# Laplace equation

Sam2000009
OP warned about not using the homework template
Consider an inﬁnitely long hollow dielectric cylinder of radius a with the electricpotential V = V0 cos φ on the surface of the cylinder where φ is an angle measured around the axis of the cylinder. Solve Laplace’s equation to ﬁnd the electric potential everywhere in space.

Do you just plug V into (del)^2 u where u=v?

I did that but it seems too simplistic

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jambaugh
Gold Member
The surface of the cylinder is a boundary condition. Typically you will want to find the series of solutions to Laplace's equation in the appropriate coordinate system to the problem at hand. (Here I would guess cylindrical coordinates). There will be an infinite series of undetermined coefficients which must be chosen to match the boundary conditions.

Since Laplace's equation is a linear equation and here it is homogeneous (away from the boundary) so any linear combination of solutions is again a solution. The trick is finding those and then finding the right linear combination to match the boundary conditions.

Some details. You should be able, with a quick search, to find Laplace's equation in various coordinate systems. You then assume separability and solve.
In cylindrical coordinates you have... well just see the Wikipedia and/or Wolfram MathWorld pages on spherical harmonics and cylindrical harmonics.

• nrqed
Sam2000009
I did all that and a got a messy non linear second order partial differential equation for the r function (radius) which Im pretty sure is not right

vela
Staff Emeritus
Homework Helper
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