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 758-4994 (o) 407-6528 (c) ecarlson@wfu.edu http://users.wfu.edu/ecarlson/eandm/index.html 1/13

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. 10-12 When in doubt, call/email first

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 1.1-1.4 0.1 Wednesday 1.5-1.8 1.1, 1.2

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 1.1-1.4 0.1 Wednesday 1.5-1.8 1.1, 1.2

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 10-12 on March 4 Final will be from 9-12 on April 29

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 email Cell: 336-407-6528 Pandemic Plans If there is a catastrophic closing of the university, we will attempt to continue the class:

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

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:

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:

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

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:

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