1. Physics 712 – Electricity and Magnetism
Everyone Pick Up:
Syllabus
Two homework passes
Materials
Classical Electrodynamics by John David Jackson
Calculator
Pencils or pens, paper
Symbolic manipulation
Eric Carlson
“Eric”
“Professor Carlson”
Olin 306
Office Hours always
(o) (c)
1/13

2. Dr. Carlson’s Approximate Schedule
Monday
Tuesday
Wednesday
Thursday
Friday
9:00
research
research
10:00
PHY 712
office hour
PHY 712
office hour
PHY 712
11:00
PHY 742
office hour
PHY 742
office hour
PHY 742
12:00
PHY 109
office hour
PHY 109
office hour
PHY 109
1:00
2:00
research
research
research
research
3:00
Collins Hall
4:00
Free food
colloquium
Collins Hall
5:00
Lab Meeting
I will try to be in my office Tues. Thurs
When in doubt, call/ first

3. Classical Electrodynamics by J.D. Jackson, 3rd Edition
Reading Assignments / The Text
Classical Electrodynamics by J.D. Jackson, 3rd Edition
I’m not writing my own textbook
Contains useful information
Reading assignments every day
ASSIGNMENTS
Day Read Homework
Today none none
Friday
Wednesday , 1.2

4. Homework http://users.wfu.edu/ecarlson/quantum
Problems assigned as we go along
About one problems due every day
Homework is due at 10:00 on day it is due
Late homework penalty 20% per day
Two homework passes per semester
Working with other student is allowed
Seek my help when stuck
You should understand anything you turn in
ASSIGNMENTS
Day Read Homework
Today none none
Friday
Wednesday , 1.2

5. Attendance and Tests Tests Attendance
I do not grade on attendance
Attendance is expected
Class participation is expected
I take attendance every day
Tests
Midterm will be from on March 4
Final will be from 9-12 on April 29

6. Percentage Breakdown:
Grades, Pandemic Plans
Grade Assigned
94% A 77% C+
90% A- 73% C
87% B+ 70% C-
83% B <70% F
80% B-
Percentage Breakdown:
Homework 50%
Midterm 20%
Final 30%
Some curving possible
Emergency contacts:
Web page
Cell:
Pandemic Plans
If there is a catastrophic closing of the university, we will attempt to continue the class:

7. 0A. Math Coordinate Systems We will generally work in three dimensions
A general coordinate in 3d will be denoted x
A general vector will be shown in bold face v
We will often work in Cartesian coordinates
Sometimes in spherical coordinates, related to Cartesian by
Coordinate x is
Sometimes in cylindrical coordinates, related to Cartesian by

8. Vector Identities
Vectors can be combined using dot products to make a scalar
Vectors can be combined using cross-products to make a vector
We often abbreviate
Some vector identities:
Symmetry/antisymmetry:
Triple scalar product:
Double cross-product:
These and many others in Jackson front cover

9. Derivatives in 3D
The vector derivative  can be used for several types of derivatives:
Gradient turns a scalar function into a vector function
Divergence turns a vector function into a scalar function
Curl turns a vector function into a vector function
There is also the Laplacian, a second derivative that can act on scalar or vector functions
Each of these has more complicated forms in non-Cartesian coordinates
See QM lecture notes or inside back cover of Jackson

10. Derivatives Product rule
The product rule for derivatives:
Numerous 3D equivalents for products
Gradient of product of scalars
Divergence of scalar times vector
Curl of scalar times vector
Divergence of a cross product
Each of these and many more in Jackson, front cover

11. Integrals in 3D 3d integrals of vector scalar functions will be common
You should know how to handle these in any coordinate system
Cartesian:
Spherical:
Cylindrical:

12. Fundamental Theorem of Calculus in 3D
Fundamental theorem of calculus says:
In general, in 3d, this theorem lets you do one integral whenever you have an integral in 3d:
Line integral:
Stokes’ Theorem:
Divergence Theorem:
Another theorem:
And another theorem:
All these and more can be found on inside front cover of Jackson

13. Sample Problem 0.1
Work in spherical coordinates
For x  0, take divergence
Tricky at x = 0 because everything is infinite there!
Integrate over a sphere of radius R using the divergence theorem:
Since the integral is zero except at the origin, we must have
You can generalize this where x is replaced by the difference from an arbitrarily chosen origin x':
Consider the vector function x/|x|3 . (a) Find the divergence for x  0. (b) By integrating over a sphere around the origin, show that the divergence does not vanish there. R

14. 0B. Units
Units is one of the most messed-up topics in electricity and magnetism
We will use SI units throughout
Fundamental units:
From these are derived lots of non fundamental units:
Equations of E and M sometimes depends on choice of units!
Distance meter m
Time second s
Mass kilogram kg
Charge coulomb C
Frequency hertz Hz s–1
Force newton N kgm/s2
Energy joule J Nm
Power watt W J/s
Current amp A C/s
Potential volt V J/C
Resistance ohm  V/A
Magnetic induction
tesla T kg/C/s
Magnetic flux weber Wb Tm2
Inductance Henry H Vs/A

15. 1A. Coulombs Law, El. Field, and Gauss’s Law
1. Electrostatics
1A. Coulombs Law, El. Field, and Gauss’s Law
Coulomb’s Law
Charges are measured in units called Coulombs
The force on a charge q at x from another charge q' at x':
The unit vector points from x' to x
We rewrite the unit vector as
For reasons that will make some sense later, we rewrite constant k1 as
So we have Coulomb’s Law:
For complicated reasons having to do with unit definitions, the constant 0 is known exactly:
This constant is called the permittivity of free space

16. Multiple Charges, and the Electric Field
If there are several charges q'i, you can add the forces:
If you have a continuous distribution of charges (x), you can integrate:
In the modern view, such “action at a distance” seems unnatural
Instead, we claim that there is an electric field caused by the other charges
Electric field has units N/C or V/m
It is the electric field that then causes the forces

17. Gauss’s Law: Differential Version
Let’s find the divergence of the electric field:
From four slides ago:
We therefore have:
Gauss’s Law (differential version):
Notice that this equation is local

18. Gauss’s Law: Integral Version
Integrate this formula over an arbitrary volume
Use the divergence theorem:
q(V) is the charge inside the volume V
Integral of electric field over area is called electric flux
Why is it true?
Consider a charge in a region
Electric field from a charge inside a region produces electric field lines
All the field lines “escape” the region somewhere
Hence the total electric flux escaping must be proportional to amount of charge in the region q

19. Sample Problem 1.1 (1)
A charge q is at the center of a cylinder of radius r and height 2h. Find the electric flux out of all sides of the cylinder, and check that it satisfies Gauss’s Law
Let’s work in cylindrical coordinates
Electric field is:
Do integral over top surface:
By symmetry, the integral over the bottom surface is the same  h h z r q h r

20. Sample Problem 1.1 (2)
A charge q is at the center of a cylinder of radius r and height 2h. Find the electric flux out of all sides of the cylinder, and check that it satisfies Gauss’s Law  h h
Do integral over lateral surface:
Add in the top and bottom surfaces: z r q h r

21. Using Gauss’s Law in Problems
Gauss’s Law can be used to solve three types of problems
Total electric flux out of an enclosed region
Simply calculate the total charge inside
Electric flux out of one side of a symmetrical region
Must first argue that the flux out of each side is the same
Electric field in a highly symmetrical problem
Must deduce direction and symmetry of electric field from other arguments
Must define a Gaussian Surface to perform the calculation
Generally use boxes, cylinders or spheres

22. Sample Problem 1.2
A line with uniform charge per unit length  passes through the long diagonal of a cube of side a. What is the electric flux out of one face of the cube?
The long diagonal of the cube has a length
The charge inside the cube is therefore
The total electric flux out of the cube is
If we rotate the cube 120 around the axis, the three faces at one end will interchange
So they must all have the same flux around them
If we rotate the line of charge, the three faces at one end will interchange with the three faces in back
So front and back must be the same
Therefore, all six faces have the same flux

23. Sample Problem 1.3
A sphere of radius R with total charge Q has its charge spread uniformly over its volume. What is the electric field everywhere?
By symmetry, electric field points directly away from the center
By symmetry, electric field depends only on distance from origin
Outside the sphere:
Draw a larger sphere of radius r
Charge inside this sphere is q(r) = Q
By Gauss’s Law,
Inside the sphere:
Draw a smaller sphere of radius r
Charge inside this sphere is only
Final answer:

24. Curl of the Electric Field
1B. Electric Potential
Curl of the Electric Field
From homework problem 0.1:
Generalize to origin at x':
Consider the curl of the electric field:
Using Stokes’ theorem, we can get an integral version of this equation:

25. Electric Field: Discontinuity at a Boundary
Consider a surface (locally flat) with a surface charge 
How does electric field change across the boundary?
Consider a small thin box of area A crossing the boundary
Since it is small, assume E is constant over top surface and bottom surface
Charge inside the box is A
Use Gauss’s Law on this small box
Consider a small loop of length L penetrating the surface
Use the identity
Ends are short, so only include the lateral part
So the change in E across the boundary is A

26. The Electric Potential
In general, any function that has curl zero can be written as a gradient
Proven using Stokes’ Theorem
We therefore write:
 is the potential (or electrostatic potential)
Unit is volts (V)
It isn’t hard to find an expression for :
First note that
Generalize by shifting:
If we write:
Then it follows that:

27. Working with the Potential
Why is potential useful?
It is a scalar quantity – easier to work with
It is useful when thinking about energy
To be dealt with later
How can we compute it?
Direct integration of charge density when possible
We can integrate the electric field
It satisfies the following differential equation:
Solving this equation is one of the main goals of the next couple chapters