4) Three identical capacitors connected in series have net capacitance \(X\). Then they are connected in parallel. The ratio of energy stored in the series configuration to that in the parallel configuration (when both configurations are connected to the same source) is:
Solution:
Let each identical capacitor have capacitance \(C\). When connected in series, the net capacitance is: \[ C_{\text{series}} = \frac{C}{3} = X. \] When connected in parallel, the net capacitance is: \[ C_{\text{parallel}} = 3C. \] For a capacitor connected to a constant voltage source \(V\), the energy stored is given by: \[ U = \frac{1}{2} C V^2. \] Thus, the energy stored in the series configuration is: \[ U_{\text{series}} = \frac{1}{2} \left(\frac{C}{3}\right) V^2 = \frac{1}{6} C V^2, \] and in the parallel configuration: \[ U_{\text{parallel}} = \frac{1}{2} (3C) V^2 = \frac{3}{2} C V^2. \] The ratio of energy stored in series to that in parallel is: \[ \frac{U_{\text{series}}}{U_{\text{parallel}}} = \frac{\frac{1}{6} C V^2}{\frac{3}{2} C V^2} = \frac{1/6}{3/2} = \frac{1}{6} \times \frac{2}{3} = \frac{1}{9}. \] Hence, the ratio is \(1:9\) and option C is the correct answer.
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