A source of potential difference V is connected to the combination of two identical capacitors as shown in the figure. When key 'K' is closed, the total energy stored across the combination is E₁. Now key 'K' is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now E₂. The ratio E₁ / E₂ will be :

Solution

Physics sir · 4 steps

Once K opens, the two capacitors behave differently: one keeps its voltage, the other keeps its charge.

1. Key closed. Both capacitors sit in parallel across the source, so each has voltage V. The total energy is E₁ = 2 × 1/2CV² = CV², and each capacitor holds charge CV.
2. Middle capacitor. It is still connected to the source after K opens, so its voltage stays V. With the dielectric its capacitance becomes 5C, and its energy is 1/2(5C)V² = 5/2CV².
3. Right capacitor. Opening K isolates it, so its charge stays Q = CV. With capacitance 5C its energy is Q²/2(5C) = C²V²/10C = 1/10CV².

   E₂ = 5/2CV² + 1/10CV² = 26/10CV² = 13/5CV²

4. So. E₁/E₂ = CV²/(13/5CV²) = 5/13, which is option (c).

The idea to keep. Before you add a dielectric, ask what stays fixed. Connected to the battery means V is fixed, so use 1/2CV². Isolated means Q is fixed, so use Q²/2C.

Answer from the NTA final answer key. Solution by PrepWiser.