Electrodynamics

Table of Contents

Equations / Theorems

Guass' Law for electrical field of closed surface

electrodynamics_de3a84c4a9062c6135a0e1c08e041e1145602b15.png

for any closed surface electrodynamics_8939ac39ed16887f3b3c033a47d4ab60d120251a.png and electrodynamics_f4e3a2bf16d265ba0831cce6738c0f83e66d1d2a.png is the surface charge-density.

Poisson's Equation

electrodynamics_ade31729f67b6965ff505a8d7905a654d66aee44.png

where we've used Gauss' Law and the fact that electrodynamics_6d56b275a706c48b63ef642237fb36d4d4b9b43e.png.

Boundary conditions

We have the standard boundary conditions in Electrostatiscs:

electrodynamics_6d190f02706e040c2114b1fea1cef3690572991b.png

electrodynamics_9d721239a3cdaeb3f5e40e483e5524e291f6873a.png

Note that since electrodynamics_6b4ba7d4f56b7261eee3449fcf619a88786c2919.png we basically have boundary conditions for electrodynamics_a3a7f43f807b9e381fc50e0fab140c0df0a03e17.png and electrodynamics_688152878982e1f3ca99e20b967e66668fedbf17.png in the "normal" ODE sense.

Properties

Uniqueness

The notes.

For any two different solutions of the Poisson's equation, the solutions only differ by a constant.

Laplace's equation

electrodynamics_d7c6a248e2e7b1e47958a3e7c3923f64a0e1d91a.png

which is a special case of the Poisson's Equation, where we have space absent of free charges .

Examples

Hollow conductor surface

The charge density within the follow region is electrodynamics_8442046eb1ddc55bd9511fb2eb202680e57216da.png, and thus we have the Laplace's equation with the boundary conditions electrodynamics_bb4cbe90c6d28a05b62cb6a4c57fc5ab2e3a241b.png where electrodynamics_aaf56bdc7d36068ce32bec925754a01a8ac80bd3.png, i.e. the potential at the boundary of the inner (non-charge) region is equal to the potential of the inner boundary of the conductor (since they're obviously the same boundary).

Method of Images

  1. "Imagine" a charge s.t. we get a homogenous PDE (equiv. Laplace's equation)
  2. Solve homogenous PDE / Laplacian using separation of variables, i.e. assume a solution of the form electrodynamics_f72cbe6a74ec288af10ac571a68840f2dd659cf3.png
  3. Then find a special solution to the non-homogenous PDE
  4. The linear combination of these two then form the general solution to the problem

There is one issue that one my bring up; is this solution unique? As it turns out, it is!

Maxwell's equations

  1. Gauss's law

    electrodynamics_3327f7b46025747ebb684b98996074ea857c9d96.png

  2. Guass's law for magnetism

    electrodynamics_b1956157ad126deffabb7190a9d733c54a8fd4c2.png

  3. Maxwell-Faraday equation

    electrodynamics_838db54e2433cdb3f23b200aa2afeba238298793.png

  4. Ampère's circuital law

    electrodynamics_9498d9a5d4d1fef603b846a0a2e8776465da10aa.png

where:

  • electrodynamics_3b8e26cf594424a1e6c422e491f40afb984a595e.png is the displacement field, which accounts for the effect of free and bound charge within materials while its sources are the free charges only
  • electrodynamics_5f6d4d49f82e9fc0155808a4e44d88841e468e7b.png is the magnetizing field
  • electrodynamics_db9d46c4e7a30615cf7a725128bbc757cb88adcb.png is the electric field
  • electrodynamics_30bdb3b857a05d1997dc59186a74b980e68d50b8.png is the magnetic field
  • electrodynamics_8848078c2c0c20a9ce19b01b79a97d7d30c6cb13.png is the free charge density