Find the ratio of the potential differences that must be applied across the parallel and series combination of two capacitors `C_1 and C_2` with their capacitances in the ratio 1 : 2 so that the energy stored in the two cases becomes the same.

Explanation

Let `C_1 = C`, then as per question `C_2 = 2C_1 =2C`
In series combination equivalent capacitance `C_s = (C_1 C_2)/(C_1 +C_2) = (Cxx 2C)/(C + 2C) = (2C)/3 ` and
in parallel differences applied across the series and parallel combinations be `V_s and V_p` respectively
such that the energy stored in the two cases are same i.e., `u_s = u_p`. Since `u = 1/2 CV^2`.
`:. 1/ 2 C_s V_s^2 = 1/2 C_pV_p^2`
`rArr V_s/V_p = sqrt(C_p/C_s) = sqrt((3C)/((2C)/3)) = 3/sqrt2 rArr V_s : V_p = 3 : sqrt2`