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Capacitor with dielectric

  1. Dec 14, 2013 #1
    1. The problem statement, all variables and given/known data

    A parallel plate capacitor consists of two plates, each of area A, separated by a small distance d. in this gap, a dielectric of relative permittivity εr and thickness d/2 is fitted tight against one of the plates, leaving an air gap of thickness d/2 between it and the other plate. Calculate the capacitance of the capacitor


    2. Relevant equations

    Gauss's law ∫D.dS = ρ

    D = ε0(1+εr)E

    C = [itex]Q/V[/itex]


    3. The attempt at a solution

    First I said the plates have a charge density σ. By using Gauss's law in the dielectric I got D = σ, and then the second equation gives

    E = D/ε0(1+εr) = σ/ε0(1+εr)

    Then (this is where i'm worried I start going wrong) I use this to figure out the potential between the plates, and I split the integral up into two integrals, one inside the dielectric with E = σ/ε0(1+εr) and another outside the dielectric with E = σ/2ε0. Altogether this ends up giving:

    V = -([itex]\int_{0}^{d/2} \frac{\sigma dl}{2\epsilon_0 (1+\epsilon)} + \int_{d/2}^{d} \frac{\sigma dl}{2\epsilon_0}[/itex])

    and going through the integrals gives

    V = [itex]\frac{(\epsilon - 1)d\sigma}{4\epsilon_0}[/itex]

    Finally, putting Q = Aσ, I ended up with

    C = [itex]4A \epsilon_0/(\epsilon -1)d[/itex]

    Could someone tell me if I made a mistake somewhere? I'm quite bad at calculating capacitance.
    Also is it possible to do this problem by thinking of the capacitor as two capacitors connected in series? Because when I try the problem that way I end up getting a d in the numerator, so I think i've slipped up somewhere.

    Thanks!
     
  2. jcsd
  3. Dec 14, 2013 #2
    This should be D = ε0εrE


    Where are those 2's coming from?
     
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