Calculate the interaction energy of two balls whose charges `q_(1)` and `q_(2)` are spherically. The distanace between teh centres of the balls is equal to `l`.

Explanation

Consider the interaction energy of one of the balls (say 1) and thin spherical shell of the other. The interaction energy can be written as `int d varphi q`
`= int (q_(1))/(4pi epsilon_(0) R) rho_(2) (r) 2pi r^(2) sin theta d theta dr = int_(0)^(pi) (rho_(2) (r) q_(1) r^(2) sin theta d theta dr)/(2epsilon_(0) (l^(2) + r^(2) + 2lr cos theta)^(1//2))`
`= (q_(1) r)/(2epsilon_(0) l) dr int_(1 - r)^(1 + r) dx rho_(2) (r)`
`= (q_(1) r)/(2epsilon_(0) l) dr . 2r rho_(2) (r) 2`
`= (q_(1))/(4pi epsilon_(0) l) 4pi r^(2) dr rho_(2) (r)`
Hence finally intergating
`U_(int) = (q_(1) q_(2))/(4pi epsilon_(0) l)` where, `q_(2) = int_(0)^(oo) 4pi r^(2) rho_(2) (r) dr`