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Сurrent through spherical capacitor

  1. Nov 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Determine the conductivity of the insulator in a spherical
    capacitor filled with weakly conductive dielectric. Specific conductivity of the dielectric is λ, the dielectric permittivity ε.

    Ansver in book is ##\Lambda = \frac{4\pi\lambda}{\epsilon} \frac{R_1R_2}{R_1-R_2}##

    3. The attempt at a solution
    My solution is to use following law's

    ##div \vec j = 0 ## (1)


    but

    ##\vec j = \lambda \vec E##(2)

    then

    ##div \vec j = \lambda div \vec E =0## (3)

    in spherical coordinates ##div \vec E =0## leads to
    ##Er^2 = const##

    then current ##I=j4\pi r^2 = 4\pi \lambda \frac{const}{r^2} r^2 = 4\pi \lambda \cdot const##.
    Now find the voltage:
    ##V = \int\limits_{R_1}^{R_2} E dr = \int\limits_{R_1}^{R_2} \frac{const}{r^2} dr = const \left(\frac{1}{R_1} - \frac{1}{R_2}\right)##

    Then, from the Ohm's law
    ##\Lambda = \frac{I}{V} =4\pi\lambda \frac{R_1R_2}{R_1-R_2}##

    My answer is differ from book, where have I missed ##\frac{1}{\epsilon}##?
     
    Last edited: Nov 8, 2015
  2. jcsd
  3. Nov 8, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Apparently "specific dielectric conductivity" (never heard of that concept) differs from the specific conductivity by this factor of ε.
     
  4. Nov 8, 2015 #3
    It must be "specific conductivity of the dielectric".
     
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