PHYS 685

Classical Electrodynamics I

The class voted on Apil 19th that hw12 will not be collected or graded. I have posted the solutions below. Please pick up the homework from my mailbox in SPACS, if you have not done so.

Final Exams:

The final exam is scheduled on May 8 from 4:30pm to 7:10pm in Robinson Hall A 249. You are allowed to bring formula sheets (up to 4 pages), but no books or notes.

Time and Location

Wednesday 4:30-7:10pm, Jan 20 – May 13, 2015, Robinson Hall A 249

Instructor & Office hours

Erhai Zhao
Contact Email: ezhao2 [at]

Office hours: 2-3pm, Wednesday, Planetary Hall, Room 207; or by appointment.

Course Goals

This course treats the static properties of electromagnetic fields systematically and in depth. It covers electrostatics, boundary values problems, multipoles, magnetostatics, macroscopic media. It then discusses dynamic electromagnetic fields and the Maxwell equations.


Classical Electrodynamics, 3rd edition, J. D. Jackson, Wiley

We will cover Chapter 1 to Chapter 6. The students enrolled in this course are assumed to be familiar with an undergraduate E&M text on the level of Introduction to Electrodynamics, 3rd ed, D. J. Griffiths, Prentice Hall. There are many physically insightful, and mathematically pedagogical books that complement Jackson such as Modern Electrodynamics by Andrew Zangwill.


Homework (40%) + Midterm (30%) + Final (30%).

Policy on homework:
1. Discussing problems with each another is encouraged. Copying solutions (including from solution manuals) can be easily spotted. Any violation of the honor code will result in zero credit.
2. You will receive partial credits provided that the solution strategy is clearly presented. Narratives and reasoning are encouraged in the answer as long as they are succinct and on-target.
3. Homework will be posted online at this website. It is due at the beginning of each class (4:30pm) on the due date.

Lecture Plan

  1. Math review

    • Scalar and vectors fields; Derivatives: grad, div, and curl.
    • Line, surface, and volume integrals; Divergence and Stokes theorem.
    • Vector identities; the Laplacian.
    • Curvilinear coordinate system.
    • Application: Maxwell equations and boundary conditions.

  2. Gauss’ Law, Poisson and Laplace equations

    • Electric field from point charge and charge distribution; Dirac delta function.
    • Derivation of Gauss’s Law from Coulomb’s law.
    • Scalar potential; Poisson and Laplace equations.
    • Surface charge; Boundary conditions for electric field and potential.
    • Dirichlet and Neumann boundary conditions; Uniqueness of the solution.
    • Energy of charge distribution; Energy density of electric fields.

  3. Potential of point charge by method of images

    • Electrostatic properties conductors; Force on the surface of conductor.
    • Uniqueness of solution in the presence of conductors.
    • Point charge above a grounded conducting plane; Method of images.
    • Point charge near a conducting sphere with fixed charge or potential.

  4. Solving boundary value problems by Green functions

    • The definition and properties of Green function.
    • Green function in free space; Plane wave representation of delta function.
    • Dirichlet Green function and formal solution of Poisson eqn with Dirichlet BC.
    • Example: Solving Lapalace eqn with G_D for conducting plane and sphere.

  5. Solving boundary value problems by separation of variables [3 lectures]

    • Orthogonal function expansion; Fourier series and Fourier integrals.
    • Solving Laplace equation in rectangular coordinates;
    • Separation of variables in spherical coordinates; Problems with azimuthal symmetry, Legendre polynomials (LP) and Legendre series.
    • Spherical harmonics (SH). Expansion of free space GF in terms of LP and SH.
    • Separation of variables in cylindrical coordinates; Bessel functions, Fourier-Bessel series.

  6. Charge distributions and dielectric media [2 lectures]

    • Field from localized charge distribution; Multipole expansion using SH.
    • Potentials and fields of electrical dipoles and quadruples.
    • Energy of charge distribution in external field.
    • Electric polarization and displacement; Susceptibility and dielectric constant.
    • Boundary conditions; Solving boundary value problems with dielectrics by method of images and separation of variables.
    • Energy in dielectric media.

  7. Magnetostatics [2 lectures]

    • Magnetic field from current, Boit-Savart Law; Force on current.
    • Vector potential; Differential equations for B field, Ampere’s Law.
    • Fields of localized current distribution; Magnetic dipole moment, its A and B; Force and torque on magnetic dipole.
    • Magnetization M and H field; Para, dia, and ferromagnetic materials.
    • Boundary conditions for B and H.
    • Boundary value problems in magnetostatics: vector potential and magnetic scalar potential.
    • Energy in magnetic field

  8. Time-dependent electromagnetic field [2 lectures]

    • Faraday’s Law of Induction.
    • Displacement current; Maxwell equations.
    • Scalar and vector potential for time-dependent fields. Gauge choices and gauge transformations.
    • Electromagnetic wave equations. Retarded Green function. Retarded fields.
    • Energy and momentum of fields; Poynting vector and Maxwell stress tensor.
    • Self and mutual inductance. Energy consideration of capacitance and inductance.
    • Classical theory of passive circuits.