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sergiokapone
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Homework Statement
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}##
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}##?
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