Question
Three identical capacitors, $P,Q$ and $S$, each of the capacitance $C$, are connected to a battery of voltage $V$, as shown in the figure. If the energy stored in the capacitor $P$ and total energy stored in the system are $U_{P}$ and $U_{T}$, respectively, then the ratio $\frac{U_{P}}{U_{T}}$ is:
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Solution
Sol. [Key Concept: Equivalent capacitance of a capacitor network]
$C_{P} = C_{Q} = C_{S} = C$
$$\begin{matrix} & U_{P} = \frac{1}{2}C\left( \frac{V}{2} \right)^{2} = \frac{1}{2}\frac{CV^{2}}{4} = \frac{CV^{2}}{8} \\ & U_{T} = U_{P} + U_{Q} + U_{S} = \frac{1}{2}C\left( \frac{V}{2} \right)^{2} + \frac{1}{2}C\left( \frac{V}{2} \right)^{2} + \frac{1}{2}CV^{2} \\ & \ = \frac{CV^{2}}{8} + \frac{CV^{2}}{8} + \frac{1}{2}CV^{2} = \frac{6}{8}CV^{2} = \frac{3}{4}CV^{2} \\ & \frac{U_{P}}{U_{T}} = \frac{CV^{2}}{\frac{8 \times 3CV^{2}}{4}} = \frac{1}{6} \end{matrix}$$
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