Two capacitors of capacitances 3 F and 6 F are connected in series across a 90 ~V DC supply. The initial energy stored in the combination is U_i . The capacitors are then disconnected from the supply and from each other, and reconnected in parallel such that plates of the same polarity are joined together. The final energy stored in this new combination is U_f . If the ratio U_i U_f = 9 x , the value of x is _____.

Step-by-step solution

In the initial series combination, the equivalent capacitance is: C_s = 3 6 3 + 6 = 2 F The initial energy stored is: U_i = 1 2 C_s V^2 = 1 2 2 (90)^2 = 8100 J In a series circuit, each capacitor acquires the same charge Q given by: Q = C_s V = 2 F 90 ~V = 180 C When disconnected, each capacitor holds a charge of 180 C . When reconnected in parallel with plates of the same polarity joined, the total charge is conserved: Q_ tot = Q₁ + Q₂ = 180 + 180 = 360 C The new equivalent capacitance in parallel is: C_p = 3 + 6