Laplace's Equation in Cylindrical Coordinates

In summary, the homework statement is a problem that has to be solved using separation of variables and Fourier analysis. The problem has two regions, one inside the pipe and one outside the pipe, and the constants A_n,B_n have to be found by Fourier analysis.
  • #1
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Homework Statement


A long copper pipe, with it's axis on the z axis, is cut in half and the two halves are insulated. One half is held at 0V, the other at 9V. Find the potential everywhere in space.

Homework Equations



[tex]\nabla^2V=0[/tex]

The Attempt at a Solution



Alright. This is a laplace's equation problem in two dimensions, since the potential should be independent of z because the pipe is infinite. Using separation of variables( [tex]V=R\Phi[/tex]) the solution to the two dimensional Laplace's EQ in cylindrical coordinates is:

[tex]R(r)=ar^k+br^-k[/tex]

[tex]\Phi(\phi)=A\sin(k\phi) + B\cos(k\phi)[/tex]

w/ [tex]V=R(r)\Phi(\phi)[/tex]I am getting confused when I try to apply boundary conditions to this problem. Here is what I think the boundary conditions should be:

1) V should go to zero as r goes to infinity. (I don't think this is right, since the pipe is infinite, but I am not sure.)

2)V=0 for phi between 0 and pi, r=radius of pipe

3)V=9 for phi between 0 and 2pi, r=radius of pipe.

I can't figure out how to actually use these to eliminate any of the constants. Any help and hints would be greatly appreciated.
 
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  • #2
First of all you have to break the problem into two regions [itex]V_{in}[/itex] inside the pipe and [itex]V_{out}[/itex] outside the pipe. In order for [itex]V[/itex] to be finite to each region you can eleminate [itex]\alpha[/itex] or [itex]\beta[/itex] in eah region.
The potential [itex]V[/itex] must be periodical for [itex]\phi[/itex] thus you know [itex]k[/itex].
Lastly you have to use the superposition principle and Fourier analysis for the remaining constants.
 
  • #3
OK. It seems the part screwing me up is finding k. The potential is periodic in Phi, but it seems more like a step function. I guess I'm saying that I don't understand how can represent that potential as a sine or cosine term, no matter what k happens to be.

Could you elaborate on that part of the problem?
 
  • #4
Since the potential must be periodic in [itex]\phi[/itex] you must have [itex]k\in \mathbb{N}[/itex]. Thus the solution is

[tex]V_{in}(r,\phi)=\frac{A_0}{2}+\sum_{n=1}^\infty r^n\left(A_n\,\cos(n\,\phi)+B_n\,\sin(n\,\phi)\right)[/tex]

The boundrary conditions read

[tex]u(R,\phi)=\left\{\begin{matrix} 0 & 0<\phi<\pi\cr 9 & \pi<\phi<2\,\pi \end{matrix}[/tex]

The constants [itex]A_n,\,B_n[/itex] can be found by Fourier analysis. Can you do that?
 
  • #5
Yes, I am fine with Fourier analysis. The boundary condition were just confusing the heck out of me. I don't quite no why either. I guess I just did not recognize that I had to use Fourier anaylsis. Thank you! I think I got it now.
 
Last edited:

Related to Laplace's Equation in Cylindrical Coordinates

1. What is Laplace's Equation in Cylindrical Coordinates?

Laplace's Equation in Cylindrical Coordinates is a partial differential equation that describes the variation of a scalar field in three-dimensional space. It is named after French mathematician Pierre-Simon Laplace and is often used in physics and engineering to model the behavior of electric and gravitational fields.

2. What are cylindrical coordinates?

Cylindrical coordinates are a system of coordinates in three-dimensional space that specify the position of a point using a distance from the origin, an angle from a reference direction, and a height from a reference plane. They are often represented using the symbols (r, θ, z) and are useful for describing objects with cylindrical symmetry.

3. What is the general form of Laplace's Equation in Cylindrical Coordinates?

The general form of Laplace's Equation in Cylindrical Coordinates is ∂²u/∂r² + (1/r)∂u/∂r + (1/r²)∂²u/∂θ² + ∂²u/∂z² = 0, where u is the scalar field and r, θ, and z are the cylindrical coordinates.

4. What are some applications of Laplace's Equation in Cylindrical Coordinates?

Laplace's Equation in Cylindrical Coordinates has various applications in physics and engineering. It is commonly used to model the electric potential and gravitational potential around a cylindrical object. It is also useful in fluid dynamics, heat transfer, and acoustics.

5. What are the boundary conditions for solving Laplace's Equation in Cylindrical Coordinates?

The boundary conditions for solving Laplace's Equation in Cylindrical Coordinates depend on the specific problem being solved. They typically involve specifying the values of the scalar field at certain points, the derivative of the scalar field at certain points, or a combination of both. These boundary conditions are essential for finding a unique solution to the equation.

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