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Capacitors with partial dielectrics

  1. May 25, 2008 #1
    [SOLVED] Capacitors with partial dielectrics

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

    Consider two parallel-plate capacitors identical in shape, one aligned so that the plates are horizontal, and the other with the plates vertical





    The horizontal capacitor is filled halfway with a material that has dielectric constant . What fraction of the area of the vertical capacitor should be filled (as shown in the figure) with the same dielectric so that the two capacitors have equal capacitance? Express your answer in terms of K

    2. Relevant equations

    e = epsilon_0

    Ch (horizontal capacitance) = 2KAe/d(K+1)

    Cv (vertical capacitance) = Ae(1-f+Kf)/d

    3. The attempt at a solution

    I've already found the capacitance of both capacitors, however it's asking for a fraction of f by itself, I've tried a few combinations, a ratio, an equivalence and I can't seem to cancel out enough variable to get f by itself in one side such that I have an answer in terms of K, if anybody could point me in the right direction, I would greatly appreciate it.
  2. jcsd
  3. May 26, 2008 #2


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    Homework Helper

    Hi clope023,

    They want Ch=Cv, so you can set the right hand sides equal to each other:

    2KAe/d(K+1) = Ae(1-f+Kf)/d

    and solve for f. When you solve for f, all of the other unknowns will cancel except for k.
  4. May 27, 2008 #3
    hey thanks! got the answer, now I actually have another dielectric problem, I've more or less solved it utlizing the meaning of C, but they want the answer in terms of C as a function of f, let me show you:


    I know the total Capacitance = Cdieletric and Cempty

    Cdielectric = KeAf/d

    Cempty = eA(1-f)/d

    C(f) = KeAf/d + eA(1-f)/d

    my original idea was to write KC_0 + C_0(1-f), would this be the correct way of writing this equation as a function of f?
  5. May 27, 2008 #4


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    Homework Helper

    I'm not certain what you are looking for here. You are missing an f in your last expression (needs to be

    KC_0 f + C_0(1-f)

    Did they just want an expression for the total capacitance with f in it? Or something else?

    If you still have questions, could you post the entire problem?
  6. May 27, 2008 #5
    they wanted Capacitance as a function of f, here's the question:

    Consider a parallel-plate capacitor that is partially filled with a dielectric of dielectric constant K. The dielectric has the same same height as the separation of the plates of the capacitor but fills a fraction f of the area of the capacitor. The capacitance of the capacitor when the dielectric is completely removed is C_0, What is the capacitance C(f) of this capacitor as a function of f?

    so as I said it was the Cdielectric + Cempty, so I think what you put will be correct, I'll check it.
  7. May 27, 2008 #6
    it was correct, thanks alphysicist!
  8. Sep 27, 2008 #7
    Re: [SOLVED] Capacitors with partial dielectrics

    ......I need help.....
    My algebra seems to have left me; I can't get the right answer whilst trying to salve for f....
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