Chapter 1 Charge & Coulomb's Law

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cbPhysicsIIb10.doc

Calculus-Based Physics II by Jeffrey W. Schnick

Copyright 2006, Jeffrey W. Schnick, Creative Commons Attribution Share-Alike License 2.5. You can copy, modify, and re-release this work under the same license provided you give attribution to the author. See http://creativecommons.org/

1 Charge & Coulomb's Law .......................................................................................................2

2 The Electric Field: Description and Effect .............................................................................10

3 The Electric Field Due to one or more Point Charges ............................................................15

4 Conductors and the Electric Field ..........................................................................................25

5 Work Done by the Electric Field, and, the Electric Potential..................................................33

6 The Electric Potential Due to One or More Point Charges .....................................................43

7 Equipotential Surfaces, Conductors, and Voltage ..................................................................49

8 Capacitors, Dielectrics, and Energy in Capacitors..................................................................55

9 Electric Current, EMF, Ohm's Law........................................................................................65

10 Resistors in Series and Parallel; Measuring I & V................................................................72

11 Resistivity, Power ...............................................................................................................86

12 Kirchhoffs Rules, Terminal Voltage...................................................................................92

13 RC Circuits .......................................................................................................................101

14 Capacitors in Series & Parallel ..........................................................................................111

15 Magnetic Field Intro: Effects .............................................................................................117

16 Magnetic Field: More Effects ............................................................................................124

17 Magnetic Field: Causes .....................................................................................................139

18 Faraday's Law, Lenz's Law................................................................................................147

19 Induction, Transformers, and Generators ...........................................................................161

20 E&M Fields in MotionFaradays Law and Maxwells Extension to Amperes Law .......180

21 The Nature of Electromagnetic Waves...............................................................................198

22 Huygens Principle and 2-Slit Interference ........................................................................204

23 Single-Slit Diffraction .......................................................................................................226

24 Thin Film Interference.......................................................................................................232

25 Polarization .......................................................................................................................238

26 Geometric Optics, Reflection.............................................................................................244

27 Refraction, Dispersion, Internal Reflection ........................................................................252

28 Thin Lenses: Ray Tracing..................................................................................................258

29 Thin Lenses: Lens Equation, Optical Power ......................................................................272

30 The Electric Field Due to a Continuous Distribution of Charge on a Line ..........................282

31 The Electric Potential due to a Continuous Charge Distribution.........................................294

32 Calculating the Electric Field from the Electric Potential ...................................................299

33 Gausss Law......................................................................................................................311

34 Gausss Law Example .......................................................................................................319

35 Gausss Law for the Magnetic Field, and, Amperes Law Revisited ..................................324

36 The Biot-Savart Law .........................................................................................................333

37 Maxwells Equations .........................................................................................................339

Chapter 1 Charge & Coulomb's Law

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1 Charge & Coulomb's Law

Charge is a property of matter. There are two kinds of charge, positive + and negative . An object can have positive charge, negative charge, or no charge at all. A particle which has

charge causes a force-per-charge-of-would-be-victim vector to exist at each point in the region of

space around itself. The infinite set of force-per-charge-of-would-be-victim vectors is called a

vector field. Any charged particle that finds itself in the region of space where the force-per-

charge-of-would-be-victim vector field exists will have a force exerted upon it by the force-per-

charge-of-would-be-victim field. The force-per-charge-of-would-be-victim field is called the

electric field. The charged particle causing the electric field to exist is called the source charge.

(Regarding jargon: A charged particle is a particle that has charge. A charged particle is often

referred to simply as a charge.)

The source charge causes an electric field which exerts a force on the victim charge. The net

effect is that the source charge causes a force to be exerted on the victim. While we have much

to discuss about the electric field, for now, we focus on the net effect, which we state simply

(neglecting the middle man, the electric field) as, A charged particle exerts a force on another

charged particle. This statement is Coulombs Law in its conceptual form. The force is called

the Coulomb force, a.k.a. the electrostatic force.

Note that either charge can be viewed as the source charge and either can be viewed as the victim

charge. Identifying one charge as the victim charge is equivalent to establishing a point of view,

similar to identifying an object whose motion or equilibrium is under study for purposes of

applying Newtons 2nd Law of motion,

m

= Fav

v. In Coulombs Law, the force exerted on one

charged particle by another is directed along the line connecting the two particles, and, away

from the other particle if both particles have the same kind of charge (both positive, or, both

negative) but, toward the other particle if the kind of charge differs (one positive and the other

negative). This fact is probably familiar to you as, like charges repel and unlike attract.

The SI unit of charge is the coulomb, abbreviated C. One coulomb of charge is a lot of charge,

so much that, two particles, each having a charge of +1 C and separated by a distance of 1 meter

exert a force of 9109 N, that is, 9 billion newtons on each other.

This brings us to the equation form of Coulombs Law which can be written to give the

magnitude of the force exerted by one charged particle on another as

2

21

rqq

kF = (1-1)

where:

k = 2

29

C

mN10998

. , a universal constant called the Coulomb constant,

1q is the charge of particle 1,

2q is the charge of particle 2, and

r is the distance between the two particles.

Chapter 1 Charge & Coulomb's Law

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The user of the equation (we are still talking about equation 1-1, 2

21

rqq

kF = ) is expected to

establish the direction of the force by means of common sense (the users understanding of

what it means for like charges to repel and unlike charges to attract each other).

While Coulombs Law in equation form is designed to be exact for point particles, it is also exact

for spherically symmetric charge distributions (such as uniform balls of charge) as long as one

uses the center-to-center distance for r .

Coulombs Law is also a good approximation in the case of objects on which the charge is not

spherically symmetric as long as the objects dimensions are small compared to the separation of

the objects (the truer this is, the better the approximation). Again, one uses the separation of the

centers of the charge distributions in the Coulombs Law equation.

Coulombs Law can be written in vector form as:

122

2112 rrrrr

qqk=F

v (1-2)

where:

12Fv

is the force of 1 on 2, that is, the force exerted by particle 1 on particle 2,

12rrrr is a unit vector in the direction from 1 to 2, and k, q

1, and q

2 are defined as before (the Coulomb consta