# Steady state boundary conditions between metal/dielectric?

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1. Jan 14, 2018

### Dor

There are few thing I'm not sure of and be happy for clarifications.
In general: at steady state, what are the electric-field,potential, and current boundary conditions between a conductor and a dielectric medium?
more specific:
a) When dealing with a perfect conductor there exist a surface charge. How can I find this surface charge? And if I can't, how can I use $\hat {\mathbf n}\cdot(D_m-D_d)=\rho_s$ as a boundary condition if I don't know this $\rho_s$?
($D_m,D_d$ are the electric displacement at the metal and dielectric medium, respectively)
* The same question holds for current density.

b) What is the difference between perfect conductor and non-perfect conductor? Why at the former there exist a surface charge but in the case of a non-perfect conductor this surface charge is zero?

c) My understanding is that at steady state, the electric field is not zero at the metal thus the electric field boundary is $\varepsilon_m *E_m=\varepsilon_d*E_d$ but then, what is the meaning of the dielectric constant of the metal $\varepsilon_m$?

d) Is the potential at the boundary always continuous or there are cases when they are not?

e) And finally, what is the boundary conditions for the currents (tangential and normal)

2. Jan 14, 2018

### Paul Colby

Most of your questions reduce to the the tangent components of the E field are continuous even at an interface, the normal component of the D is continuous even at an interface, like are the tangent components of H and the normal B. Continuity of charge and current follow from Maxwell's equations and the above continuity requirements.

3. Jan 14, 2018

### Dor

So, in the case of steady-state, the boundary conditions are the same as in electrostatic?
My issue with the tangent component arises when looking at a one-dimensional problem. In this case, I can only "work" with the normal component.
A second issue is with the displacement (D)? If D is continuous how do I define D in the metal side?

4. Jan 14, 2018

### Paul Colby

Well, in the limit anything is truly static. Any real device being modeled as static came into being and will be discarded in the fullness of time. Maxwell's equations remain valid for all cases. For example if I glue plexiglass to granite and apply a 100 volts across the stack, D will be continuous since there are no free charges to accumulate on the interface. On the other hand I could imagine painting on some coulombs of charge on the plexiglass prior to gluing. Even in this case D would still be continuous on the boundary (there would be a zero crossing in D).

As the name implies, a perfect conductor is perfect and therefore an idealization. Normal metals have a finite resistivity and so D is continuous and $\epsilon$ for metals while big is not infinite.