Physics 33 (Electricity and Magnetism) Home Page, Spring 2002

Physics 33: Electricity and Magnetism


12/04/02 I've added the NSF REU website to the list of interesting links at the bottom.

12/03/02 I've posted the final problem set below.

11/20/02 I've posted problem set 11 below, due after break. As we agreed in lab, the final lab report will be due on Friday after break, let's say at 2 pm (the start of the usual lab time).

11/13/02 I've extended the due date on PS 10 to Wednesday 11:59 pm. Read the AM Radio lab for Wednesday.

11/13/02 I've posted problem set 10 below.

11/05/02 I've posted problem set 9 below.

10/30/02 I've posted problem set 8 below.

10/22/02 I've posted problem set 7 below.

10/11/02 I've posted problem set 6 below.

10/7/02 The deadline on problem set 5 has been extended. It's now due Wednesday, 11:59 pm.

10/1/02 I've posted problem set 5 below.

9/30/02 I added the solution to the electric dipole problem, which had been omitted from the hmwk #2 solutions.

9/28/02 I've updated the syllabus and added a list of key derivations that you should commit to memory.

9/16/02 The homework will now be due on Tuesdays at 11:59 pm.

8/29/02 The syllabus that's forming at the bottom of the page is only tentative. If we need more or less time on a subject we'll take it without remorse.


Course Information

Course Description:

Physics 33 is traditionally the second semester of the majors sequence in physics. In this course we treat electromagnetism its phenomenology both in vacuum and in matter. We start with Coulomb's Law, but quickly introduce field concepts in the treatment of electrostatics and magnetostatics. Through Faraday's Law the dynamic relationship between electric and magnetic fields enters, and the interrelations between electricity and magnetism becomes manifest in Maxwell's equations, which also provide an explanation for light as an electromagnetic wave. Throughout, vector calculus will be the mathematical language that will permit us to express the ideas in elegant geometric simplicity.

In both lab and lecture we will also consider the application of these principles in the context of basic electronics.


Times and places:


Math 12 and Physics 32 or the equivalent, or instructor's permission.

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (should be available at the Jeffrey Amherst bookstore): I've asked the library to order copies of these books. They'll be placed on reserve when they arrive.

Additional useful references (if the library doesn't have them, I'll try to get them):

On electromagnetism in general: On circuits: Lab Work: Specialty books:

Key derivations / chains of logic / results to commit to memory
From lecture

Important ideas from lab

Lecture/Lab Schedule
Week Notes Hmwk Lab
1. September 2 Electrostatics I

Sept 3: Introduction to the course
General course information. Brief overview of history and significance of electromagnetism.
[Handouts: Course information sheet, first problem set, lab manual ]

Sept 4: Electrostatics: charge, force and energy
Properties of electric charge: quantized and conserved. Coulomb's law. Brief review of conservative forces and the connection between conservative force, work and potential energy (recall: gravity). Potential energy of an electrostatic system.
[Handouts in lab: Student information sheet, diagnostic questionnaire, page on triboelectricity]

Sept 6: Electrostatic potential energy
Calculate the work to bring two point charges together using Coulomb's law. Define potential energy. With superposition, obtain work to assemble a system of charges and an associated potential energy. Write down expression for potential energy of 1D crystal.
[Handout: How van de Graaff generators work]

Sept 7: Stability of matter / Electric fields
Calculate energy of a 1D crystal. While it has lower potential energy than segregated charges, energy can be further decreased by shrinking collapsing + and - charges on each other. Stability of matter problem: why is matter made up of compact charges stable? (Earnshaw's theorem generalizes this beyond our crystal). Resolution is QM.
Define electric field via force. Auxiliary variable now, but will later be a dynamical, local object. Visualizing electric fields: arrows, field lines.
Read: Purcell, 1.1-1.8
Problems: problems
Lab 0: Questionnaires, Lab overview, and Lab 0. EI
2. September 9 Electrostatics II: Electric Fields

Sept 9: Calculating electric fields
Recap on field lines as visualization tool. E fields for continuous charge distributions. EXAMPLE: E field of electric dipole along dipole axis. EXAMPLE: Field of a uniform line of charge.

Sept 10: Calculating electric fields II (more examples)
EXAMPLE: E field of uniform ring of charge on symmetry axis. EXAMPLE: E field of uniform shell of charge -- Shell Theorem.

Sept 11: Electric flux and Gauss's law
Motivation--why Gauss's law when we've got Coulomb's law?: Recap advantage to thinking directly about fields rather than the sources (charges) which set up the fields. Given the fields, what can we deduce about the sources?
Define flux, first through a patch then as an integral over a general surface.
State and prove Gauss's law: (i) single point charge and spherical surface, (ii) single point charge and arbitrary surface.
[Handout: Supplemental notes on the derivation of the shell theorem]

Sept 13: Gauss's law
Finish proof of Gauss's law: (ii) single point charge and arbitrary surface, (iii) arbitrary charge distribution and arbitrary surface. EXAMPLE: Use Gauss to prove shell theorem. EXAMPLE: Field of a line of charge. EXAMPLE: Field of a sheet of charge. EXAMPLE: Field around two sheets of charge.
Read: Purcell, 1.9-1.15
Problems: problems

solution to dipole problem
Lab 1: Simple DC Circuits EI
3. September 16 Electrostatic Energy and Electric Potential (I)

Sept 16: Energy of electric fields (I)
Questions from class about Gauss's law. Force on a layer of charge.

Sept 17: Energy of electric fields (II)
Finish force on a layer of charge. Potential energy of two sheets of charge. Electrostatic energy per unit volume for this configuration (which generalizes to all configurations). Conceptual issue: is the potential energy to be associated with the charges or with the field?

Sept 18: Electric potential
Potential defined as the work per unit charge to move a charge from a reference point to a final point. Write as a line integral of E. E is a central force, so line integrals are path independent. Because of path independence, the scalar potential has a single value at each point in space and is a scalar field. Equivalently, line integral of E around any loop is zero, which implies E is a conservative force.
Visual representation of potential in terms of surfaces of constant potential (equipotential surfaces). Equipotential surfaces are normal to E fields. Analogous to contour maps, where potential is the height, E is vector in direction of steepest descent.

Sept 20: From potential back to E field
Going from potential to E field: differentiating to get a vector. Define gradient of a field, illustrate with simple examples. From Taylor series, show that gradient gives a vector that points in direction of steepest ascent, has magnitude given by derivative of function in that direction. Geometrically, the gradient of a function is orthogonal to constant surfaces of that function.
Relate E and the potential via the gradient. Superposition holds for potentials. Expression for potential for continuous charge distributions.
Read: Purcell 2.1-2.6
Problems: problems
Lab 2: Simple Resistor Networks Formal
4. September 23 Electric Potential (II) / Understanding Electrostatics Locally

Sept 23: Calculating the potential / Local formulations
EXAMPLE: potential of uniform ring of charge, on symmetry axis. EXAMPLE: line of charge. (Illustrates problem of divergence of potentials for infinite charge distributions.) EXAMPLE: electric dipole potential.
Can we formulate our relationships between potential, charge density, and E field locally?

Sept 24: Local formulation of Gauss's law
Derive local form of Gauss's law using a small surface. Define divergence of a vector field. Prove divergence theorem. Obtain explicit expression for divergence in cartesian coordinates using rectangular boxes.

Sept 25: Examples of divergence, and Poisson's equation
EXAMPLE: sphere of uniform charge density.
Use relation between E field and potential with Gauss's law to obtain local relation between potential and charge density: Poisson's equation. Define Laplacian of a field. Introduce Laplace's equation. Three results for functions which obey Laplace's equation (harmonic functions): Mean value theorem, Earnshaw's theorem, no equilibrium for charged particles in empty space due to electrostatic fields. Set up calculation for proof of mean value theorem.

Sept 27: Three results for harmonic functions
Prove mean value theorem: explicit calculation for point charge, then invoke superposition. Earnshaw: proof by contradiction. Prove no stable equilibrium points (although UNSTABLE equilibrium points may abound). Problem for stability of matter!
Can we formulate path independence of line integrals of a vector field locally? Yes...
Read: Remainder of chapter 2.
Problems: problems
Lab 3: Semiconductor Diodes EI
5. September 30 Electrostatics, Conductors and Capacitors

Sept 30: Path independence locally
Define circulation and comment on its geometrical meaning. Note path independence implies zero circulation for all loops. Define curl as the circulation per unit area at a point. Claim: path independence if and only if curl is everywhere zero. Obtain expression for curl in terms of cartesian coordinates by taking the appropriate tiny loops and surfaces.

Oct 1: Wrapping up path independence (and review)
Recap of how to obtain expression for curl in cartesian coordinates. Summary/recap of connections between path independence, curl, and gradients (in the form of a triangle). A vector with zero curl is expressible as a gradient of a scalar, and if a function is the gradient of a scalar it has zero curl. Stokes theorem. Use Stokes theorem to prove that zero curl everywhere implies path independence of line integrals. Review of course to date: lightning recap of main ideas and results.

Oct 2: Electrostatics on, in, and near conductors
[Started with some review questions prior to the exam.] Properties of conductors, insulators and semiconductors. Facts about conductors: (1) E=0 inside conductor. (2) E perpendicular to surface of conductor at the surface. (3) Conductor is an equipotential surface. (4) Gauss's law relates surface charge density to E field at surface. (5) Net charge resides on surface of a conductor.

Oct 4: Boundary value problems
Recap facts about electrostatics on and in conductors. Specifying boundary conditions. Boundary value problems. Claim existence of solutions. Prove uniqueness of solutions using Earnshaw's theorem. E=0 inside of a hollow conductor.

Read: Purcell, Chap. 3
Problems: problems
Lab: Exam #1 na
6. October 7 Conductors, capacitors and boundary value problems / DC Circuits

Oct 7: Solving potential problems
[Some initial questions from the class: Why does free charge accumulate on the surface of a conductor? If there are multiple cavities in a conductor, is the potential the same in each of them?] Recap of the principles of superposition and the uniqueness theorem. Using Gauss's law to solve for the charge in hollow conductors. Using the principle of superposition to obtain the potential for concentric spherical shells of charge. [Question: Does the principle of superposition hold for conductors?] Problem: point charge and conducting plane. Solution via image charge (method of images) and uniqueness theorem.

Oct 8: More method of images / capacitors
Solve problem of a point charge near a grounded sphere using method of images. Comments on solving boundary value problems numerically using relaxation method. Define capacitance. Capacitance of an isolated sphere. Parallel plate capacitor: solve for the E fields, obtain capacitance. Example: Capacitance of concentric spherical shells.

Oct 9: More about capacitors / Current in circuits
Recap on the calculating the charge distributions on surfaces of parallel plate capacitors. Capacitors in series and in parallel. Energy in capacitors. DC circuits: general discussion. Define current, current density.

Oct 11: Currents and Ohm's law
Calculate drift velocity of electrons in a typical household circuit: very small! Local formulation of conservation of charge. Ohm's law in terms of current density. Translating Ohm's law to V=IR. Resistance of a uniform wire.

Read: Purcell, Chapter 4
Problems: problems
addendum to 4.30
Lab 4: RC Transients and Sinusoidal Response Formal
7. October 14 DC Circuits

Oct 14: Fall Break!
Oct 15: Fall Break!
Oct 16: Current and resistance: microscopic and practical
Microscopic model of current: Ohm's law and expression for conductivity in terms of model parameters. Caveat: collision time seems too long in real metals. Adding resistors in series and parallel. Seval comments about conductors and E-fields. Cliffhanger: Kirchoff's laws necessary for analyzing complicated networks.

Oct 18: Kirchoff's laws and voltage sources
Kirchoff's laws and the physics behind them. Voltage sources: pumps of electric current. Charged capacitor + my hand as a simple model of a voltage source. Kirchoff's law for loops with voltage sources.

Read:Purcell 5.1-5.4, Chap. 6
Problems: (no new problems this week)
Lab 5: Faraday's Law and Inductors in Circuits EI
8. October 21 Circuits / Magnetostatics

Oct 21: Analyzing circuit loops
Loop method. Branch method. RC circuits: loop equations and solutions. Charging the capacitor. Power, both generally and for a resistor.

Oct 22: Magnetostatic fields: history and phenomenology
Capsule history of magnetism. Basic phenomenology of magnetism. Magnetic force on charged particles. Magnetic force on current-carrying wire. Current loops: force and torque in uniform magnetic fields.

Oct 23: Forces and torques due to static magnetic fields
Torque on a current loop in a uniform B field. Magnetic moment defined. Energy of a current loop in a B-field--analogous to electric dipole in an external E-field. Motion of charged particle in uniform B field.

Oct 25: Applications of magnetic fields / Sources of magnetic fields
Velocity selector. Mass spectometer. Thompson's discovery of the electron. DC motor. Biot-Savart law: statement and general features. Biot-Savart law for volume current distributions.
Read: Purcell Chap. 6
Problems: problems
Problem 2
Problem 3
Problem 4
Lab 6: LC Circuits and Resonance Formal
9. October 28 Magnetostatics

Oct 28: Applications of Biot-Savart Law
B field around an infinite wire. B field due to a curved segment of wire. B field along the symmetry axis of a current loop.

Oct 29: Magnetic dipoles and solenoids / Magnetostatics vs. electrostatics
More on magnetic dipoles and magnetic dipole moments. B field of an infinite solenoid on the symmetry axis. Parallels between our progress in magnetostatics vs. our progress in electrostatics.

Oct 30: Deriving Ampere's law
Start derivation of Ampere's law in integral form.

Nov 1: Deriving Ampere's law (cont'd) / Using Ampere's law
Complete derivation of Ampere's law in integral form. Obtain statement of Ampere's law in differential form. Application: B field of infinite wire (inside and outside). B field of a sheet of current. Gauss's law for magnetic fields, differential and integral forms.

Read: Purcell Chap. 6
Problems: problems
problem 1
problem 2
problem 6
Lab 7: RLC Circuits EI
10. November 4 Magnetostatics / Faraday's law

Nov 4: Solenoids / Potentials for Magnetic fields?
Magnetic field of ideal solenoid using Ampere's law. Maxwell's equations for magnetostatics. Can we write potentials for B fields? Pertinent observations: (1) Line integrals of B fields are path dependent, so no scalar potential. (2) Relation between conservative forces and path independence. Magnetic forces do no work!

Nov 5: Vector potential
Lack of path independence precludes a scalar potential, but a vector potential can be formulated. Make div B = 0 automatic by writing B = curl A. Plug into Ampere's law to relate A to J. Gauge invariance permits simplifying assumption: div A = 0. Resulting equation relating A and J is a Poisson's equation for each vector component. Can write integral relation between J and A, just as for rho and phi in electrostatics. Final observations.

Nov 6: Motional EMFs
Magnetostatic triangle. Recap about EMFs. Motional EMF: conducting rod moving through a uniform B field. Closed circuit moving through a uniform B field at constant v. Magnetic flux. Flux rule for motional EMF in this simple case. Start the generalization of the flux rule for arbitrary wire loops in nonuniform fields.

Nov 8: Faraday's Law
Generalize flux rule to arbitrary wire loops in nonuniform fields. Observe that there is no new physics here. Faraday's "three experiments". Universal flux rule. Force on stationary charges leads Faraday to propose: changing magnetic flux induces electric field. Faraday's law, integral and local forms. Observe: this new E field has a curl!

Read: Purcell Chap. 7
Problems: problems
Lab 8: Transistors EI
11. November 11 Faraday's Law / AC circuits

Nov 11: Exam Review and Faraday's Law
Recap of Faraday's law. Contrast Faraday electric field with electrostatic field. Compare of Faraday electric field equations with magnetostatic field equations: can use same techniques to solve problems. Lenz's law.
Demonstrate proportionality between current in one wire and magnetic flux through another. Define mutual inductance. Write (but don't yet prove) Neumann formula.
Exam review.

Nov 12: Inductance and RL circuits
Derive Neumann formula.
EXAMPLE: mutual inductance of coaxial solenoids.
EXAMPLE: self-inductance of solenoid.
RL circuits: Kirchoff's law and solution.

Nov 13: RL and LC circuits
General solution for RL circuit. Charging the inductor. Energy stored in inductors and magnetic fields. Comparison between LC oscillator and mass-on-spring system.

Nov 15: Damped RLC circuits
Analog with mechanical system with viscous drag. Solution to Kirchoff's law equation. Q factor.
Read: Purcell Chap. 7 and 8
Problems: problems
Lab: Exam #2, 2-5 pm na
12. November 18 AC Circuits / Maxwell's equations

Nov 18: Sinusoidally-driven R, L and C
General solution for RLC circuits. Overdamped and critically damped cases. Obtain expressions for the current through R, L and C when driven by a sinusoidal voltage source. Write expressions for I and V in terms of sinusoidal functions and in terms of phasors. Power (instantaneous and average).
[Handout: Magnetic Forces Doing Work? AJP 42 (1974) 205]

Nov 19: Driven RLC circuits and complex numbers
Recap relation between current and voltage for R, L and C. Current-voltage relation in series RLC circuit. Resonance. Power dissipation at and away from resonance. Width of resonance peak in terms of Q.
Complex numbers: Definition of i, complex numbers. Complex conjugate. Addition, subtraction, multiplication and division of complex numbers. Graphical representation of complex numbers.

Nov 20: Complex numbers in AC circuits
Polar representations of complex numbers. Euler relation: statement and proof. Multiplication and division in polar rep. Physical quantities as real parts of complex numbers. Trig identities using complex numbers. Driven inductor using complex numbers. Complex admittance and impedance defined.
Lab Intro:
Driven capacitor using complex numbers. Impedance of capacitor. Generalized Ohm's law. Driven RLC circuit using complex number approach. Impedances add like resistances in DC circuits.

Nov 22: Complex numbers approach to AC circuits / Completing Maxwell's equations: Displacement current
Driven RLC circuit: Connect complex number expression for current to relation derived using phasors. Another driven RLC circuit example.
Completing Maxwell's equations: recap of E&M equations. Inconsistent! Inconsistency appears when (1) taking divergence of Ampere's law or (2) applying Ampere's law to a charging capacitor circuit. Maxwell's reasonable fix: modify Ampere's law by adding extra term suggested by continuity equation. This is displacement current, and it works: changing E field induces a B field.

Read:Purcell, Chap. 8 and 9
Problems: problems
partial solutions
Lab 8: AM radio receiver Formal
November 25 Thanksgiving Break!
Read: Purcell, Chap. 9 and 10
Problems: see previous week
13. December 2 Maxwell's equations / Electric fields in matter

Dec 2: Displacement currents / Maxwell's equations
Dec 3: Solutions to Maxwell's equations: light as electromagnetism
Dec 4: Electromagnetic waves / Dielectrics and dipoles
Dec 6: Maxwell's equations in dielectrics
Read: Purcell, Chap. 10
Problems: problems
Lab 9: none! ?
14. December 9 Electric fields in matter

Dec 9:
Dec 10: (Last day)
Dec 11: Reading period
Dec 13: Reading period
Lab: Exam review or bonus lecture EI
December 17 Final exam: Dec. 17, 9-12 am, Merrill 4


I'll keep scheduling information on this site primarily. I may occasionally use Blackboard as well.

Useful Links

I'll post interesting or useful links pertinent to the course here as they I come across them. If you come across any others, please let me know.

Interesting talks in the Five-College area:

You should start attending the departmental colloquia early and often. They are intended primarily for you, to broaden your exposure to current physics in ways that the department faculty alone cannot. They'll give you an overview of what exciting work is going on in physics and who's doing it. In the beginning you won't always understand all of the talks, but you'll be surprised by how much you can understand even now. In addition, the colloquium food here is better than anywhere else I've ever been. Plus, I organize the colloquia, and it warms my heart to see you there.

Area Seminars and colloquia

Interesting and useful papers:

Interesting and useful websites: