1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Nov 15, 2011 #1
    1. The problem statement, all variables and given/known data

    1. The problem statement, all variables and given/known data
    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. Relevant 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
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Tough Concentric Spheres with mixed Dielectrics and a air-gap Problem!
  1. Mixing Gases (Replies: 2)

  2. L.I.H Dielectric (Replies: 0)

Loading...