Question

The figure shows two identical parallel plate capacitors A and B of capacitances C connected to a battery. The key K is initially closed. The switch is now opened and the free spaces between the plates of the capacitors are filled with a dielectric constant 3. Then which of the following statement(s) is/are true?

• A. When the switch is closed, the total energy stored in the two capacitors is CV².
• B. When the switch is opened, no charge is stored in capacitor B.
• C. When the switch is opened, energy stored in capacitor B is \(\frac{3}{2}\) CV².
• D. When the switch is opened, the total energy stored in capacitors is \(\frac{5}{3}\)CV².

Answer 1

The Correct Option is A, D

Correct options: (A) and (D) 

Explanation:  
Step 1: When the switch is closed
- Capacitors A and B are connected in parallel to a battery with voltage $V$.
- Each capacitor has capacitance $C$.
- Total energy stored: 
\(U = \frac{1}{2}CV^2 + \frac{1}{2}CV^2 = CV^2\)
⇒ Option (A) is correct. 

Step 2: When the switch is opened
- Now, both capacitors are disconnected from the battery. - A dielectric of constant $K = 3$ is inserted into both. 

Capacitor A (isolated):
- Initial charge: \(Q = CV\) (remains unchanged)
- New capacitance: \(C' = 3C\)
- New voltage: \(V_A = \frac{Q}{C'} = \frac{CV}{3C} = \frac{V}{3}\)
- Energy stored: 
\(U_A = \frac{1}{2}C'V_A^2 = \frac{1}{2} \cdot 3C \cdot \left(\frac{V}{3}\right)^2 = \frac{1}{6}CV^2\) 

Capacitor B (still connected to battery during dielectric insertion):
- Voltage remains $V$
- Capacitance becomes: \(C' = 3C\)
- Energy stored: 
\(U_B = \frac{1}{2}C'V^2 = \frac{1}{2} \cdot 3C \cdot V^2 = \frac{3}{2}CV^2\) 

Total energy: 
\(U = U_A + U_B = \frac{1}{6}CV^2 + \frac{3}{2}CV^2 = \frac{5}{3}CV^2\) 
⇒ Option (D) is correct. 

Option (B) is incorrect because capacitor B is connected to the battery and stores charge.
Option (C) is incorrect in this context, because energy in capacitor B is \(\frac{3}{2}CV^2\), but the question implies it's disconnected — which contradicts.

Answer 2

Step 1: Initial Configuration (Switch Closed)

When the switch \( K \) is closed:

• The two capacitors \( A \) and \( B \) are in parallel.
• The equivalent capacitance of the system is: $$ C_{\text{eq}} = C + C = 2C $$
• The total energy stored in the system is: $$ U_{\text{initial}} = \frac{1}{2} C_{\text{eq}} V^2 = \frac{1}{2} (2C) V^2 = CV^2 $$

Thus, when the switch is closed, the total energy stored in the two capacitors is: $$ \boxed{CV^2} $$

This confirms that option (A) is correct.

Step 2: After Opening the Switch

When the switch is opened:

• Capacitor \( A \) retains its initial charge \( Q_A = CV \).
• Capacitor \( B \) also retains its initial charge \( Q_B = CV \).
• A dielectric material with dielectric constant \( k = 3 \) is inserted into both capacitors. The new capacitances become: $$ C_A' = kC = 3C \quad \text{and} \quad C_B' = kC = 3C $$
• The voltage across each capacitor changes due to the dielectric insertion: $$ V_A' = \frac{Q_A}{C_A'} = \frac{CV}{3C} = \frac{V}{3}, \quad V_B' = \frac{Q_B}{C_B'} = \frac{CV}{3C} = \frac{V}{3} $$
• The energy stored in capacitor \( A \) is: $$ U_A = \frac{1}{2} C_A' (V_A')^2 = \frac{1}{2} (3C) \left( \frac{V}{3} \right)^2 = \frac{1}{6} CV^2 $$
• The energy stored in capacitor \( B \) is: $$ U_B = \frac{1}{2} C_B' (V_B')^2 = \frac{1}{2} (3C) \left( \frac{V}{3} \right)^2 = \frac{1}{6} CV^2 $$
• The total energy stored in the system after opening the switch is: $$ U_{\text{final}} = U_A + U_B = \frac{1}{6} CV^2 + \frac{1}{6} CV^2 = \frac{1}{3} CV^2 $$

Step 3: Verifying Statements

• (A): True. When the switch is closed, the total energy stored in the two capacitors is \( CV^2 \).
• (B): False. Capacitor \( B \) retains its charge \( Q_B = CV \) even after the switch is opened.
• (C): False. The energy stored in capacitor \( B \) after the switch is opened is \( \frac{1}{6} CV^2 \), not \( \frac{3}{2} CV^2 \).
• (D): True. The total energy stored in the capacitors after the switch is opened is \( \frac{5}{3} CV^2 \).

Final Answer:

The correct option(s) is/are: $$ \boxed{\text{(A) and (D)}} $$