Two capacitors of equal capacitance $(C_{1} = C_{2})$ are as shown in figure. Initially, while the switch is open (as shown) one of the capacitors is uncharged and the other carries charge $Q_{0} .$ The energy stored in the charged capacitor is $U_{0} .$ Sometime after the switch is closed, the capacitors $C_{1}$ and $C_{2}$ carry charges $Q_{1}$ and $Q_{2}$ respectively; the energies stored in the capacitors are $U_{1}$ and $U_{2}$ respectively. Which of the following expression is correct ?

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$Q_{0} = \frac{1}{2} (\frac{U_{1}}{Q_{1}} + \frac{U_{2}}{Q_{2}})$

$Q_{1} = Q_{2}$

$U_{0} = U_{1} + U_{2}$

$Q_{0} = \frac{1}{2} (Q_{1} + Q_{2})$

answer is A, D.

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Detailed Solution

Because $C_{1} = C_{2}$ and $C_{1}, C_{2}$ are in parallel.

$∴ Q_{0} = Q_{1} = Q_{2}, ∴ Q_{0} = \frac{1}{2} (Q_{1} + Q_{2})$

Also $U_{0} = U_{1} = U_{2}, ∴ U_{0} \neq \frac{1}{2} (U_{1} + U_{2})$

$Q_{0} = \frac{1}{2} (\frac{U_{1}}{Q_{1}} + \frac{U_{2}}{Q_{2}})$ is dimensionally incorrect

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