Question

When two capacitors are connected in parallel and connected to a battery, the total energy stored is 5 times greater than when they are in series and connected to the same battery. What is the ratio of the two capacitances?

Answer 1

There can be two different ratios, C1/C2 or C2/C1 , C1 and C2 being individual capacitances

When in parallel equavalent capacitance Cp = C1 + C2
and when in series , Cs = C1*C2 / (C1 + C2)

So, energy stored in parallel is Ep = 1/2 * Cp * V^2
= 1/2 * (C1 + C2) * V^2

And  energy stored in series is Es = 1/2 * Cs * V^2
= 1/2 * (C1*C2 / (C1 + C2)) * V^2

Now, Ep = 5*Es

So, 1/2 * (C1 + C2) * V^2 = 5 * 1/2 * (C1*C2 / (C1 + C2)) * V^2

(C1 + C2) = 5*C1*C2 / (C1 + C2)  
(C1 + C2)^2 = 5C1C2
C1^2 + 2C1C2 + C2^2 = 5C1C2
C1^2 - 3C1C2 + C2^2 = 0

Now, dividing by C2^2; it becomes

C1^2 / C2^2 - 3C1 / C2 + 1 = 0
x^2 - 3*x + 1 = 0 { where x = C1 / C2 = ratio of the two capacitances }

So, ratio of the two capacitances is x= 2.618 or 0.381