Graduate Electrodynamics

Mathematica Examples

(c) R. Johnson 1996 - All rights reserved

The following are Mathematica notebooks which illustrate points made in a first year graduate course in Electrodynamics. The course uses Jackson's Electrodynamics as a text but introduces additional material. In particular, the first quarter's coverage of conformal mappings and finite element analysis are outside of Jackson.

Quarter 1
- Green's functions
  - One dimensional Green's function: Calculating the one dimensional Green's function for Helmholz's equation subject to the boundary conditions that the function is zero at both ends of the interval. Green's function is expanded as a Fourier series and also calculated directly.
  - Green's function for a spherical cavity: The Green's function for Laplace's equation for inside a spherical cavity is used to find the potential inside the cavity if the surface is either
    1. V=+V_0 if z>0 and V= -V_0 if z<0; or
    2. V= V_0 Cos[theta]
  - See separation of variables solution for Green's function for Laplace's equation inside a square box given below under separation of variables section.
- Introduction to Mathematica: Examples use during Mathematica lab sessions.
  - Guant coefficients: Expanding the product of two spherical harmonics in terms of spherical harmonics. Calculation of the expansion coefficients directly from Mathematica's Wigner3J functions is also done. The Gaunt coefficients are used later as the classical approximatiuon for multipole radiation from atoms.
  - Vector analysis package: Example of using Mathematica's vector analysis package to compute various differential operations starting with a 1/r function.
- Conformal Mapping
  - Cylinder: A cylinder in a uniform electric field along the x-axis, rotated by Pi/4 from the x-axis, and then the cylinder is transformed into an elliptical cylinder in a unifrom electric field along the diagonal.
  - Schwarz-Christoffel transforms
    - Strip: Jackson illustration of the potential in the region bounded by y>0 and |x| < a when the sides x=+/-a are grounded and the side y=0 is held at V0 volts. Jackson solves the problem by an expansion in orthonormal polynomials and then sums the series to get the result. Here the result is obtained directly by conformal mapping. Problem is extended to solve the problem of a single wire between two grounded planes and then multiple wires between two grounded planes.
    - Square-theory: Field inside a square tube with the top and bottom held at 100 volts and the sides at 0 volts.
    - Capacitor: The field at the edge of a capacitor.
- Finite Elements
  - Mlaplace: Finite element solution to Laplace's equation for a square tube with a voltage V0 on two opposite faces and zero voltage on the other two. Various finite element techniques, including direct matrix inversion and various relaxation methods, are illustrated and compared to the complex analysis solution to the same problem.
- Separation of Variables - 2 Dimensions
  - Rectangular Coordinates
    - Square-poly: The problem of the square tube with the two opposite walls held at 100 volts and the other two walls grounded is solved by a Fourier expansion of the boundary surface potential. The result is compared with the result obtained from conformal mappings. The field lines are plotted by finding the imaginary part of the analytic function whose real part is given by the Fourier expansion.
  - Polar Coordinates
    - Split cylinder: Upper half of a cylinder is held at +V0, the bottom half at -V0. The voltage is found inside. The equivalent analytic function is found for this problem.
- Separation of Variables - 3 Dimensional
  - Cartesian
    - Cartesian_Green: Cartesian coordinate expansion of the Green's function.
  - Spherical with azimuthal symmetry
    - Eloop: The potential of a loop of charge which lies in the xy plane and centered on the origin is calculated along the z axis and then expanded in Legendre polynomials for other angles.
    - Split Sphere: Solution to the problem of a spherical cavity with the walls held at +V0 for the upper half of the cavity and -V0 for the lower half. Result through the center of sphere is compared with the result from the split cylinder given above. This same problem is solved with the Green's function technique above.
  - Spherical
    - Split Sphere II : Same problem as above except the voltage is given in term of phi. V= +V0 for 0 < phi < Pi and V=-V0 for Pi < phi < 2 Pi.
    - Example: Find the voltage inside a spherical cavity if the voltage on the surface is V=V0 Sin[theta] Cos[2 phi].
  - Cylindrical Coordinates
    - Example 1: The voltage inside a cylinder whose top is held at V0 volts and whose side and bottom are grounded.
    - Example 2: The voltage inside a cylinder whose top and bottom are grounded and whose side has a voltage pattern V(z,theta)=V0 Cos[theta].
Quarter 2
- Magnetostatics
  - Vector Potential
    - Aloop: Calculate the vector potential for a current loop by direct integration. Then expand in spherical harmonics and compare results.
  - Scalar Potential
    - Loop: Calculate the magnetic scalar potential along the z-axis for a loop of current centered on the origin and situated in the xy plane. The scalar potential is then expanded in Legendre polynomials for other angles.
    - Sphere: Find the scalar potential for a sphere of magnetic material in an external magnetic field.
    - Shield: calculate scalar potential for a cylindrical magnetic shield in an external magnetic field. Surface current, potential lines and field lines are plotted for both paramagnetic and diiamagnetic substances.
- Induced Electric Fields
  - Square.Magnet: find the electric field induced by increase the magnetic field between the poletips of a square horseshoe magnet.
- Plane Waves
  - Pulse Precursors: Numeric Fourier transform of a propogating sine wave. Result is compared with the Sommerfeld and Brillouin precursors given in Jackson.
- Relativity
  - Electric Dipole. Transforming the fields from a moving electric dipole into the lab. (Solution to Spring '95 final problem 1.) Also, obtaining frequency spectrum of impulse of electric dipole.
Quarter 3