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The Seperation of Variables

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Solve Laplace's equation by separation of variables in cylindrical coordinates. Assume symmetry about the z axis and no z dependence.
Check the result by comparing to an infinite line charge.

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Solution Summary

This solution solves Laplace's equation for the separation of variables.

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Laplace's equation in polar coordinates is:
(1.1)
We assume that the potential can be written as a product of independent single-variable functions, namely:
(1.2)
Then, substituting this back in (1.1) the partial derivatives become full derivatives and we obtain:

Dividing both sides by we separate the equation:
(1.3)
Since the functions are completely independent, the only way equation (1.3) can be true for all values of r and  is if both sides equal the same constant k.

So from the angular equation we get:
(1.4)

And the radial equation becomes:
(1.5)

Case 1:
The ...

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