# The Laplace Equation in Cylindrical Coordinates

Please see the attached file for a question taken from the "Classical Electrodynamic 3rd by Jackson".

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#### Solution Preview

Laplace equation in cylindrical coordinates is:

(1.1)

The boundary conditions are:

(1.2)

And

(1.3)

Implicitly, we require that the solution will be invariant under full rotations:

(1.4)

And inside the cylinder we require a finite (physical) solution - no singular points.

We start by setting the solution as a product of three independent univariate functions

(1.5)

Then the partial differentials become full differentials:

(1.6)

Dividing both sides by we get:

(1.7)

The left hand side of (1.7) depends on while the right hand side depends on z.

the variables are independent, hence for (1.7) to be true for any both sides must be equal the same constant:

(1.8)

We now solve the equation for

from the boundary conditions we see that

(1.9)

Case 1:

The equation becomes

(1.10)

This is a simple second order equation with constant coefficients, so its general solution is:

(1.11)

Where (A,B) are some arbitrary constant ...

#### Solution Summary

The 10-pages solution describes in detail how to find the potential inside a cylinder where both caps are grounded and the envelope is held on some potential V(phi,z). It includes full derivations of the differential equations and the derivation of the expansion coefficients.