Find the ratio of potential difference that must be applied across the parallel and series combination of two capacitors `C_(1) and C_(2)` with their capacitance in the ratio 1:3 so that energy stored in the two cases becomes the same.

Answer

`sqrt(3) : 4`

Explanation

` :3`
`U_(P) = U_(s)`
`(1)/(2) C_(P) V_(P)^(2) = (1)/(2) C_(S) V_(S)^(2)`
`(V_(P)^(2))/(V_(S)^(2)) = (C_(S))/(C_(P)) = (C_(1) C_(2)//(C_(1) + C_(2)))/(C_(1)+C_(2))`
`= (C_(1) C_(2))/((C_(1) + C_(2))^(2)) = ((C_(1))/(C_(2)))/(((C_(1))/(C_(2)) + 1)^(2)) = ((1)/(3))/(((1)/(3) + 1)^(2)) = (3)/(16)`
`(V_(P))/(V_(S)) = (sqrt(3))/(4)`