Question

A series combination of $10$ capacitors, each of value ' $C_1$ ' is charged by a source of potential difference ' $4\text{ V}$ '. When another parallel combination of $8$ capacitors, each of value ' $C_2$ ' is charged by a source of potential difference ' $V$ ', it has the same total energy stored in it as in the first combination. The value of ' $C_2$ ' is

• A. $\frac{C_1}{5}$
• B. $\frac{8}{5} C_1$
• C. $\frac{64}{5} C_1$
• D. $\frac{C_1}{40}$

Hint:

N/A

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Energy stored in a combination of capacitors depends on equivalent capacitance and voltage square.
Step 2: Key Formula or Approach:
Energy $U = \frac{1}{2} C_{eq} V^2$.
Series: $C_{eq} = \frac{C}{n}$. Parallel: $C_{eq} = nC$.
Step 3: Detailed Explanation:
Case 1 (Series): $C_{s} = \frac{C_1}{10}$, $V_1 = 4V$.
\[ U_1 = \frac{1}{2} \left(\frac{C_1}{10}\right) (4V)^2 = \frac{16 C_1 V^2}{20} = 0.8 C_1 V^2 \]
Case 2 (Parallel): $C_{p} = 8C_2$, $V_2 = V$.
\[ U_2 = \frac{1}{2} (8C_2) V^2 = 4 C_2 V^2 \]
Equating $U_1 = U_2 \implies 0.8 C_1 V^2 = 4 C_2 V^2 \implies C_2 = \frac{0.8}{4} C_1 = 0.2 C_1 = \frac{C_1}{5}$.
Step 4: Final Answer:
The value of $C_2$ is $C_1/5$.