Two capacitors each having area a and plate separation d are connected as shown in the circuit, each capacitor carries a charge Q/2. The plates of one capacitor are slowly pulled apart by an external agent till the separation between them becomes 2d , the other capacitor is not disturbed.

Find the work done by the external agent in this process.

To find the work done by the external agent while pulling apart one of the capacitors, we need to analyze how the energy stored in the capacitor changes as the plate separation increases. Let's break this down step by step.

Understanding Capacitors and Energy Storage

A capacitor stores electrical energy in the electric field created between its plates. The energy (U) stored in a capacitor can be expressed using the formula:

• U = (1/2) * C * V²
• or U = (Q² / (2C))

Where:

• U is the energy stored in the capacitor.
• C is the capacitance.
• V is the voltage across the capacitor.
• Q is the charge on the capacitor.

Capacitance and Charge Relationship

The capacitance (C) of a parallel plate capacitor is given by:

• C = ε₀ * (A/d)

Where:

• ε₀ is the permittivity of free space.
• A is the area of the plates.
• d is the separation between the plates.

Initially, each capacitor has a charge of Q/2 and a separation of d. Therefore, the initial capacitance (C₁) is:

• C₁ = ε₀ * (A/d)

Energy Before the Change

The initial energy stored in the capacitor before the separation is changed can be calculated as:

• U₁ = (Q/2)² / (2 * C₁) = (Q² / 8C₁)

Changing the Plate Separation

When the external agent pulls the plates apart to a new separation of 2d, the new capacitance (C₂) becomes:

• C₂ = ε₀ * (A/(2d)) = C₁ / 2

Since the charge on the capacitor remains constant (Q/2), we can find the new energy stored (U₂) using:

• U₂ = (Q/2)² / (2 * C₂) = (Q² / 8C₂) = (Q² / 8 * (C₁ / 2)) = (Q² / 4C₁)

Calculating the Work Done

The work done (W) by the external agent is equal to the change in energy of the capacitor:

• W = U₂ - U₁

Substituting the values we found:

• W = (Q² / 4C₁) - (Q² / 8C₁)
• W = (2Q² / 8C₁) - (Q² / 8C₁) = (Q² / 8C₁)

Final Expression for Work Done

Thus, the work done by the external agent in pulling the plates apart from separation d to 2d is:

• W = Q² / (8C₁)

This result shows how energy changes in a capacitor when the physical configuration is altered, emphasizing the relationship between charge, capacitance, and energy storage. If you have any further questions about capacitors or related concepts, feel free to ask!