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Lorentz Transformation and Charge Conservation Elisha Huggins and Drew Milsom Citation: Phys. Teach. 45, 328 (2007); doi: 10.1119/1.2768683 View online: http://dx.doi.org/10.1119/1.2768683 View Table of Contents: http://tpt.aapt.org/resource/1/PHTEAH/v45/i6 Published by the American Association of Physics Teachers Related Articles The Combination of Just-in-Time Teaching and Wikispaces in Physics Classrooms Phys. Teach. 51, 44 (2013) Association of American Universities (AAU) initiative to improve the quality of STEM teaching and learning, www.insidehighered.com/news/2011/09/15/qt#270450 Phys. Teach. 49, 526 (2011) Banish “weight”? Phys. Teach. 49, 196 (2011) Newton and projectile motion Phys. Teach. 49, 196 (2011) Reaching the peaks of teaching Phys. Teach. 49, 132 (2011) Additional information on Phys. Teach. Journal Homepage: http://tpt.aapt.org/ Journal Information: http://tpt.aapt.org/about/about_the_journal Top downloads: http://tpt.aapt.org/most_downloaded Information for Authors: http://www.aapt.org/publications/tptauthors.cfm Downloaded 08 Sep 2013 to 205.133.226.104. Redistribution subject to AAPT license or copyright; see http://tpt.aapt.org/authors/copyright_permission

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Page 1: Lorentz Transformation and Charge Conservation

Lorentz Transformation and Charge ConservationElisha Huggins and Drew Milsom Citation: Phys. Teach. 45, 328 (2007); doi: 10.1119/1.2768683 View online: http://dx.doi.org/10.1119/1.2768683 View Table of Contents: http://tpt.aapt.org/resource/1/PHTEAH/v45/i6 Published by the American Association of Physics Teachers Related ArticlesThe Combination of Just-in-Time Teaching and Wikispaces in Physics Classrooms Phys. Teach. 51, 44 (2013) Association of American Universities (AAU) initiative to improve the quality of STEM teaching and learning,www.insidehighered.com/news/2011/09/15/qt#270450 Phys. Teach. 49, 526 (2011) Banish “weight”? Phys. Teach. 49, 196 (2011) Newton and projectile motion Phys. Teach. 49, 196 (2011) Reaching the peaks of teaching Phys. Teach. 49, 132 (2011) Additional information on Phys. Teach.Journal Homepage: http://tpt.aapt.org/ Journal Information: http://tpt.aapt.org/about/about_the_journal Top downloads: http://tpt.aapt.org/most_downloaded Information for Authors: http://www.aapt.org/publications/tptauthors.cfm

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Page 2: Lorentz Transformation and Charge Conservation

No-Shock AlternativeWhen I read the article “Watt-

age: Household Appliances” on p. 249 of the April 2007 issue, the comment that “a grounded student would get a shock” caught my eye. Since a student would have the op-portunity to get shocked, I would never use the apparatus as described, even with a ground fault adaptor. I would enclose the meter inside a box so that the meter could be operated and read, but the wires could not be gotten to by inquisitive fingers, or I would use a P4400 Kill A Watt™ me-ter. It measures volts, current, watts, frequency, power factor, and VA, and I’ve seen it as low as $20.90. I suggest doing a web search for “Kill A Watt” to find the best price.

I have used the meter to check power usage of items that are “turned off ” but still consuming energy, i.e., 5 watts for my “turned off ” 35-in TV. This idea came from an article in PCMagazine, which can be found online.11. See TPT Letters to the Editor, “The

Electron Leak,” July 2007; http:// scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH0000450000L10000L2000001&idtype=cvips.

Dick HeckathornCuyahoga Valley Christian Academy

4687 Wyoga RoadCuyahoga Falls, OH 44224

Physics First, Chemistry and Biology

I admire and respect teacher Bob Bessin’s arguments (“Why Physics First?” March 2007, p. 134) sup-porting the early high school study of physics even as I disagree with his

dismissal of its “usefulness” in chem-istry and biology. He misses a crucial fact—that in a deep sense, chemis-try is physics, explaining a range of fascinating applications. That the connections between the core disci-plines “leaves him cold” worries me about the state of his students. What is more wondrous to students than the molecular explanation of the gas laws, or the qualitative interatomic analysis of the van der Waals law, or the lovely Bohr analysis of the spec-trum of hydrogen, the activity of so-dium, the inertness of helium—and at a slightly more elevated level, the life-giving process of photosynthesis? Ninth-grade physics indeed allows students to “sample the broad range of physics topics” and among these are the atomic and molecular tri-umphs of chemistry and biology.Leon Lederman

Resident ScholarIllinois Math and Science Academy

Aurora, IL 60506

European-Type Physics Plan

Bob Bessin’s argument for Physics First is really only a plea for Physics for All, since he is unconcerned with its place in the biology-chemistry-physics sequence. I agree that every-one should take physics, but I remain unconvinced that ninth grade is the best year for them to do it. I don’t see how a ninth-grade course could possibly prepare students for col-lege physics or an AP course in high school.

I prefer a European-type plan in which students get some physics all four years, beginning with things that are relatively directly acces-

sible, such as mechanics, optics, and electrical circuits, and ending with topics that are less accessible, more mathematical, and more abstract, such as rotational mechanics, electro-magnetic waves, and energy levels in crystals.

More advanced topics require the application of earlier concepts, re-inforcing the earlier learning as that learning becomes more general and the broader connections between topics can be drawn more clearly. It may become apparent at some point that certain students are unlikely to appreciate the subtleties of later topics. At that point, they could be shifted to a class that would turn in more applied directions, such as automobile engines, surveying tools, and house wiring.

Physics is not easy no matter when you begin studying it, and it takes more than one year to adjust your world view as much as physics asks you to. It also takes some recycling through the same concepts. If we can only hope for one year for everyone in the high school curriculum, why not distribute that over two years, leaving time for an additional year of physics for the few who want more? If your year is 180 days, you could use approximately 90 periods a year for two years. Better still might be even fewer, double periods, so that labs can be completed without undue anxiety. If you can make a case for both the importance and the diffi-culty of physics, it might be possible to schedule two days a week of lon-ger-than-usual classes.Bill Franklin

7016 Edmond Ave.Waco, TX 76710

Lettersto the Editor

326 The Physics Teacher ◆ Vol. 45, september 2007

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Page 3: Lorentz Transformation and Charge Conservation

The Elastic Constant I was very excited to see the article

“A Simple Experiment for Determin-ing the Elastic Constant of a Fine Wire.”1 This is a very important and underutilized subject in many introductory physics labs. However, after reading the article there were a few points that I felt needed clarifica-tion. One might argue the semantics of the phrase “the elastic constant” as described in the article. Although the parameter (M, units N/m) does remain constant when measured, it would not typically be called an elastic constant (units N/m2) and calling it as such may subsequently lead to confusion. The calculated pa-rameter in the article might be more accurately termed an effective spring constant. Also, any solid will have at least two (and up to 21) independent elastic constants.

Elastic constants are typically written in terms of Young’s modulus and Poisson’s ratio, or in terms of the stress-strain relationship

si = cij j ,

Where the stress (si ) has units of force per unit area, the strain (j) is the dimensionless stretch per unit length, and cij is the elastic stiffness matrix whose elements have dimen-sions of force per unit area.

In general, Young’s modulus can be written as a linear combination of other elastic constants, and thus the elastic constants must have the same physical dimensions (force per unit area, N/m2). For example, two independent elastic constants are needed to fully describe any isotropic material (such as copper metal). For isotropic materials, the stress-strain relationship is commonly written in terms of the two elastic constants c11 and c12. Young’s modulus (Y) is then

given by

Y cc

c c= = −

+sε

1

111

122

11 12

2.

Thus it is clear that two elastic constants are required to fully specify the elastic properties of an isotropic solid and that the elastic constants must have physical dimensions of N/m2.1 W. Larry Freeman and Ronald F.

Freda, “A simple experiment for determining the elastic constant of a fine wire,” Phys. Teach. 45, 224–227 (April 2007).

Kenneth A. Pestka IIDivision of Natural Sciences and

MathematicsDalton State College

650 College Drive Dalton, GA 30720

[email protected]

Tacoma Narrows BridgeAt the prodding of my wife Val,

I just had to respond to the Guest Editorial in the April 2007 issue of TPT.1 In it six high school teachers from the Chicago suburbs describe the thrill of visiting the Tacoma Nar-rows Bridge.

About 10 years ago, while still a scientist in a government laboratory, I took Val to Seattle and made the pilgrimage to The Bridge. I drove Val over the bridge, photographed her in front of the bridge, and took her to the Tacoma Narrows Museum, where she saw the classic film of the col-lapse. Whether she really understood the momentous nature of the expe-rience or not, she certainly under-stood me: For Christmas that year, I received the AAPT-produced VCR tape of the bridge collapse.

Fast forward to 2001. I was about to retire from my government job and was applying for the position of director of the physics and math

laboratories at a small liberal arts col-lege. The interview was demanding; not only was I to meet with each of the faculty in both departments, I also had to give an hour presenta-tion. The topic was “Technology in the Classroom.” So I diligently prepared a PowerPoint presentation that enumerated my reasons for us-ing technology in the classroom. My premise: Technology is not a goal in itself and must not get in the way of teaching. But the use of technology in classroom demonstrations can:

• Speed up the taking of data• Eliminate busywork in recording

data• Clarify the display and highlight

the importance of results• Do tedious calculations• Break the routine• Present phenomena that are too

large (or too small) for live demosIn my presentation to the assem-

bled faculty and students, I talked through the above litany, clarifying each point by a demonstration. What better way to talk about technology in the classroom than by applying it to the talk? And what better example of a phenomenon too large for an in-class recreation than the collapse of the Tacoma Narrows Bridge? So upon reaching the last point, I segued to my VCR tape. But upon starting the tape, I noticed a general look of amusement on the faces in the audience, especially on the face of one rather elderly gentleman.

Yep. The college was Kenyon Col-lege and the gentleman was Emeritus Professor of Physics Franklin Miller, who almost 40 years earlier had as-sembled the movies, produced, and narrated the very film I was using in an attempt to impress my interviewers!

A classic film—and a classic ex-

The Physics Teacher ◆ Vol. 45, september 2007 327

Letters

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Page 4: Lorentz Transformation and Charge Conservation

ample of “coals to Newcastle.”Incidentally, the same April issue

of TPT also contains a wonderful article by Miller on the disagreement between Millikan and Bearden on the charge of the electron,2 especially insightful through the eyes of one who was a colleague of many of the people involved.1. Scott Beutlich et al., “Taking advan-

tage of opportunities,” Phys. Teach. 45, 200 (April 2007).

2. Franklin Miller Jr., “Two kinds of electron?” Phys. Teach. 45, 210–216 (April 2007).

J. Terrence KlopcicDirector of Laboratories

Depts. of Physics and Mathematics Kenyon College, Gambier, OH

The Real StoryI enjoyed reading the Guest Edito-

rial “Taking Advantage of Opportu-nities,” which appeared in a recent issue of The Physics Teacher. I am sure that most of your readers, like myself, were envious of the “Holy Moment” that those six physics teachers expe-rienced on their visit to the site of the Tacoma Narrows Bridge. But do those teachers know the real reason for the bridge’s premature collapse?

For years in my physics classes I have been showing the classic video of “Galloping Gertie,” complete with the explanation of the phys-ics behind the collapse. This year, however, while I was watching the Turner Movie Classics channel on TV, I discovered that the bridge col-lapse was entirely the result of human interference. Specifically, it was the sound waves that were generated by a machine built by Superman’s nemesis Lex Luthor that caused the bridge’s demise! In Episode 1 of the movie serial Atom Man vs Superman (1950), Superman (Kirk Alyn) uses his su-

perpowers to detect the source of Lu-thor’s (Lyle Talbot) destructive sound waves while he uses his superstrength to temporarily stop the bridge from swaying so that a woman can be res-cued. After the woman is safe, Super-man lets go of a supporting column, the bridge collapses, and Superman brings Lex Luthor to justice.

If only the physics teachers who made the pilgrimage to the famous site had known all of this before their visit. In addition to artifacts from the original bridge, perhaps they would have also found pieces of Lex Luthor’s sound wave generator!Raymond H. Hahn

Massapequa High School4925 Merrick Road

Massapequa, NY [email protected]

Physics RespresentationsThe “Physics with a Smile”

(p. 158, March 2007) study shows the importance of teaching good physics representations. If the stu-dents in the study had not learned an effective representation of the third law, it is unlikely that they would have been successful understanding the third law.

The study provides an example of the importance of including physics representations in state physics stan-dards. At present, physics representa-tions are often not included in state physics standards. Not having repre-sentations in state physics standards makes the novice physics teacher believe that memorizing physics con-cepts is learning physics.

Including representations in the state physics standards guides the novice physics teacher into emphasiz-ing the physics representations that provide students with lifetime prob-lem-solving skills. By teaching good

328 The Physics Teacher ◆ Vol. 45, september 2007

Let

ters

physics representations, the novice physics teacher can make his or her students as successful as the students in the “Physics with a Smile” study.Bob Baker

El Camino Fundamental High School4300 El Camino Ave.

Sacramento, CA [email protected]

Lorentz Transformation and Charge Conservation

In his article “Special Relativity and Magnetism in an Introductory Physics Course,” R.G. Piccioni1 dis-cusses how the Lorentz contraction can have a significant impact even for particles moving as slowly as a fraction of a millimeter per second (10–12c).

His example is a neutral wire car-rying a current i. To an observer mov-ing along the wire, the wire appears to have net electric charge due to the difference in the Lorentz contraction of the positive and negative charge spacing. As Piccioni pointed out, two textbooks, by Purcell2 and Huggins,3 used this phenomena to introduce magnetic fields and derive the mag-netic force law.

The fact that a neutral wire appears charged in a different coordinate system raises a question about conser-vation of electric charge. Purcell han-dled this problem by modeling the neutral electric current as two lines of equally spaced, equal but opposite magnitude, charges moving at equal speeds in opposite directions.The lines have equal Lorentz contractions so that if they were started from rest, there would clearly be no net charge.

To handle the appearance of charge in a coordinate system where the lines do not have equal and op-posite velocities, Purcell points out in a footnote that total charge within a closed boundary is Lorentz invari-

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Page 5: Lorentz Transformation and Charge Conservation

ant, but if the two lines of charge were infinitely long, there would be no boundary that could enclose all the charges. Huggins took a different approach and modeled the neutral current more or less as you would see it in the lab. The line of positive charges, represent-ing copper ions, would be at rest, while the line of negative charges, representing the conduction elec-trons, would be moving at a speed v. The observation that the wire is electrically neutral requires that the equal magnitude positive and negative charges have equal spacing as seen in the lab. Piccioni used the Huggins model with the sign of the charges reversed so that one deals with positive electric currents and positive test particles.

In an unpublished letter to The Physics Teacher, Drew Milsom argued that the Purcell approach should be used because the Purcell model can be created starting from no net charge. Begin with two lines with equal but opposite charge at rest and start moving them equally fast in opposite directions. From this point of view there never was any net charge and therefore there can be no electric field. In the Huggins-Piccio-ni approach, the two lines at rest have different charge densities, and when you start moving one line to create a neutral current, there still has to be a net electric charge somewhere.

Milsom argued that this net charge would always have an electric field that you could not get rid of by a coordinate transformation.

In discussion between Huggins and Milsom, a very simple solution to the problem of charge conserva-tion emerged. Use finite length rods, include end effects, and charge conservation becomes obvious. In Fig.1(a,) we show two finite length

rods, with equal but opposite charge, in storage before the lecture. The negative rod is longer than the posi-tive rod because the charge density |l–| is less than l+ . During the lec-ture, a negative current is modeled by having the minus rod move paral-lel to the stationary positive rod at a speed v. This speed is adjusted so that, due to the Lorentz contraction, the rods have the same length. At the instant that the ends line up as shown in Fig.1(b), there is precisely no electric field if we picture the rods as merged together like a wire carry-ing a current. To someone moving with the negative rod, as in Fig. 1(c), the short positive rod is even more contracted, the negative rod is back to its uncontracted length, and from a distance we end up with a dipole electric field. However, if the moving observer places a test particle very

Letters

The Physics Teacher ◆ Vol. 45, september 2007 329

near the rods at the center, the test particle will feel the radial electric field described in the approach of Huggins and Piccioni. 1. R.G. Piccioni, “Special relativity

and magnetism in an introductory course,” Phys. Teach. 45, 152–156 (March 2007).

2. E.M. Purcell, Electricity and Mag-netism: Berkley Physics Course, Vol. 2 (McGraw-Hill, Boston, 1st ed. 1965, 2nd ed. 1985, p. 174).

3. E. Huggins, Physics I (W.A. Benja-min Co., New York, 1968), p. 315; and Physics2000 text at www. physics2000.com, p. 28-1.

Elisha HugginsDepartment of Physics

Dartmouth CollegeHanover, NH 03750

[email protected] Drew Milsom

Department of PhysicsUniversity of Arizona

Tucson, AZ [email protected]

Fig. 1. By analyzing finite length rods, we have no problem with conservationof charge. A test particle, in close to the center of the rods, sees the radial electric field of a positively charged “wire.” There the net charge density turns

out to be simply λl = iv/c2.

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