Monday, October 03, 2005

Using the Green's functions method.

Recently my graduate Electricity and Magnetism course covered Green's functions for solving electrostatics problems. It's an unusual geometric/mathematical approach to solving problems for which physicists would normally go straight to the hard core tools which may make some problems unnecessarily ugly.

Green's functions are not a set of functions, in the way sine, cosine, polynomial, exponential, and Bessel functions are, but rather a method for finding a function which is useful for solving the problem.

The real meat of the method is to take any convenient, geometrically identical problem, and mix it with your real problem. Generally in these problems, you will have the potential at certain positions given to you, and you will want to find the potential at some other arbitrary point.

Solving the Poisson equation directly may be difficult, but Green's reciprocation theorem comes to the rescue. You can take *any* other geometrically identical problem, find whatever you can about the charge and potential distributions anywhere in the analogy problem, and mix it with what you know about the original problem to get the result.
In some ways, this may appear to be a 'free lunch', but really, the lunch was free the whole time, solving the problem in a different way is just paying for your 'free lunch'.

I'm going to post a link to a pdf up here for my review of Green functions eventually.

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