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Laplace equation in cylindrical coordinates

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In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... <i>Please see attached</i>... Let us assume that the potential is known at z = 0, so that
V(r, theta, z)= f(r, theta)
is a given function.
Because of the symmetry in the problem, V(r, theta, -z) = V(r, theta, z), so we only have to consider the semi-infinite cylinder in the domain z>0. Since the cylinder is grounded, the potential there must be zero, meaning that V(1, theta, z) = 0...

Please see attached for full question.

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Hi there!

The solution is attached both in Word format and in pdf format (the last one for the benefit of the solution library).

Check out the following site:
http://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html

For a nice solution of the Helmholtz equation in cylindrical coordinates.

The Laplace equation:

Can be solved using separation of variables:

Substituting it back into the equation one obtains:

Multiplying by the equation becomes separated:

Since we require periodic solution for the polar function Θ we can write:

Which is the harmonic equation:

With the ...

Solution Summary

The 6 pages solution shows how to solve Lapalace equation in cylindrical coordinates and how to apply the boundary conditions.

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