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Dielectric Capacitor

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data
    A parallel plate cap of cross sectional area A and thickness d is filled with a dielectric material whose relative permittivity varies from [tex]\epsilon_r = 1 to \epsilon_r = 10[/tex]
    Find the capacitance.


    2. Relevant equations



    3. The attempt at a solution

    C = Q/V

    E = [tex]\sigma / e0 \epsilon_r[/tex]

    I am thinking that since V = [tex]- \int{}{} E dot dl[/tex] I need to integrate, I can't quite figure out the relationship between this though.

    I kind of would like to just evaluate the from 10 to 1 and get C = 9 A e0 / d but that isn't quite it.
    C = 9A e0 / d is close to the answer, but somehow a ln sneaks in there too?
     
  2. jcsd
  3. Mar 2, 2008 #2

    olgranpappy

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    how does the dielectric vary from 1 to 10? linearly?
     
  4. Mar 2, 2008 #3

    olgranpappy

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    no
    [tex]
    V=\int \vec E \cdot d\vec \ell=\int_{x=0}^{x=\ell}\frac{D}{\epsilon(x)}dx=D\int_{x=0}^{x=\ell}\frac{1}{\epsilon(x)}dx
    [/tex]
    and if eps varies linearly one does end up with a log term in the capacitance. (here D=\epsilon E is the electric displacemnt field)

    I kind of would like to just evaluate the from 10 to 1 and get C = 9 A e0 / d but that isn't quite it.
    C = 9A e0 / d is close to the answer, but somehow a ln sneaks in there too?[/QUOTE]
     
  5. Mar 2, 2008 #4
    Yes, sorry it's linearly.


    Cool thanks, I'll give it a shot now.
     
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