Electric potential of a charged dielectric-coated conducting cylinder

[rsheldon86]

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 ε₂ and radius b. The combined conductor/dielectric cylinder is embedded in an infinite space (Region 1) of permittivity ε₁. The core has a constant static potential of K. What is the electric potential at every point within the two regions surrounding the core?

Homework Equations

Polar Laplace equation:
$$\nabla^2V = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left( \rho\frac{\partial V}{\partial\rho}\right)+\frac{1}{\rho^2} \frac{\partial^2V}{\partial\phi^2}=0$$

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:
$$V(\rho,\phi)=P(\rho)\Phi(\phi)$$
When this is divided from Laplace's equation and the result multiplied by ρ², we get:
$$\frac{\rho}{P} \frac{d}{d\rho} \left(\rho\frac{dP}{d\rho}\right)=-\frac{1}{\Phi} \frac{d^2\Phi}{d\phi^2}$$
This equation can only be true if both sides are both equal to a nonzero constant m². This means that:
$$\frac{d}{d\rho}\left(\rho\frac{dP}{d\rho}\right)-\frac{m^2}{\rho}P=0$$
$$\frac{d^2\Phi}{d\phi^2}+m^2\Phi=0$$
It can be shown that:
$$V=\sum_{m=1}^{\infty}\left(C_{1m}\rho^{m}+\frac{C_{2m}}{\rho^{m}}\right)\left[C_3\cos(m\phi)+C_4\sin(m\phi)\right]$$,
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 C₃ and C₄ are equal to each other and can be absorbed by the constants of the P term. Therefore the general solution becomes:
$$V=C_1\rho+\frac{C_2}{\rho}$$.
As ρ goes to infinity V goes to 0, so the C₁ for Region 1 must equal 0. The final solution becomes:
$$V_1=\frac{A}{\rho}$$
$$V_2=B\rho+\frac{C}{\rho}$$
$$V_3=K$$

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?

 

[anathema]

Your derivation and solution appear to be correct. However, I would like to offer a few suggestions and clarifications:

1. In your solution, you have used the term "m" to represent the constant that arises from separation of variables. It is more common to use the term "n" for this constant, as it is usually referred to as the mode number.

2. In the final solution, you have not explicitly stated the values of the constants A, B, and C. These constants can be determined by applying boundary conditions at the interfaces between the different regions. For example, at the interface between Region 1 and Region 2, the potential must be continuous, which means that the potential at the interface (ρ = b) must be the same for both regions. This allows you to solve for the value of B in terms of A and the permittivities of the two regions.

3. Similarly, at the interface between Region 2 and Region 3, the electric field must be continuous, which means that the derivative of the potential with respect to ρ (i.e. the electric field) must be the same for both regions. This allows you to solve for the value of C in terms of A and the permittivities of the two regions.

4. It is also worth noting that the solution you have derived is for the potential in the two regions surrounding the core (Region 1 and Region 2), not within the core itself (Region 3). This is because the core is a perfect electrical conductor, which means that the potential inside it is constant and equal to the given value of K.

I hope this helps to clarify any doubts you may have had about your solution. Good luck with your further studies!