# Laplace equation in cylindrical coordinates

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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#### Solution Preview

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.