Tough Concentric Spheres with mixed Dielectrics and a air-gap Problem

[jajay504]

Homework Statement

I have concentric spheres with mixed dielectrics. There is an air-gap between the spheres which consist of a permittivity ε0. The radius' are a, b and c and the permittivities of the dielectric portions are ε1 and ε2. An image is attached! What are the potentials in the 4 regions of the image.

Homework Equations

Laplace's equation in spherical coordinates 1/r^2 ∂/∂r (r^2 ∂V/∂r) = 0

The Attempt at a Solution

So, I know from Laplace's equation that r^2 (∂V/∂r) = 0
V = A∫dr/r^2 + B = -A/r + B

V(I) (r,θ)= Ʃ A_l*r^l * P_l*(cosθ), where Ʃ goes l=0 to ∞
V(II) (r,θ)=Ʃ (A_l*r^l + B_l/ r^(l+1)) * P_l*(cosθ)
V(III) (r,θ)=Ʃ B_l(1/ r^(l+1) - r^l/(r^(2l+1)) * P_l*(cosθ)
V(IV) (r,θ)= Ʃ ( B_l/ r^(l+1)) * P_l*(cosθ) - Eo*rcosθ

Set up boundary conditions:
(I) ε1 ∂V(I)/∂r (a,θ)= ε0 ∂V(II)/∂r (a,θ)
(II) V(II) (b,θ)= V(III) (b,θ)
(III) ε2 ∂V(III)/∂r (c,θ)= ε0 ∂V(IV)/∂r (c,θ)

Went through the process of applying the boundary conditons.
Got A1 (I)= -Eo
B1 (I)= (Eo R^3 (ε1 - εo))/(ε1 + 2εo)

This problem got extremely tough after this! I am completely lost now!
Is there a simpler way of approaching a problem like this

 

[mark726]

?



Thank you for posting your question about concentric spheres with mixed dielectrics. This is a complex problem that requires a thorough understanding of Laplace's equation and boundary conditions. Your approach is correct so far, and I would recommend continuing with the same method to solve for the potentials in the remaining regions.

One suggestion I have is to try simplifying the problem by considering only one region at a time. For example, for region I, you can assume that the potentials in regions II, III, and IV are zero, and use the boundary condition at the interface between regions I and II to solve for the potential in region I. Then, you can repeat this process for the other regions.

Another approach is to use the method of images, where you introduce a fictitious charge at the center of the spheres to create a simpler problem with only one dielectric. You can then solve for the potentials using this simplified problem and adjust for the presence of the air gap afterwards.

I hope these suggestions are helpful in approaching this problem. Keep in mind that it may be challenging, but with persistence and a clear understanding of the underlying concepts, you will be able to solve it. Best of luck!