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Laplace equation

  • #1
Sam2000009
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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
 

Answers and Replies

  • #2
jambaugh
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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.
 
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  • #3
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
 
  • #4
vela
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I did that but it seems too simplistic.
You mean "too simple." Simplistic means "oversimplified," and it doesn't really make sense to say something is "too simplistic" because there's no right level of oversimplification. If a situation were simplified the right amount, it wouldn't be oversimplified, would it?

I did all that and a got a messy non-linear second-order partial differential equation for the r function (radius) which I'm pretty sure is not right.
Telling us you tried something and got the wrong answer isn't very helpful. We need to see what you did to be able to give advice. Please post your work if you want help.
 
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