Homework Help: Capacitor filled with two conducting materials

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1. Dec 29, 2017

Alettix

1. The problem statement, all variables and given/known data
a) State the boundary conditions for the electric field strength E and electric flux density D at a planar interface separating two media with dielectric constants ε1 and ε2.

b) A parallel plate capacitor with a plate separation d is filled with two layers of different materials which have dielectric constants ε1 and ε2, conductivities σ1 and σ2 and layer thicknesses d1 and d2, respectively. The two materials completely fill the region between the plates so that d1 + d2 = d. A constant voltage V is applied between the plates. By considering the voltage drop across the capacitor and the current flow though the capacitor, find expressions for the magnitudes of the electric fields in both materials.

c) Derive an expression for the current density flowing through the capacitor.

d) Find expressions for the total surface charge density on the interface between the two materials and for the free surface charge density on the interface between the two materials.

2. Relevant equations
Ohm's law: $\vec{J} = \sigma \vec{E}$
Capactiance: $C = \frac{\epsilon \epsilon_0 A}{d}$
Electric field intensity: $D = E \epsilon \epsilon_0$

3. The attempt at a solution
a) The boundary conditions are:
• Perpendicular component of D continuous → perpendicular E discontinuous
• Parallel component of E continuous → parallel component of D discontinuous

However, in the derivation of the first condition, it is assumed that there is no free charge on the boundary so my first question is: are these equations valid in this case?

b) Using $V = E_1d_1 + E_2d_2$ and conservation of current density: $\sigma_1 E_1 = \sigma_2 E_2$ we find:

$E_1 = \frac{\sigma_2 V}{\sigma_1d_2 + \sigma_2d_1}$
$E_2 = \frac{\sigma_1 V}{\sigma_1d_2 + \sigma_2d_1}$

I also tried using current conservation with $I = \frac{V_n}{C_n\omega}$, giving expressions involving $\epsilon_n$ instead of $\sigma_n$, but I am not sure if the capactance formula is valid in this case.

c) The current density is hence: $J = \frac{\sigma_1\sigma_2 V}{\sigma_1d_2 + \sigma_2d_1}$

d) From this point on I am a bit really I think we have:
$\nabla \cdot E = \frac{\rho_{total}}{\epsilon_0}$ and $\nabla \cdot E = \frac{\rho_{free}}{\epsilon_0\epsilon}$ from Gauss's law.

The first relation is easy to apply to find the total charge, but what about the free charge, which $\epsilon$ ought to be used?

Many thanks!

UPDATE: It goes under the integral - doesn't it? Like $E_1\epsilon_0\epsilon_1 - E_2\epsilon_0\epsilon_2 = \rho_{free}$?
However, I am still interested in finding out if the E-D boundary conditions are applicable in conducting media.

Last edited: Dec 29, 2017
2. Dec 29, 2017

Staff: Mentor

How would a free charge get there?
I agree (same for the other one).
The capacitance formulas assume no current flow.

3. Jan 1, 2018

Alettix

I am not quite sure how it get's there in the first place, but I think that if there is free charge in the system it should be on the boundaries? Like consider the extreme case of a metal-air boundary? Or are the boundary conditions only valid for strictly non conducting media? What if we put free charge on a non-conducting dielectric, wouldn't it end up on the surface?

4. Jan 1, 2018

Staff: Mentor

In equilibrium free charges can only exist at surfaces to ideal insulators.

5. Jan 1, 2018

Alettix

Really? Why? I thought free charges were found on conducting surfaces.

6. Jan 1, 2018

Staff: Mentor

Surfaces - that is the point. On one side you have the conductor, on the other side you have the insulator.

7. Jan 7, 2018

Alettix

Alright, if I understand it correctly: free charge can only reside on boundaries between ideal insulators and conductors.

How then do we deal with non-ideal media, ie. materials which have a low conductivity for instance?
Are there any common surface imperfections which could lead to free charge getting stuck on dielectric-dielectric boundaries? Eg suppose we place free charge on an insulating material, where will it end up if there is no conductor connected to it?

8. Jan 7, 2018

Staff: Mentor

It can also be anywhere within the volume of insulators, something I forgot before - but we don't have this case in your problem anyway. This happens when semiconductors get irradiated, for example.
See if you get an equilibrium within the timescale you are interested in. If not, things get complicated.

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