Two small metallic balls of radii `R_(1) & R_(2)` are kept in vacuum at a large distance compared to the radii. Find the ratio between the charges on the two balls at which electrostatic energy of the system is minimum.

Answer

`Q_(1)/Q_(2)=R_(1)/R_(2)`

Explanation

Electrostatic energy of sytem = Interaction Energy+ self Energy
(Let the total charge of balls be Q)
`U=0+(KQ_(1)^(2))/(2R_(1))+(K(Q-Q_(1))^(2))/(2R_(2))`
here for it's minimum value `(dU)/(dQ_(1))=0`
`=K[(2Q_(1))/(2R_(1))+(2(Q-Q_(1)))/(2R_(2))(0-1)]=0`
`rArr Q_(1)/R_(1)-Q_(2)/R_(2)=0 [ :' Q-Q_(1)=Q_(2)]rArr Q_(1)/Q_(2)=R_(1)/R_(2)`