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Electromagnetism - Dielectric and Various Cavities

  1. May 1, 2013 #1
    1. The problem statement, all variables and given/known data
    A large block of dielectric contains small cavities of various shapes whose sizes are much smaller than the block. The dielectric has uniform polarization P except those cavities which are assumed not to disturb the polarization of dielectric.
    (a) Show that, inside a needle-like cavity parallel to P, the electric field E(inside the cavity) is the same as in the dielectric.
    (b) Show that, inside a thin crack perpendicular to P, the electric field E(inside the cavity) is εr times larger than in the dielectric.
    (c) Show that, at the center of a small spherical cavity, E=P/3*μ0.

    Attempt at Solution:

    a) q = σbA = P*A
    Working this out E = [itex]\frac{PA}{∏εl^2}[/itex] - [itex]\frac{-PA}{∏εl^2}[/itex]

    l is the length of the slit

    E = [itex]\frac{2PA}{∏εl^2}[/itex] Z

    However as A is very small in that direction, E needle is close to zero. So E0 is the E-field in the cavity.

    b) This is where I'm stuck.
    I assumed that the slit acted like a plate capacitor and attempted to go from there with σ+ on the top and σ- on the bottom. However, I get the E field of the perpendicular cavity is E0 + P/ε, which is not a multiple of εr


    2. Relevant equations

    E = - ∇V

    E = [itex]\frac{V}{d}[/itex], where d should be considered extremely small in part (b)
     
    Last edited: May 1, 2013
  2. jcsd
  3. May 4, 2013 #2

    rude man

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    Homework Helper
    Gold Member

    For a) and b) I would use continuity rules of E and D fields parallel and perpendicular to an air-dielectric interface. What are they? You might want to be in a position to defend them if called upon ...

    Hints:
    a) what would discontinuity of the E field imply that we know is incorrect?
    b) what would discontinuity of D imply that we know is incorrect? Realize there is no free charge at the air-dielectric interface. (D = epsilon*E for both air and dielectric).
    c) gotta think about that one.
     
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