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...
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The solution is attached both in Word format and in pdf format (the last one for the benefit of the solution library).
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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 ...
The 6 pages solution shows how to solve Lapalace equation in cylindrical coordinates and how to apply the boundary conditions.