Question

Two capacitors of capacitances \( 2C_0 \) and \( 6C_0 \), are first connected in series and then in parallel across the same battery. The ratio of energies stored in series combination to that in parallel is:

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{2}{15} \)
• D. \( \frac{3}{16} \)

Hint:

The energy stored in a capacitor depends on its capacitance. When capacitors are connected in series, the equivalent capacitance is smaller, while in parallel, it is larger, affecting the energy stored.

Answer 1

The Correct Option is D

Let the battery voltage be \( V \). Energy Stored in a Capacitor:
The energy stored in a capacitor is given by the formula:
\[ E = \frac{1}{2} C V^2 \] where \( C \) is the capacitance and \( V \) is the voltage across the capacitor. Energy Stored in Series Combination:
For capacitors connected in series, the equivalent capacitance \( C_{\text{eq, series}} \) is given by: \[ \frac{1}{C_{\text{eq, series}}} = \frac{1}{C_1} + \frac{1}{C_2} \] where:
\( C_1 = 2C_0 \),
\( C_2 = 6C_0 \).
Substituting the values:
\[ \frac{1}{C_{\text{eq, series}}} = \frac{1}{2C_0} + \frac{1}{6C_0} = \frac{3}{6C_0} + \frac{1}{6C_0} = \frac{4}{6C_0} \] Thus: \[ C_{\text{eq, series}} = \frac{6C_0}{4} = 1.5C_0 \] The energy stored in the series combination is:
\[ E_{\text{series}} = \frac{1}{2} C_{\text{eq, series}} V^2 = \frac{1}{2} \times 1.5C_0 \times V^2 = 0.75C_0 V^2 \] Energy Stored in Parallel Combination:
For capacitors connected in parallel, the equivalent capacitance \( C_{\text{eq, parallel}} \) is given by:
\[ C_{\text{eq, parallel}} = C_1 + C_2 \] Substituting the values: \[ C_{\text{eq, parallel}} = 2C_0 + 6C_0 = 8C_0 \] The energy stored in the parallel combination is:
\[ E_{\text{parallel}} = \frac{1}{2} C_{\text{eq, parallel}} V^2 = \frac{1}{2} \times 8C_0 \times V^2 = 4C_0 V^2 \] Ratio of Energies:
The ratio of the energy stored in the series combination to that in the parallel combination is: \[ \text{Ratio} = \frac{E_{\text{series}}}{E_{\text{parallel}}} = \frac{0.75C_0 V^2}{4C_0 V^2} = \frac{0.75}{4} = \frac{3}{16} \] Thus, the ratio of energies stored in series combination to that in parallel is \( \frac{3}{16} \).
% Correct Answer Correct Answer:} (D) \( \frac{3}{16} \)