizontal axis, so you will have to think about the shape of the graph.

Appendix – Supercapacitors For a parallel plate capacitor with each plate having a surface area A, the plates separated by a distance d, and the space between the plates filled with a dielectric of constant ε, the capacitance C is:

dAC ε=

So there are three ways to increase the capacitance:


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1. Use a dielectric with a higher constant. 2. Increase the surface area of the plates 3. Decrease the distance between the plates.

If the space between the plates is air and the plates are separated by 1 mm, then a 1 Farad capacitor must have a plate surface area of about 108 m2. For square plates this means the plates must be 10 km × 10 km. Supercapacitors use nanotechnology to achieve extremely high effective surface areas. Often highly porous carbon is used, which can achieve effective areas of as much as 2000 m2 per gram. Although the basic idea was known as early as 1957, it is only since the mid-1990’s that advances in material science have led to reliable inexpensive supercapacitors. Over time chemical redox reactions can occur on the electrodes which will degrade performance, and some manufacturers minimize this effect by making the two electrodes somewhat differently. Thus supercapacitors have their two terminals marked to indicate which is positive and which is negative. Some other more conventional capacitors, such as electrolytic types, also have a polarity. Using the correct polarity for supercapacitors extends their life somewhat but is not terribly important for our purposes. Electrolytic capacitors can explode if they are wired incorrectly. This Guide was written in December 2007 by Kimberly Strong, Dept. of Physics, Univ. of Toronto. The Appendix was written by David M. Harrison, Dept. of Physics, Univ of Toronto in February 2008. The activities are based on Randall D. Knight, Student Workbook (Pearson, 2004) and Priscilla W. Laws, Workshop Physics Activity Guide, Module 4: Electricity and Magnetism (John Wiley & Sons, 2004). First version: December 11, 2007. Last revision: February 6, 2008.


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Enlarged version of the diagram in Activity 1

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• Interactions of permanent magnets with other magnets, conductors, insulators, and electric charges.

• Magnetic fields of permanent magnets, current carrying coils, and a current carrying straight wire.

• Introduction to induction. NOTE: for most setups the North poles of the magnets are colored red.

Activity 1

A. You are supplied two bar magnets, with the North and South poles marked. Do the same poles attract or repel each other? Do the opposite poles attract or repel each other?

B. Imagine that you have two electric dipoles, each of which consists of equal and opposite electric charges +q and –q separated by a constant distance d. Would the positive “poles” (i.e. charges) of the dipole attract or repel each other? What about the opposite “poles”? Is there any difference between the interaction of two electric dipoles and two bar magnets?

Activity 2

In Electricity and Magnetism Module 1 you may have used a white pith ball, which is an insulator, and a black Aluminum coated pith ball, which is a conductor, both hanging by strings from stands. In this Activity you are given those two balls again plus a small carbon steel ball hanging by a string from a stand. The steel is also a conductor. In this Activity you will use one of the two supplied bar magnets.

A. Does the North Pole of the magnet exert a force on the pith ball? How about the South Pole of the magnet? Repeat for the Aluminum coated pith ball.


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B. Does the North Pole of the magnet exert a force on the carbon steel ball? Is the force attractive or repulsive?

C. Predict whether or not the South Pole of the magnet will exert a force on the carbon steel ball. Will it be attractive or repulsive?

D. Check your prediction. Were you correct? Explain. E. Does the magnet interact with other insulators and conductors in the room? What

about the whiteboard? Generalize your observations about the interaction of the magnet with insulators and conductors.

Activity 3

In addition to the bar magnet of Activity 2, you are supplied an unmagnetised soft iron rod. Imagine that you don’t know which object is the magnet. Using only these two objects find a way to determine which object is the magnet.

Activity 4

In Electricity and Magnetism Module 1 Activity 1 you may have put an electric charge on a length of sticky tape by suddenly peeling it off the tabletop and hanging it from the cupboard. You may wish to refer to the Student Guide for that Module and/or your lab book to refresh your memory. Repeat that procedure.

1. Does the magnet exert a force on the charged tape? Is there a difference between the North and South poles?

2. You are supplied with an unmagnetised soft iron rod. Does it exert any forces on the charged tape? Are there any differences between the interaction of the tape with magnet and with the metal rod?

3. What can you conclude about the interaction of stationary electric charges with magnetic fields?

Activity 5

We can describe the fact that the Earth exerts a gravitational force on all objects near it by saying:

1. The Earth creates a gravitational field in all regions of space around it. 2. The gravitational field exerts a gravitational force on all objects in the field.


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A convenient definition of the gravitational field is that it is equal in magnitude and direction to the acceleration due to gravity gr at that point in space. Then the gravitational force exerted on a mass m in the gravitational field is:

gmF rr=

We could measure the direction of the gravitational field by taking a mass hanging from a string and observing in what direction the mass hangs. The figure to the right illustrates. A compass is a little bar magnet that is free to rotate. In this Activity we will use a simple compass the measure the direction of the magnetic field around various objects. The procedure is similar to the one we described for determining the gravitational field. We will use the following standard convention:

The direction of an external magnetic field is parallel to the orientation of the compass, and points from the South to the North direction as marked on the compass.

All of the fields you will map are sufficiently strong that the Earth’s magnetic field is negligible. Be sure when you are mapping a magnetic field that all other magnets and metals are as far away as possible.

A. Map the magnetic field around one of the bar magnets using the compasses. Use

your results to sketch the magnetic field “lines” around the bar magnet. Be sure to include arrows on the lines to indicate the direction of the magnetic field.

B. Place the two bar magnets together, with the North pole of one in contact with the South pole of the other, as shown. Map the magnetic field of the combination and sketch the field lines including the direction of the field. Do you think the field any different from that of a single long bar magnet?

C. In Activities 11 and 12 below you may be using the pair of wire coils shown to the right. Here you will use only the outer coil, and you may place the inner coil to the side. You are supplied with a plastic plate with a rectangular hole cut in it. Place the outer coil in the hole. You will place the compasses on the Plexiglas You will connect the 6V battery to the red sockets on the outer coil, with a switch to avoid draining the batter.


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The switch you will use is called a Contact Key: this type of switch is only on when you hold it down. The symbol for a coil is shown to the right. So the circuit diagram of the experiment you will do is: Use the compasses to map the magnetic field around the coil, and draw field lines including the direction. Compare your results to Parts A and B.

D. In terms of its magnetic field we can model the Earth as a big bar magnet. Is the magnetic North pole located at the geographic North pole (where Santa’s workshop is located) or at the geographic South pole (where the penguins live)?

Activity 6

A. In Activity 5 Part C you found the magnetic field of a coil of wire. Here you will investigate the magnetic field of a single straight wire carrying a current. You will orient the wire vertically through a hole in a Plexiglas stand, and place compasses on the stand. Sketch the fields lines around the wire.

B. In Electricity and Magnetism Module 2 you may have used a clamp meter to measure the current in a wire. Can you now explain how that meter works? If yes, do so.


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C. In Part A you used a vertical current carrying wire passing through a hole in a Plexiglas stand to map the magnetic field. Here you will investigate how the magnitude of the magnetic field varies with distance from the wire. You will measure the magnetic field with a Giant Magnetoresistance (GMR) sensor. These devices use a quantum mechanical effect which makes resistances vary when it is in an external magnetic field; for the keen some further details appear in the Appendix.

The rest of this Activity is still under development.

Activity 7

In Activity 5 you mapped the magnetic field of a coil carrying an electric current. In Activity 6 you explored the magnetic field of a single straight current carrying wire. Here is a figure of a few of the wires of a coil.

The current in the upper wires is out of the plane of the paper, and the current in the lower wires is into the plane of the paper.

A. Sketch the total magnetic field at point α due to currents A1 and A2 indicating its magnitude by the length of the vector line. You do not need to do a detailed calculation of the magnitude of the field, just a rough estimate. You may wish to label this line “A upper”. On the same sketch add the total magnetic field at point α due to currents A3 and A4, perhaps labeling the line “A lower”. So what is the direction and magnitude of the total magnetic field at α due all four A currents?


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B. Repeat Part A for the B currents. So for a real coil of many “turns” of wire, what is the direction of the total magnetic field at a point reasonably far from the ends of the coil?

C. Repeat Parts A and B for the point β, which is on the axis of the coil and far from the ends.

D. Repeat Parts A and B for the point γ. Compare to your result from Part C.

Activity 8

A. In Part B of Activity 5 you made a big bar magnet by joining two smaller magnets together and determined its magnetic field. In Electricity and Magnetism Module 3 Activity 9 you may have determined the electric field of an electric dipole, which is two equal and opposite charges separated by some distance; the figure for Activity 1 of this module shows two such dipoles. Are there any differences between the shape of the magnetic field of a big bar magnet and the shape of the electric field of an electric dipole?

B. If you have an electric dipole and cut it in half along its axis you end up with an isolated positive charge and an isolated electric charge. If you take the big bar magnet of Part B Activity 4 and take it apart into two smaller magnets, do you end up with an isolated magnetic North “charge” and an isolated magnetic South “charge”? What about if you took one of the small bar magnets and sawed it in half [don’t actually do this!]? Can you think of any reason why we never observe an isolated magnetic “monopole”?

Activity 9

The force exerted on a moving charge is given by:

BvqFrrr

×= (1)

A. The direction of this force is always perpendicular to the velocity of the charge. Can this force ever do work on the charge?

B. Here is a figure of a typical electric motor:


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A current in the loop experiences the forces shown due to the magnet. Eqn. 1 describes the forces. Is this fact consistent with your answer to Part A? Do motors do work? Explain.

Activity 10

A galvanometer is a type of ammeter. Typically, they have a coil of wire through which the current to be measured flows. A magnetic field from a permanent magnet exerts a torque on the coil of wire that is proportional to the current. A spring exerts a torque opposite the torque due to the current-magnetic field interaction. So at equilibrium the deflection of the pointer of the meter is proportional to the current. Many galvanometers are capable of detecting very small currents, but often are not as accurate as more modern instruments. Some galvanometers do not even attempt quantitative measurements of the value of the current but are just used to detect their presence and direction. Although sometimes a galvanometer is represented in circuit diagrams as an ammeter, it is usually drawn as shown to the right. In 1831 Faraday asked an interesting question. His reasoning was:

Electric currents cause magnetic fields. Is the converse true: do magnetic fields cause electric currents?


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A diagram of the apparatus he used to try to answer this question is shown to the right. A battery is connected to a coil of wire, which will generate a magnetic field when the switch is closed. A second coil of wire is connected to a galvanometer. If the magnetic field from the first coil causes electric currents in the second coil, then the galvanometer could show a current provided it is sensitive enough. In the actual experiment, the two coils of wire were concentric with each other. A photograph of the coils you will be using appears as part of Activity 5 above.. In the photograph the inner coil is slid partly out from the outer one. In this experiment the inner coil should be slid all the way into the outer coil. The battery and switch should be connected to the outer coil using the red banana sockets, and the galvanometer should be connected to the inner coil using the black banana sockets. In Activities 3 and 4 you may have used a unmagnetised soft iron rod. This rod comes with the coils, but is not part of the apparatus for this Activity. Please be sure to set it aside and do not use it here. The switch you will use is called a Contact Key: this type of switch is only on when held down. Please do not leave the switch closed for too long: this will drain the battery. Also please do not connect the battery to the inner coil: the resistance of that wire is so small it will either quickly drain the battery or trip the circuit breaker on the battery. What is the answer to Faraday’s original question? Is there anything else going on or is this just a failed experiment?

Activity 11

You are supplied a soft iron cylinder, which you may have used in Activities 2 and 3. Insert the rod completely into the hole in the inner coil of the setup you used in Activity 10 and repeat the experiment. What happened? Why is the result different from the earlier one?


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Activity 12

A U-shaped conducting wire has a straight vertical wire resting on and making electrical contact with it. There is a uniform magnetic field B

r directed into the page. The vertical

bar is being pulled to the right at a constant velocity vr . At some moment in time the contact between the U-shaped and the vertical wires is at points a and b.

A. What is the force, magnitude and direction, exerted on the electrons in the vertical wire by the magnetic field?

B. If an electric field in the vertical wire were to exert the same force on the electrons as the answer to Part A, what would be its magnitude and direction?

C. If the electric field of Part B is constant, what is the potential difference ΔV between points a and b?

D. The magnetic flux ΦB through the loop consisting of the U-shaped wire and the vertical wire is the magnitude of the magnetic field B times the area A of the loop. What is the rate of change of the magnetic flux?

E. Compare your answers to Parts C and D. Explain.

Activity 13

Lenz’s Law is often stated: “The direction of an induced current is such that the induced magnetic field opposes the change in the flux.” However, an alternative statement is: “The direction of induced current opposes its cause.” You may find that the second form is useful for this Activity.

A. A bar magnet is falling towards a circular loop of wire as shown. The falling magnet induces a current in the wire. The current in the wire exerts a force on the falling magnet. What is the direction of the force exerted on the magnet?


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B. The magnet has fallen past the loop of wire and is now accelerating away from it. Now what is the direction of the force exerted on the magnet?

C. The magnet is falling away from the loop as in Part B, but the poles of the magnet are reversed. What is the direction of the force exerted on the magnet?

D. If your answers to Parts A, B, and C were opposite to the correct answers, would this violate any physical principles?

Appendix Here we discuss Giant Magnetoresistance (GMR) and how the effect is used to measure a magnetic field. GMR was discovered independently in 1988 by Grünberg and Fert; they were awarded the Nobel Prize in Physics in 2007 for their discovery. It is now used extensively in read heads of computer disc drives. GMR is a quantum mechanical effect that produces significant decreases of resistance in the presence of an external magnetic field. GMR materials are commonly comprised of alternating very thin layers of various metallic elements. As described by IBM:

The key structure in GMR materials is a spacer layer of a non-magnetic metal between two magnetic metals. Magnetic materials tend to align themselves in the same direction. So if the spacer layer is thin enough, changing the orientation of


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one of the magnetic layers can cause the next one to align itself in the same direction. Increase the spacer layer thickness and you'd expect the strength of such "coupling" of the magnetic layers to decrease. But as [an IBM research team] made and tested some 30,000 different multilayer combinations of different elements and layer dimensions, they demonstrated the generality of GMR for all transition metal elements and invented the structures that still hold the world records for GMR at low temperature, room temperature and useful fields. In addition, they discovered oscillations in the coupling strength: the magnetic alignment of the magnetic layers periodically swung back and forth from being aligned in the same magnetic direction (parallel alignment) to being aligned in opposite magnetic directions (anti-parallel alignment). The overall resistance is relatively low when the layers were in parallel alignment and relatively high when in anti-parallel alignment.

A GMR sensor that measures magnetic fields uses four such resistors in an arrangement called a Wheatstone bridge. The two resistors R are shielded from the magnetic field, and the two resistors r are exposed to it. The sensor is driven with an applied voltage ΔV, which is 5 volts in our case. Then we measure the voltage V. It is fairly easy to show that:

)()(

rRrRVV

+−

Δ=

Further information on GMR sensors is at: http://www.nve.com/Downloads/lowfield.pdf. This Guide Sheet was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in January 2008. Activity 4 is from Lillian McDermott et al, Tutorial in Introductory Physics, Magnets and Magnetic Fields I.B, (Prentice-Hall, 2002), pg. 113. Parts of Activity 5 are similar to Priscilla W. Laws et al., Workshop Activity Guide, Module 4, Unit 28, (John Wiley, 2004), pg. 725. The figure of the electric motor is slightly modified from Fig. 32.48 of Randall D. Knight, Physics for Scientists and Engineers (Pearson, 2004). The figure of the mechanism of a galvanometer is slightly modified from a version at Wikipedia: http://en.wikipedia.org/wiki/Galvanometer. The quotation from IBM in the Appendix is from http://www.research.ibm.com/research/gmr.html, accessed February 5, 2008. The date stamp of the html file is March 16, 2004. Last revision: February 14, 2008.

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PHY131 Practicals Manual Relativity Module

Relativity Module Student Guide


• Events that can cause other events • Synchronising clocks • Simultaneity • Time dilation and inertial reference frames • Rigid bodies and relativity • Tachyons • Geometric approaches to relativity • Photons and relativity • Mass-energy equivalence • The Equivalence Principle • Geometry

The Activities

Activity 1

Event 1 occurs at x = 0 and t = 0.

A. Event 2 occurs at x = 1200 m and is caused by Event 1. When is the earliest that Event 2 can occur. Explain.

B. Event 0 occurred at x = -4,000 m at time t = -1.0 μs. Could Event 0 have caused Event 1? Explain.

Activities 2 and 3 refer to the following situation. Assume Toronto and Montreal are exactly 510 km apart in the Earth frame of reference. Assume that Kingston is exactly half way between Toronto and Montreal. They are all, of course, stationary relative to each other and the Earth. Ignore any effects due to the Earth’s rotation on its axis. Ignore any effects due to the Earth’s gravitational field. Assume the surface of the Earth is flat. Assume that the speed of light in the air is exactly equal to c.


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A powerful searchlight is on top of the CN Tower in Toronto, pointed towards Montreal. A second searchlight is on top of 1000 de La Gauchetière in Montreal, pointed towards Toronto. Initially the two searchlights are turned off. Assume that both the CN Tower and 1000 de La Gauchetière are visible from each other and from any point between Toronto and Montreal. There are also two identical atomic clocks with an accuracy of at least 1 μs, one on top of the CN Tower and the other on top of 1000 de La Gauchetière.

Activity 2

A. You are stationary in Kingston. With two powerful telescopes you are looking at

the two clocks. You see that they read the same time. Are the clocks synchronized? If yes, explain. If no, which clock is ahead and by how much?

B. You are on top of the CN Tower right beside its clock. With a powerful telescope you look at the clock in Montréal, and see that it reads exactly the same time you see on the clock on the CN Tower. Are the clocks synchronized? If yes, explain. If no, which clock is ahead and by how much?

C. You are on top of 1000 de La Gauchetière right beside its clock and see that it reads a time of 12.000000 s. With a powerful telescope you simultaneously look at the clock in Toronto. If the two clocks are synchronized, what time should you see the Toronto clock read?

Activity 3

A. You are in a rocket ship traveling from Toronto to Montreal at 0.8 c relative to the

Earth. For you what is the distance between Toronto and Montreal? For you what is the distance between Toronto and Kingston?

B. The two searchlights on the CN Tower in Toronto and 1000 de La Gauchetière in Montreal are quickly turned on and off, both emitting flashes of light. For you, the two flashes were emitted simultaneously. Imagine you are right beside the CN Tower when its light is turned on. You see the flash from the searchlight on the CN Tower instantaneously. Sketch the two flashes of light, from the CN Tower and from 1000 de La Gauchetière, a very brief moment after they are emitted, indicating their speeds and the distance between them for you. How long before you see the flash from the searchlight on 1000 de La Gauchetière? Be sure to clearly indicate how you arrived at your answer.

C. Another member of your Team also is in a rocket flying at 0.8 c from Toronto to Montreal. What is your Teammate’s speed relative to you? Imagine your Teammate is just over Kingston when the flashes are emitted. Are the two flashes of light emitted simultaneously for your Teammate? Will he/she see the flashes


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simultaneously? If yes, how long after the flashes will he/she see them? If no, which flash will he/she see first and by how much? You may find it useful to add your Teammate to your sketch from Part B.

D. Your Instructor has gone to Kingston, and is stationary relative to Kingston and the Earth. What is your Instructor’s speed relative to you? Will your Instructor see the flashes from the two searchlights of Part B simultaneously? If no, which flash will he/she see first and by how much as measured by you? Explain. You may find it useful to add your Instructor to the sketch from Parts B and C.

E. Imagine your Instructor has a cassette player that will begin playing when it receives a flash of light, and quits when it receives a second flash of light. Is there any music? Does the tape move? Do you hear any music? Does your Instructor hear any music?

Activity 4

In 1971 Hafele and Keating tested the predictions of the Theories of Relativity by flying cesium beam atomic clocks around the world on regularly scheduled commercial airline flights. One clock remained in their laboratory outside Washington DC, one was flown to the East, and the other to the West. They got the data on speeds and altitudes of the planes flying the Eastbound and Westbound clocks from the flight recorders. When the got all three clocks back in the laboratory they compared the measured times. Here are the predicted and experimental results for the elapsed times compared to the clock that stayed in the laboratory.

Eastbound clock (ns) Westbound clock (ns)

Special Relativity prediction lose 184 ± 18 gain 96 ± 10

General Relativity prediction gain 144 ± 14 gain 179 ± 18

Total predicted effect lose 40 ± 23 gain 275 ± 21

Measured lost 59 ± 10 gained 273 ± 7

The Special Relativity prediction is because that theory predicts that moving clocks run slowly. The General Relativity prediction is because that theory predicts that clocks in gravitational fields run slowly.

A. Why does Special Relativity predict that the Eastbound clock gains time and the Westbound clock loses time? Shouldn’t they both either gain time or lose time?

B. Do the predicted effects agree with the experimental data? Explain. C. Are the calculated errors in the total predicted effect correct? How were these

errors calculated?


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Activity 5

We have a 25 m long pole and a 20 m long barn, both as measured at rest relative to the pole and the barn. We will assume the back wall of the barn is very very strong. If the pole is moving towards the barn at 70% of the speed of light, its length will be contracted to about 18 m. Thus it clearly fits in the barn, and we can slam the door shut (and run!). But if we are riding along with the pole, its length is not contracted and is 25 m long. But the barn is contracted and is now about 14 m long. Clearly the pole does not fit in the barn. Does the pole fit into the barn or not? Explain.

Activity 6

For a long time people interpreted the Special Theory of Relativity to mean that nothing can travel faster than the speed of light. In 1967 Feinberg showed that this is not correct. There is room in the theory for objects whose speed is always greater than c. Feinberg called these hypothetical objects tachyons; the word has the same root as, say, tachometer. Attempts have been made to observe tachyons: so far all such experiments have failed.

A. The relation between energy E and mass m for an object traveling at speed u relative to us is given by:

22

2

/1 cumcE−

=


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If the energy of a tachyon is a real number, what kind of number must its mass m be?

B. Draw energy versus speed axes with the speed going from 0 to 5 c. Sketch the relation between the energy E and speed u for an ordinary object. On the same plot sketch the relation between E and u for a tachyon. For ordinary objects we say: “It takes infinite energy to speed the object up to a speed equal to the speed of light.” What is the equivalent statement for a tachyon? For ordinary objects we can also say: “The minimum value of the energy is when it is at rest, and has a value equal to mc2.” What is the equivalent statement for a tachyon?

C. A tachyon is produced by some apparatus in the laboratory at x = 0 at t = 0 and travels at ux = 20 c in the +x direction to a detector at x = 1,000 m. You are traveling at v = 0.1 c in the +x direction relative to the laboratory. The formula for

addition of velocities is: 2/1'

cvuvuu

x

xx −

−= . What is the speed of the tachyon

relative to you? D. For Part C describe in words what you see the tachyon doing. What does this say

about the tachyon being produced by some apparatus causing its later detection by the detector.

Activity 7

Sue and Lou are moving relative to each other at speed v. We can relate their spacetime diagrams as shown.


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In the above diagram:

cv

=)tanh(θ

Similar to the Parable of the Surveyors, the axes are rotated relative to each other, but they are rotating in opposite directions. Also shown in yellow is the worldline for light traveling at c relative to both Sue and Lou. You will be using this diagram below, and may find it useful to print a few copies of it to staple into your lab book.

A. Explain how the diagram shows that the speed of light is the same value for both Sue and Lou.

B. Place two dots on the diagram representing the positions and times of two events that are simultaneous for Lou. Are the two events simultaneous for Sue? If no, which event occurs first? Place two more dots on the diagram representing the positions and times of two events that are simultaneous for Sue. Are the two events simultaneous for Lou? If no, which event occurs first?

C. Sketch the worldline of an object that is stationary relative to Lou. What is the direction of motion of the object relative to Sue? Explain.

D. Sketch the worldline of an object moving at some speed u < c relative to Lou. Show geometrically that the object is moving at speed u’ < c relative to Sue.

E. Activity 6 introduced tachyons, objects whose speed is always greater than the speed of light. Draw the worldline of a tachyon moving at some speed u > c relative to Lou. Show geometrically that the direction of motion of the tachyon for Sue, u’, can be negative but with a magnitude > c.

Activity 8

As you may know, in some circumstances we can treat light as a particle called a photon.

A. Relativistic time dilation means that an unstable particle which decays in time Δτ when it is rest relative to some observer will live a longer time Δt for an observer for whom it is moving with speed v where:

ττΔ>

−

Δ=Δ

22 /1 cvt

If you apply this formalism to a photon, what does it predict about its lifetime relative to any observer?

B. The relation between energy E and mass m for an object traveling at speed u relative to us is given by:


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22

2

/1 cumcE−

=

A photon has a real non-zero energy. What does this equation say about the value of its mass m? What do you think your high school math teacher would say about your answer?

Activity 9

Four elementary particles, an electron, a muon, a proton, and a neutron have the rest energies and relativistic total energies as shown.

Particle Rest energy Total energy Electron 0.511 MeV 0.511 Mev Muon 106MeV 212 MeV Proton 938 MeV 4,690 MeV

Neutron 940 MeV 2,820 MeV Rank in order from the largest to the smallest the particle’s speeds.

Activity 10

In Jules Verne’s From the Earth to the Moon (1865) a huge cannon fires a projectile at the moon. Inside the projectile was furniture, three people and two dogs. The figure is from the original edition. Verne reasoned that at least until the projectile got close to the Moon it would be in the Earth’s gravitational field during its journey. Thus the people and dogs would experience normal gravity, and be able to, for example, sit on the chairs just as if the projectile were sitting on the Earth’s surface. One of the dogs died during the trip. They put the dog’s body out the hatch and into space. The next day the people looked out the porthole and saw that the dog’s body was still floating just beside the projectile.


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A. Is there a contradiction between the inhabitants inside the projectile experiencing normal gravity and the dog’s body outside the projectile not falling back to the Earth?

B. If your answer to Part A is yes, where did Verne make his mistake? If your answer is no, explain.

Activity 11

A bucket of water has a spring soldered to the bottom, as shown. A cork is attached the spring, and is therefore suspended under the surface of the water.

You are on top of the CN tower, holding the bucket, and step off. While falling towards the ground, do you see the cork move towards the top of the water, towards the bottom of the bucket, or stay where it is relative to the bucket and the water?

Explain your answer using Einstein’s Equivalence Principle.

Activity 12

As you know, according to plane geometry the circumference C0 of a circle is related to its radius r according to:

rC π20 = Imagine we have a wheel of radius r whose circumference when it is stationary is C0. But the wheel is rotating with constant angular velocity ω. According to Special Relativity, the rim of the wheel will be contracted. Assume the rim has negligible thickness.

A. Will the length of the spokes of the wheel be contracted too? Explain.

B. What is the circumference C of the wheel according to Special Relativity? Express your answer in terms of the centripetal (radial) acceleration a of each point on the rim.

C. Following Einstein, we say that the geometry of the rotating wheel is not the geometrical equation given above. Express the difference in the geometries of the rotating and non-rotating wheel, i.e. C – C0.


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Activity 13

The word geometry literally means the measure of the Earth. As you know, according to plane geometry the sum of the angles of a triangle is 180º.

A. Assume the Earth is a perfect smooth sphere. Imagine you construct a triangle on the surface of the Earth. What is the sum of the angles of this triangle? You may wish to consider an isosceles triangle with the apex at the North Pole and the base along the equator.

B. From Part A, you may have discovered that geometry on a spherical surface is different than the geometry on a plane. To describe the geometry around a massive object, which geometry is the best bet to be correct? Explain.

This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in January 2009. Activity 3 is based on Rachel E. Scherr, Peter S. Shaffer, and Stamatis Vokos, “The challenge of changing deeply held student beliefs about the relativity of simultaneity,” American Journal of Physics 70 (12), December 2002, 1238 – 1248. I learned about the geometric approach to Special Relativity of Activity 7 from Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (W.H. Freeman, 1963). This classic is highly recommended. Activities 1, 6 and 8 also appear in David Harrison and William Ellis, Student Activity Workbook that accompanies Hans C. Ohanian and John T. Markert, Physics for Engineers and Scientists, 3rd ed. (W.W. Norton, 2007). Activities 10 and 11 also appear in Mechanics Module 3. Last revision: March 24, 2009

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PHY132H1S Summer Practicals Manual - U of T Physicsjharlow/teaching/... · PHY132H1S Summer Practicals Manual Department of Physics July to August, 2009 University of Toronto Welcome