Is Solving Laplace's Equation for a Dielectric Cylinder Straightforward?

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Homework Help Overview

The discussion revolves around solving Laplace's equation for an infinitely long hollow dielectric cylinder with a specified electric potential on its surface. The subject area includes electrostatics and partial differential equations, particularly in cylindrical coordinates.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the application of boundary conditions and the need for a series of solutions to Laplace's equation. There are questions regarding the appropriateness of the methods used, including concerns about the complexity of the resulting equations and the interpretation of terms like "simplistic."

Discussion Status

The discussion is ongoing, with participants providing insights into the nature of Laplace's equation and the importance of boundary conditions. There is a recognition of the need for clarity in the original poster's attempts, as well as a call for more detailed information to facilitate further assistance.

Contextual Notes

Participants note the potential confusion arising from terminology and the implications of the equations derived. There is an emphasis on the need for clear communication of the work done to better guide the discussion.

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

Sam2000009 said:
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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