Find the capacitance of a system of two identical metal balls of radius a if the distance between their centres is equal to `b`, with `b gt gt a`. The system is located in a uniform dielectric with permittivity `K`.

Explanation

Let `q` and `-q` be the charges on two balls. Then
`V_(1) = V_(ball) - V_(infty) = V`
`V_(2)=V_(ball) - V_(infty) = -V`
The potential difference between the balls

`V_(1)-V_(2)=2V`
`= 2 underset(a) overset(b-a) int E dr`
`=2 underset(a) overset(b-a) int ((1)/(4 pi in_(0) K)(q)/(r^(2)))dr =(2q)/(4 piin_(0) K)[(1)/(a)-(1)/(b-a)]`
`C=(q)/(V_(1)-V_(2))=(q)/([(2q)/(4 pi in_(0)K)((b-2a))/(a(b-a))])`
`=(2 pi in_(0) Ka(b-a))/((b-2a))`
For `b gt gta`, we can write `C = 2 pi in_(0) Ka`.