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Electromagnetism - Boundary conditions for Polarization field at interface

  1. Oct 30, 2011 #1
    Not actually a homework question, this is a question from a past exam paper (second year EM and optics):

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

    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.


    2. Relevant equations


    D = ε0E + P,

    [itex]\nabla[/itex].P = -ρb

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

    3. The attempt at a solution

    Lack of a solution is the reason for posting?

    Thanks in advance.
     
  2. jcsd
  3. Oct 31, 2011 #2
    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 [itex]\delta[/itex]Px/[itex]\delta[/itex]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.
     
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