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## Homework Statement

We have an infinitely-long perfect electrical conductor (PEC) cylindrical core (Region 3) of radius

*a*surrounded by a dielectric layer (Region 2) of permittivity

*ε*and radius

_{2}*b*. The combined conductor/dielectric cylinder is embedded in an infinite space (Region 1) of permittivity

*ε*. The core has a constant static potential of

_{1}*K*. What is the electric potential at every point within the two regions surrounding the core?

## Homework Equations

Polar Laplace equation:

[tex]\nabla^2V = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left( \rho\frac{\partial V}{\partial\rho}\right)+\frac{1}{\rho^2} \frac{\partial^2V}{\partial\phi^2}=0[/tex]

## The Attempt at a Solution

To solve this, I first used separation of variables to define

*V*as a product of functions of variables

*ρ*and

*ϕ*:

[tex]V(\rho,\phi)=P(\rho)\Phi(\phi) [/tex]

When this is divided from Laplace's equation and the result multiplied by

*ρ*, we get:

^{2}[tex]\frac{\rho}{P} \frac{d}{d\rho} \left(\rho\frac{dP}{d\rho}\right)=-\frac{1}{\Phi} \frac{d^2\Phi}{d\phi^2}[/tex]

This equation can only be true if both sides are both equal to a nonzero constant

*m*. This means that:

^{2}[tex]\frac{d}{d\rho}\left(\rho\frac{dP}{d\rho}\right)-\frac{m^2}{\rho}P=0[/tex]

[tex]\frac{d^2\Phi}{d\phi^2}+m^2\Phi=0[/tex]

It can be shown that:

[tex]V=\sum_{m=1}^{\infty}\left(C_{1m}\rho^{m}+\frac{C_{2m}}{\rho^{m}}\right)\left[C_3\cos(m\phi)+C_4\sin(m\phi)\right][/tex],

however, only the

*m*= 1 term contributes to the solution. Also, because of the circular symmetry of the regions, the sine and cosine terms will add to one at any angle and the constants

*C*and

_{3}*C*are equal to each other and can be absorbed by the constants of the

_{4}*P*term. Therefore the general solution becomes:

[tex]V=C_1\rho+\frac{C_2}{\rho}[/tex].

As ρ goes to infinity V goes to 0, so the

*C*for Region 1 must equal 0. The final solution becomes:

_{1}[tex]V_1=\frac{A}{\rho}[/tex]

[tex]V_2=B\rho+\frac{C}{\rho}[/tex]

[tex]V_3=K[/tex]

I believe I started the problem off correctly, but I'm not entirely convinced this is the correct solution. Does anyone see any problems with my derivation or solution?