Coloumb's Law Lecture

Lecture 1

97.315 Basic E&M and Power Engineering Topic: COULOMB FORCECOULOMB FORCE

TITLETITLE

97.315 Basic E&M and Power Engineering Topic: COULOMB FORCE

• Theory• Two point charges, N = 2

• Units

• Examples of calculation

• Two point charges on axis

• Two point charges, one off axis

• Theory

• Multiple point charges, N > 2

• Examples of calculation

• Three point charges

• Force balancing

• Assignment

• References

• SummaryLecture 1 OUTLINEOUTLINE


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+Lecture 1 THEORYTHEORY


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The force exerted by one point charge on another acts along the line joining the charges. It varies inversely as the square of the distance separating the charges and is proportional to the product of the charges. The force is repulsive if the charges have the same sign and attractive if the charges have opposite signs.

Lecture 1 THEORYTHEORY


Two point charges q1 and q2q1

q2 F�

origin

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[F]-force; Newtons {N}

[q]-charge; Coulomb {C}

[r]-distance; meters {m}

[�]-permittivity; Farad/meter {F/m}

Property of the medium

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� UNIT VECTORCOULOMB FORCE

Lecture 1 UNITSUNITS


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Permittivity is a property of the medium.Also known as the dielectric constant.

Permittivity of free spacemFo / 10854.8 12�

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Coulomb’s constant

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Lecture 1 FORCE IN MEDIUM SMALLER THAN FORCE IN VACUUMTHEORYTHEORY


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PROBLEM SOLVING STRATEGYPROBLEM SOLVING STRATEGY

1. Consistent units are essential. Distances must be in meters,charge in coulombs and force in newtons. If you are given distances in centimeters, inches, or furlongs, don’t forget to convert! When a charge is given in microcoulombs, remember that 1 �C = 1x10-6 C.

2. Remember that the electric force is a vector quantity. You may want to go back and review vector algebra. It’s often useful to use components in an (x, y, z) coordinate system. Be sure to use the correct vector notation. If a symbol represents a vector quantity, underline it or place an arrow over it. If youget sloppy with your notation, you will get sloppy with your thinking. It is essential to distinguish between vector quantities and scalar quantities and to treat vectors properly as vectors.

Lecture 1 ASSIGNMENTASSIGNMENT


Example (Question):

A negative point charge of 1�C is situated in air at the origin of a rectangular coordinate system. A second negative point charge of 100�C is situated on the positive x axis at the distance of 500 mm from the origin. What is the force on the second charge?

Lecture 1 EXAMPLEEXAMPLE


Example (Solution):

A negative point charge of 1�C is situated in air at the origin of a rectangular coordinate system. A second negative point charge of 100�C is situated on the positive x axis at the distance of 500 mm from the origin. What is the force on the second charge?


mmr 50012 ��

Y

q1 = -1 �C F�

Xorigin q2= -100 �C


Example (Solution):

q1 = -1 �C

q2= -100 �C

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origin

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ENDLecture 1 EXAMPLEEXAMPLE


Example (Question):

Two point charges of opposite sign are placed in air. One charge (q1 = +1 C) is located at the origin while the other (q2 = -1 C) is at x = 1.0 cm, y = 3.0 cm. What is the magnitude and direction of the attractive force for each charge?



Example (Solution):

Two point charges of opposite sign are placed in air. One charge (q1 = +1 C) is located at the origin while the other (q2 = -1 C) is at x = 3.0 cm, y = 1.0 cm. What is the magnitude and direction of the attractive force for each charge?

Yq2= -1 C3 cm

1 cm

Xq1 = +1 C




Example (Solution):

q1 = +1 C

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Yq2= -1 C3 cm

Force on q2 1 cm

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YExample (Solution):

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q2= -1 C3 cm

Force on q1 1 cm

Xq1 = +1 C

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ENDLecture 1 EXAMPLEEXAMPLE


Example (Question)

A hydrogen atom consists of an electron of mass me = 9.1094 X 10-31 kg, moving about a proton of mass mp = 1.6726 X 10-27 kg at an average distance of 0.53 X 10-10 m. Determine the ratio of the electrical and gravitational forces acting between the two particles.



Example (Solution)

A hydrogen atom consists of an electron of mass me = 9.1094 X 10-31 kg, moving about a proton of mass mp = 1.6726 X 10-27 kg at an average distance of 0.53 X 10-10 m. Determine the ratio of the electrical and gravitational force acting between the two particles.

Given: � qe = �1.602 X 10-19 C, me = 9.1094 X 10-31 kg, mp= 1.6726 X 10-27 kg and r = 0.53 X 10 -10 m. Find Fe and Fg.

From Coulomb’s law:

From Newton’s law:

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Example (Solution)

Given: � qe = �1.602 X 10-19 C, me = 9.1094 X 10-31 kg, mp= 1.6726 X 10-27 kg and r = 0.53 X 10 -10 m. Find Fe and Fg.

From Coulomb’s law:

From Newton’s law:

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So Fe/Fg = 2.3 X 10 39 The electric force is far stronger than the gravitational force.

END


Multiple point charges

It has been confirmed experimentally that when several charges are present, each exerts a force given by

on every other charge. The interaction between any two charges is independent of the presence of all other charges.

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The net force on any one charge is the vector sum off all The net force on any one charge is the vector sum off all the forces exerted on it due to each of the other charges the forces exerted on it due to each of the other charges interacting with it independently.interacting with it independently.



5 point charges: Net force on q5

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The fact that the electric forces add vectorially is called the

SUPERPOSITION PRINCIPLESUPERPOSITION PRINCIPLE



Example (Question)

The charging of individual raindrops is ultimately responsible for the electrical activity in thunderstorms. Suppose two drops with equal charge q are located on the x axis at � a. Find the electric force on a third drop with charge Q located at an arbitrary point on the y axis.



Example (Solution)

The charging of individual raindrops is ultimately responsible for the electrical activity in thunderstorms. Suppose two drops with equal charge q are located on the x axis at � a. Find the electric force on a third drop with charge Q located at an arbitrary point on the y axis.

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Y

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y

q q

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Example (Solution)

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Charge Q is the same distance r from the two other charges, so the force from each has the same magnitude:

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The direction of the two forces are different.Lecture 1 EXAMPLEEXAMPLE


Example (Solution)Y

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The x components cancel, while the y components add.

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Example (Question)

A positive charge +2q lies on the x axis at x = -a, and a charge -q lies at x = +a. Find the point where the electric force on a third charge Q would be zero.



Example (Solution)


YQ


-q2q

Xa a

The point must lie on the x axis since off axis points cannot have force vectors which point in opposite directions.


Example (Solution)


Y


Q -q2q

Xa a

Between the two charges the repulsion from one and the attraction of the other would add and give a non-zero effect.


Example (Solution)


Y


-q2qQ

Xa aOn the left of 2q, the Q charge would be closer to the 2q charge than the -q charge. The force from 2q on Q would be larger partially because it is closer and partially because it is a greater charge.


Example (Solution)


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Q-q2q

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So the only location where the two forces might cancel is to the right of both charges.


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Example (Solution)

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magnitude but different directions.

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Example (Solution)

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Since x > a, this point is to the right of the -q and 2q charges and therefore is indeed a point of zero force.


Y Consider x - a � 0. Force from -q charge dominates.


Example (Solution)

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Consider x - a >> 0. Force appears to originate from one charge q only. netF

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At x= 5.83 a = 0 netF�

END


Coulomb’s Law: Multiple point charges

• Questions (16-30)

Coulomb’s law: Two point charges

• Questions (1-15)

Assignment goal:Be able to solve the 75% and 100% questions

without making reference to your notes.

Start with the 25% questions and work your way up.




These questions are straight forward. Plug in the numbers and get your answer. Being able to solve this type of question ensures you of at least a grade of 25% on a quiz or final exam containing questions related to this lecture.

25

These questions require a few manipulations of equations or numbers before the answer can be obtained. Being able to solve this type of question ensures you of at least a grade of 50% on a quiz or final exam containing questions related to this lecture.

50

These question are quite involved and requires a thorough understanding of the topic material. Being able to solve this type of question ensures you of at least a grade of 75% on a quiz or final exam containing questions related to this lecture.

75

These questions are the most difficult and require a thorough understanding of the topic material and also pull in topics from other lectures and disciplines. Being able to solve this type of question ensures you an A grade on a quiz or final exam containing questions related to this lecture.

100

These form excellent review questions when preparing for the quiz and final exam.10075

25 10050 75

SELL EVALUATION SCALESELL EVALUATION SCALE


(1) A typical lightning flash delivers about 25 Coulombs of negative charge from clouds to ground. How many electrons are involved?

25



(2) How many electrons are needed to produce a charge of -1.0C?25



(3) The electron and proton in a hydrogen atom are 52.9 pm apart. What is the magnitude of the electric force between them?

25



(4) Two protons are fired directly at each other in a vacuum chamber. What is the force of repulsion at the instant 1.1 X 10-14 m apart?

25



(5) Two point-charges of +0.50 �C are 0.10 m apart. Determine the electric force they each experience in air.

25



(6) Two charges, one twice as large as the other, are located 15 cm apart and experience a repulsive force of 95 N. What is the magnitude of the larger charge.

50

Cans 510259.1: �

�Lecture 1 ASSIGNMENTASSIGNMENT


(7) How many excess electrons must be present on each of two small spheres spaced 15.0 cm apart if the spheres have equal charge and if the magnitude of the force of repulsion between them is 5.00 X 10-9 N?

50

610698: �ansLecture 1 ASSIGNMENTASSIGNMENT


(8) A proton is at the origin and an electron is at the point x = 0.41 nm, y = 0.36 nm. Find the electric force on the proton.

50

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(9) Two tiny spheres carrying the same charge are 1.0 m apart in vacuum and experience an electrical repulsion of 1.0 N. What is their charge?

50

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(10) A very small conducting sphere in air carries a charge of 5.0 picocoulombs and is 0.20 m from another such sphere carrying a charge Q. If each sphere experiences a mutual electrical repulsion of 2.0 �N, find Q.

75

nCans 78.1:Lecture 1 ASSIGNMENTASSIGNMENT


(11) Two equally charged small spheres repel each other with an electric force of 1.0 N when 0.50 m apart, center-to-center, in air. What is the charge on each sphere.

75

Cans 61027.5: �



(12) A 9.5 �C charge is at x = 16 cm, y = 5.0 cm, and a -3.2 �C charge is at x = 4.4 cm, y = 11 cm. Find the force on the negative charge.

75

� �Nyxans ˆ36.7ˆ24.14: �Lecture 1 ASSIGNMENTASSIGNMENT


(13) Earth caries a net charge of -4.3 X 105 C. The force due to this charge is the same as if it were concentrated at earth’s center. How much charge would you have to place on a 1.0 g mass in order for the electrical and gravitation forces on it to balance?

100



(14) Compute the gravitational attraction between two electrons separated by 1.0 mm in vacuum, and compare that with the electrical repulsion they experience.

100


(15) Two small spheres with the same mass m and charge q are suspended from massless strings of length , as shown in the figure. Each string makes and angle � with the vertical. Show that the charge on each sphere is

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100



(16) Three small negatively charged spheres in vacuum are fixed on a horizontal straight line, the x axis. One (-12.5 �C) is at the origin, another (-5.0 �C) is at x = 2.0 m, and the third (-10.0 �C) is 1.0 m beyond that at 3.0 m. Compute the net electric force on the last sphere due to the other two.

25



(17) Two charges +q and -q reside in vacuum on the y axis at location of -1/2 d and +1/2 d respectively. Determine the force on a third charge +Q located at a distance of +x from the origin on the x axis.

50

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22

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(18) The figure shows three tiny uniformly charged spheres. Determine the net force on the middle sphere due to the other two. Take q1 = +5 �C, q2 = -4.0 �C and q3 = +10.0 �C.

50

q1 q3q2


6.0 cm2.0 cm

xNans ˆ350:�


(19) A 60 �C charge is at the origin, and a second charge in on the positive x axis at x = 75 cm. If a third charge placed at x = 50 cm experiences no net force, what is the second charge?

50

Cans �15:Lecture 1 ASSIGNMENTASSIGNMENT


(20) A charge 3q is at the origin, and a charge -2q is on the positive x axis at x = a. Where would you place a third charge so it would experience no electric force.

50

axans 45.5: �Lecture 1 ASSIGNMENTASSIGNMENT


(21) In the figure take q1 = 68 �C, q2 = -34 �C, and q3 = 15 �C. Find the electric force on q3.50

Y (m)


4

3

X (m)1 2 3 4

1

2q3

q1

q2

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(22) The figure depicts three small charged spheres at the vertices of a 3-4-5 right triangle. Calculate the force exerted on q3 by the other two charges. Take q1= +50 �C, q2 = -80 �C and q3 = +10 �C. Express the forces using the unit vectors.� �yx ˆ,ˆ

75

Y

40 cm

30 cm

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q2

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(23) Two point charges are located on the y axis as follows: Charge q1 = +3.80 nC is at y = 0.600 m, and charge q2 = -2.50 nC is at the origin (y = 0). What is the total force (magnitude and direction) exerted by these two charges on a third point charge q3 = +5.00 nC located at y = -0.400 m?

75

yNans ˆ10531: 9��



(24) You have two charges +4q and one charge -q. (a) How would you place them along a line so there’s no net force on any of the three? (b) Is this equilibrium stable or unstable?

75



(25) Two point charges are placed on the x axis as follows: Charge q1 = +3.00 nC is located at x = 0.400 m, and charge q2 = +5.00 nC is at x = -0.200 m. What are the magnitude and direction of the total force exerted by these two charges on a negative point charge q3 = -8.00 nC at the origin?

75

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(26) Four identical charges q form a square of side a. Find the magnitude of the electric force on any of the charges.

75

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(27) Three identical charges +q and a fourth charge -q form a square of side a. (a) Find the magnitude of the electric force on a charge Q placed at the center of the square. (b) Describe the direction of this force.

75



(28) Two identical small metal spheres initially carry charges q1 and q2, respectively. When they’re 1.0 m apart they experience a 2.5 N attractive force. Then they’re brought together so charge moves from one to the other until they have the same net charge. They’re again placed 1.0 m apart, and now they repel with a 2.5 N force. What were the original values of q1 and q2?

100



(29) A shown in the figure a positive charge +Q is located at the origin and an array of equally spaced negative charges (-q) are placed along the x axis. Obtain a compact expression for the total force on the charge +Q due to all the other charges when: (a) N = 1, (b) N = 2, (c) N = 10, (d) N = 100, (e) N = 1000, (f) N = infinity. You may find it instructive to plot total force versus charge number N.

100

+Q -q-q -q-q -q -q -q

Xa a a a a a a



(30) A shown in the figure a positive charge +Q is located at the origin and an array of equally spaced alternating sigh charges (�q) are placed along the x axis. Obtain a compact expression for the total force on the charge +Q due to all the other charges when: (a) N = 1, (b) N = 2, (c) N = 10, (d) N = 100, (e) N = 1000, (f) N = infinity. You may find it instructive to plot total force versus charge number N.

100

+Q -q+q -q-q +q -q +q

Xa a a a a a a



Lecture 1 REFERENCESREFERENCES

(0) Inan p. 246 - 255(1) Kraus p. 12 - 15(2) Reitz p. 27 - 31(3) Plonus p. 2 - 4(4) Winch p. 258 - 266(5) Lorrain p. 40 - 42(6) Duckworth p. 5 - 8(7) Jackson p. 27 - 28(8) Ulaby p. 7, 143 - 144

(0) Textbook: U. S. Inan, A. S. Inan “Engineering Electromagnetics”

(1) J.D. Kraus, K. R. Carver “Electromagnetics” 2nd

(2) Reitz, Milford, Christy “Foundations of Electromagnetic theory” 4th

(3) M. Plonus “Applied Electromagnetics”(4) R. P. Winch “Electricity and Magnetism”(5) P. Lorrain, D. Corson “Electromagnetic fields and Waves”

2nd

(6) Duckworth “Electricity and Magnetism”(7) J.D. Jackson “Classical Electrodynamics” 2nd

(8) F. Ulaby, “Fundamentals of applied Electromagnetics”


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SUPERPOSITION PRINCIPLESUPERPOSITION PRINCIPLELecture 1 SUMMARYSUMMARY


Lecture 1 ENDEND

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Coloumb's Law Lecture