Electromagnetism
Physics 15b
Lecture #1
Coulomb’s Law
Electric Charge, Force, and Energy
Purcell 1.1–1.6
Today’s Goals
Electric charge
Quantization, charge symmetry
  Conservation

Coulomb’s Law
Forces between two charged objects
 Inverse-square law
  CGS units vs. SI units

Superposition Principle

2-body  N-body of charges
Work and energy

Conservative force
Charles-Augustin de Coulomb
(1736–1806)
1
Four Forces
There are 4 fundamental forces in Nature
Gravity
  Long distance. Keeps planets and satellites in orbits
  Classical model (Newtonian) simple and accurate
  Modern model (General Relativity) complex and more accurate
  Electromagnetic force
  Long distance. Responsible for most daily things
  Classical model (Maxwell) simple, accurate, and identical to the
modern model
  Strong nuclear force
  Short distance. Keeps protons and neutrons in atomic nuclei
  Weak nuclear force
  Short distance. Responsible for nuclear b-decays

Electrical Charge
Objects don’t always respond to electricity

Something must be added  or, they must be “charged”
Experiments have told us:
Charges can be positive or negative (Franklin)
  They come in small same-sized units (Millikan)
  Each unit is so small, and the number of units in human-size
object so large (~1023), it looks like continuous

Intrinsic property of the elementary particles
Proton is positively charged
  Electron is negatively charged by the same amount

e = “Elementary Charge”
2
Charge Symmetry
What is “positive” (or “negative”) is just a convention

Franklin could have named them oppositely
Elementary particles have oppositely-charged anti-particles
Particle
Proton
Antiparticle
+e
Electron –e
Neutron
0
Antiproton
–e
Positron
+e
Antineutron
0
Physical laws are almost
unchanged if the positive
and negative charges are
replaced
Small catch: Universe made exclusively of particles
Just so, i.e. it started that way and stayed that way?
  Only in our neighborhood, i.e. this galaxy?
  Small violation of the charge symmetry?

Conservation of Charge
Charge cannot be created or destroyed

You can only move them from
one object to another
Total charge of an isolated
system is conserved
True even in extreme conditions
where particle are destroyed
and created
  Example from high-energy collision
of an electron and a positron 

e + e − → π +π −π +π −π +π −
Demo: charge conservation
An event from
the CLEO Experiment
Courtesy of A. Foland
3
Coulomb’s Law
Electric force (F) between two charges (q1 and q2) can work
over a distance (r)
Smaller r  stronger F
  Larger r  weaker F

F1
Coulomb found
F2 = k

q1q2
r212
q1
q2
r21
F2
Proportional to both charges
r̂21
Along the line connecting the charges
Inversely proportional to the (distance)2
NB: definition of the unit vector
r̂ ≡
r
r
Parallel to the original r, with the length
r̂ = 1
Units
Two systems of units: CGS vs. SI (Systèm Internationale)
CGS
SI
Fundamental Units cm, g, s m, kg, s, ampere (A)

In CGS, Coulomb’s Law (with k = 1) defines the unit of charge
F[dyne] =

q1[esu]q2 [esu]
r21[cm]2
1esu = 1cm 1dyne
electrostatic unit
In SI, the unit of charge is coulomb (C) = ampere x second
F[N] = k

q1[C]q2 [C]
More often:
r21[m]2
k≡
1
4πε 0
k = 8.9875 × 109 N ⋅ m 2 C2
≈ 9.0 × 109 N ⋅ m 2 C2
ε 0 = 8.8542 ×10−12 C2 N ⋅ m 2
permittivity of vacuum
4
Conversion Table
SI Units
CGS Units
Energy
1 joule (J)
= 107 erg
Force
1 newton (N)
= 105 dyne
Electric Charge
1 coulomb (C)
= “3” x 109 esu
Electric Current
1 ampere (A)
= “3” x 109 esu/sec
Electric Potential
“3” x 102 volt (V) = 1 statvolt
Electric Field
“3” x 104 V/m
= 1 statvolt/cm
Elementary Charge e
1.6 x 10–19 C
= 4.8 x 10–10 esu
“3” = 2.99892458 (exact)
  See Table E.1 (p.475) of Purcell for more

We will stick with CGS most of the time (as Purcell does)

Real-world examples may have to use SI
Inverse-Square Laws
Electrostatic force and gravity share the r-dependence
Fe = k

q1q2
r
2
Fg = G
m1m2
r
Inverse-square
laws
2
Much of what you know about gravity will apply to electricity
⎧k = 1
Difference 1: constants ⎨
−8
2
2
⎪⎩G = 6.672 × 10 dyne ⋅ cm g
1 esu
1 esu
1g
1g
1 dyne
0.00000007 dyne
1 cm
5
Electrostatic Force vs. Gravity
Difference 2: signs
Charge or Mass
Force
Electrostatic positive or negative attractive or repulsive
Gravity
only positive
only attractive
Since electrostatic force is so strong, large objects tend to
contain roughly equal numbers of protons and electrons

Net charge is zero, or quite small
Electrostatic force is important inside or between atoms

That’s physics and chemistry
Gravity is important for astronomical objects
Superposition Principle
Suppose we have many charges distributed in space

What is total force on charge Q
from all the other charges?
Principle of Superposition
Just add them up as vectors
  For 3 charges (picture right)
q1q3
r312
r̂31 +
q2 q3
r322
N
Fj = ∑ Fjk = q j ∑
k =1
k≠ j
Force on the j-th charge
k =1
k≠ j
q3
F31
r̂32
F32
Generally for N charges
N
r32
r31

F3 = F31 + F32 =
q2
q1
qk
rjk2
r̂ jk
F3
Sum over k, skipping j
6
Work against Gravity
“Work” is defined (in physics) by W = ∫ force × distance

Slope and gravity from 15a:
© J.E. Hoffman
W = mg sin θ1 ⋅ L21 + h 2
= mg
h
L +h
2
1
2
W = mg sin θ 2 ⋅ L22 + h 2
L21 + h 2
= mgh

= mg
h
L +h
2
2
2
L22 + h 2
= mgh
Work against gravity does not depend on the angle of the slope
 Gravity is a conservative force
Conservative Force
With a conservative force (such as gravity)
Work is path-independent : depends on the initial and final positions,
but not on the path taken between
  Work is reversible : WAB = –WBA
  You can “store” your work and get it back later
  Energy U can be defined as a function of the positions, so that
WAB = U(B) – U(A)
  For gravity, U(h) = mgh

Electrostatic force is conservative
It must be  Same inverse-square law as gravity!
In fact any force in the form F = f (r) r̂ is conservative
  Will have some more to say in Chapter 2


7
Two Charges
Bring two charges from very far away to a distance r12
q1

r12
q2
ds
F
r=∞
r
Work is
W = ∫ F ⋅ ds =
r12
q1q2
qq
r̂ ⋅ (− drr̂) = 1 2
2
r
r12
r=∞
∫
Using r = ∞ as the reference point (U = 0) we can define the
electrostatic energy of the two-charge system as
U=


q1q2
r12
Note that it doesn’t matter which charge is moved
It doesn’t matter if the path is not straight
One More Charge
Bring a third charge to the system
q1
r13
r12
q3
r23
q2


ds
F32
F31
r=∞
W = ∫ (F31 + F32 ) ⋅ ds = ∫ F31 ⋅ ds + ∫ F32 ⋅ ds
Superposition Principle allows us to do the integration in pieces
Each piece is the energy of a two-charge system, which is pathindependent
Total energy for the three-charge system is
U=
q1q2 q1q3 q2 q3
+
+
r12
r13
r23
8
N-Charge System
Electrostatic energy of an N-charge system is
q j qk
1 N
U = ∑∑
2 j =1 k ≠ j rjk


The double sum runs j = 1 … N and k = 1 … N, but skipping j = k
½ takes care of the double-counting of, e.g. (j, k) = (1, 2) and (2, 1)
Remember: U = 0 is defined as “when all charges are very
far away from each other”
Summary
Electric charge = Source and recipient of electric forces

Positive and negative; conserved; and quantized
Coulomb’s Law F2 =

q1q2
r212
r̂21
F1
q1
q2
r21
Inverse-square law same as gravity
Multiple charges  Superposition Principle Fj =
Electrostatic force is conservative
∑F
k≠ j
F2
jk
Electrostatic energy depends only on the positions of the charges
  For 2-body:
For N-body:
N
qq

U=
q1q2
r12
U=
1
j k
∑
∑
2 j =1 k ≠ j rjk
9