A spherical capacitor with the radii of the plates a and b, where a < b, is filled with an isotropic heterogeneous dielectric whose permittivity depends on the distance r from the centre of the system as ε = α/r, where α is a constant. Find the capacitance of the capacitor.
Text Solution
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Sol. In accordance with the definition of the capacitance of a capacitor (C = q/U), the problem is reduced to determination of the potential difference U for a given charge q:
U = 
dr, ….. (1)
where we assume that the charge of the inner plate is q > 0. Let us find E with the help of the Gauss theorem for vector D:
4πr 2 D = q, E = 
= 
= 
.
Substituting the latter expression into (1) and integrating, we find
U = 
ln 
, C = 
.
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