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. What is the potential difference between the two balls ? total charge of balls is constant.

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)`