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

[jajay504]

1. Homework Statement

1. 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.

2. Homework Equations

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

3. 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 the potentials in the different regions of your concentric spheres with mixed dielectrics. This is a complex problem, but it can be solved using some mathematical techniques and understanding of electrostatics.

Firstly, let's start with the basics. The potential difference between two points in a region with a uniform permittivity can be calculated using the equation V = E*d, where V is the potential difference, E is the electric field and d is the distance between the two points. In your case, we have different permittivities in the different regions, so we need to take that into account.

Based on the boundary conditions you have set up, we can divide the problem into three parts: the region between the inner and middle spheres (Region I), the region between the middle and outer spheres (Region II), and the region outside the outer sphere (Region III).

In Region I, the potential is a combination of the potential due to the inner sphere and the potential due to the air-gap. We can use the equation V = E*d to calculate the potential due to the inner sphere, and then use Laplace's equation to calculate the potential due to the air-gap. Once we have both potentials, we can add them together to get the total potential in Region I.

In Region II, we can use the same approach as in Region I, but we need to take into account the different permittivities of the dielectric materials in this region.

In Region III, we can use the same approach as in Region I, but the potential due to the outer sphere will be constant and equal to the potential at the surface of the outer sphere.

I hope this helps you in approaching this problem. If you need further assistance, please do not hesitate to ask. Good luck with your calculations!