Calculating E-field Through Layers w/ Diff. Permittivities & Conductivities

[ThomasAnderson]

I'm trying to understand how the total electric field changes as it passes through layers with different electrical permittivities and conductivities (as shown in the linked figure). The rectangular prism layers are assumed to be very thin. The conductivities $\sigma$ and relative permittivities $\epsilon_r$ for the 5 layers as well as the surrounding medium are labeled. The external field $E_0$ is uniform. I'm hoping to find the profile of the total normal component of the E-field along the line A-A'.

I assumed that the profile would have a stepped pattern as shown in the right panel. For example, the electric field normal component inside layer 2 would be $E_{n,2}=E_0/\epsilon_{r,2}$, and similarly for the other layers.

However, I also recall learning that there is a surface charge accumulation that occurs at the interfaces between each layer, due to the difference in conductivities. If I understand correctly, these surface charges create a secondary electric field. I don't know if I need to find this secondary field, and I'm not sure how I would calculate it. I would appreciate some guidance on this problem!

https://i.stack.imgur.com/UHQ7L.png

 

[hutchphd]

Do things vary with time or are you interested in steady state (after system settles)?

 

[ThomasAnderson]

Do things vary with time or are you interested in steady state (after system settles)?

I was interested in the steady state

 

[hutchphd]

For any finite conductivities, the charge will migrate to the opposing surfaces and the field inside will be zero (ignoring edge effects). Not very interesting?

For all conductivities zero the Displacemant field D will be E/ε₀ and the local E will be εD =ε_rE₀ in each slab. I think Griffiths does a nice job on this general subject.