Find the capacitance of an isolated ball-shaped conductor of radius `R_(1)` surrounded by an adjacent concentric layer of dielectric with permittivity `epsilon` and outside radius `R_(2)`.

Explanation

Let us mentally inmpart a charge `q` on the conductor, then
`varhpi_(+) - varphi_(-) = int_(R_(1))^(R_(2)) (q)/(4pi epsilon_(0) epsilon r^(2)) dr + int_(R_(2))^(oo) (q)/(4pi epsilon_(0) r^(2)) dr`
`= (q)/(4pi epsilon_(0) epsilon) [(1)/(R_(1)) - (1)/(R_(2))] + (q)/(4pi epsilon_(0) (1)/(R_(2))`
`= (q)/(4pi epsilon_(0) epsilon) [((epsilon -1))/(R_(2)) + (1)/(R_(1))]`
Hence the sought capacitance,
`C = (q)/(varphi_(+) - varphi_(-)) = (q 4pi epsilon_(0) epsilon)/(q [((epsilon -1))/(R_(2)) + (1)/(R_(1))]) = (4pi epsilon_(0) epsilon R_(1))/((epsilon - 1) (R_(1))/(R_(2)) + 1)`