Electromagnetism - Boundary conditions for Polarization field at interface

[lmcelroy]

Not actually a homework question, this is a question from a past exam paper (second year EM and optics):

Homework Statement

Use a Gaussian surface and Amperian loop to derive the electrostatic boundary conditions for a polarization field P at an interface between media 1 and 2 with relative permittivities εr1 and εr2.

Homework Equations

D = ε0E + P,

$\nabla$.P = -ρb

where D is the displacement field, E the electric field and ρb the bound volume charge density.

The Attempt at a Solution

Lack of a solution is the reason for posting?

Thanks in advance.

 

[lmcelroy]

Ok, after obtaining a copy of Feynman's lectures and referring to Volume 2, Chapter 33-3, I think I have found a solution.

Treating the boundary as a separate region (media 3 with a relative permittivity that begins at εr1 and changes continuously to εr2) then since the P field is different in each region, in region 3 (the boundary) there is a $\delta$Px/$\delta$x where Px is the P field in the x direction. So;

d/dx (Dx) = d/dx ( (ε0Ex) + (P)) ... (partial derivatives with respect to x [can't use latex very well]

D doesn't change in materials so:

-d/dx (Exε0) = d/dx (Px)

Integrating each side with respect to x over region 3 and letting P2 be the polarisation in εr2 region and P1.. :

Px2-Px1 = -ε0(Ex2 - Ex1)


Also there is no B divergence from Maxwell so:
B1 = B2 (for all directions)

Also from Maxwell:

curl(E) = -dB/dt (again partials)

Which gives the following set:

dEz/dy -dEy/dz = -dBx/dt
dEz/dx - dEx/dz = -dBy/dt
dEy/dx - dEx/dy = -dBz/dt

The E field only changes in the direction so only the following is considered:

dEz/dx = -dBy/dt
dEy/dx = -dBz/dt

Now if E were to change the right hand side of the equation would have to be balanced by a change of B with respect to time which does not happen so:

Ez1 = Ez2 and Ey1 = Ey2

So I have the boundary conditions of a dielectric interface. The boundary conditions on P are then just found by substituting P = E(εr-1)ε0:

(1) Px2-Px1 = -ε0(Ex2 - Ex1)
Px2 - Px1 = (-Px2/(εr2-1))+(Px1/(εr1-1))
Px1εr1 / (εr1-1) = Px2εr2 / (εr2-1)

(2) B1 = B2

(3) (P1/(εr1-1))y = (P2/(εr2-1))y

(4) (P1/(εr1-1))z = (P2/(εr2-1))z


Hoping someone can verify this for me. Fairly sure about the derivation of boundary conditions; it is essentially Feynman's derivation modified for two materials (he does it for vacuum to material).

However, not sure if the substitution of P is the right thing to do - does this form constitute an answer to the question or is it asking something else?

Again thanks in advance.