Find the capacitance of a spherical capacitor whose electrodes have radii `R_(1)` and `R_(2) gt R_(1)` and which is filled with istropic dielectric whose permittivity varies as `espilon = a//r`, where `a` is a constant , and `r` is the distance from the centre of the capacitor.

Explanation

Let, us mentally impart a charge `q` to the conductor. Now potential difference between the plates,
`varphi_(+) - varphi_(-) = int_(R_(1))^(R_(2)) vec(E). vec(dr)`
`= int_(R_(1))^(R_(2)) (q)/(4pi epsilon_(0) a//r) (1)/(r^(2)) dr = (q)/(4pi epsilon_(0) a) In R_(2)//R_(1)`
Hence, the sought capaictance,
`C = (q)/(varphi_(+) - varphi_(-)) = (q 4pi epsilon_(0) a)/(q In R_(2)//R_(1)) = (4pi epsilon_(0) a)/(In R_(2)//R_(1))`