Question

When the battery across the plates of a charged condenser is disconnected and a dielectric slab is introduced between its plates, then the energy stored

• A. becomes infinity
• B. does not change
• C. increases
• D. decreases

Hint:

Always track the state of the battery! If the battery stays connected, voltage remains constant ($V = \text{constant}$), and energy increases ($U = \frac{1}{2}CV^2 \propto C$). But if the battery is disconnected, charge remains constant ($Q = \text{constant}$), and energy decreases ($U = \frac{Q^2}{2C} \propto \frac{1}{C}$). Keeping this distinction clear avoids a common trap!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A capacitor is fully charged by a battery, and then the battery is completely disconnected, isolating the system. A dielectric slab is then inserted between the plates. We need to find what happens to the total electrostatic potential energy stored in the capacitor.

Step 2: Key Formula or Approach:
1. Since the battery is disconnected, the electrical charge ($Q$) trapped on the capacitor plates remains completely constant ($Q = \text{constant}$). 2. The insertion of a dielectric slab with a dielectric constant $K > 1$ increases the capacitance from $C$ to $C' = KC$. 3. Use the energy formula that contains the constant charge parameter: $$U = \frac{Q^2}{2C}$$

Step 3: Detailed Explanation:
Let's write down the initial energy expression stored in the air-filled capacitor: $$U_1 = \frac{Q^2}{2C}$$ After inserting the dielectric slab, the new capacitance is $C' = KC$. Let's write the new energy expression $U_2$: $$U_2 = \frac{Q^2}{2C'} = \frac{Q^2}{2(KC)} = \frac{1}{K} \left(\frac{Q^2}{2C}\right)$$ Substitute $U_1$ back into the equation: $$U_2 = \frac{U_1}{K}$$ Since the dielectric constant for any real material is always strictly greater than one ($K > 1$), dividing by $K$ reduces the value: $$U_2 < U_1$$ Therefore, the total potential energy stored inside the electric field decreases.

Step 4: Final Answer:
The stored energy decreases, which corresponds to option (D).