A capacitor is made from two parallel plates separated by a distance d, and each has area A. The capacitor holds a charge Q. (a) What is the energy stored in the capacitor? (b) A mechanical force pulls the plates apart, without changing the stored charge, until they are 2d apart. Now how much energy is stored? (c) Explain the difference between (a) and (b).

Question

A capacitor is made from two parallel plates separated by a distance d, and each has area A. The capacitor holds a charge Q. (a) What is the energy stored in the capacitor? (b) A mechanical force pulls the plates apart, without changing the stored charge, until they are 2d apart. Now how much energy is stored? (c) Explain the difference between (a) and (b).

Answer 1

(a)

C=εε₀A/d
Energy=Q²/2C
(b)
C₁=εε₀A/2d=C/2
Energy=Q²/2(C/2)= Q²/C

Answer 2

To calculate the energy stored in a capacitor, we can use the formula:

(a) U = (1/2) * Q^2 / C.

Where U is the energy stored, Q is the charge, and C is the capacitance.

The capacitance of a parallel plate capacitor can be calculated as:

C = ε * A / d.

Where ε is the permittivity of free space, A is the area of the plates, and d is the distance between the plates.

(a) To find the energy stored in the capacitor before the plates are pulled apart, we need to calculate the capacitance first. Assuming the plates are initially at a distance d, we can substitute the values into the capacitance formula:

C = ε * A / d.

Then, we can calculate the energy using the formula:

U = (1/2) * Q^2 / C.

(b) After the plates are pulled apart to a distance of 2d, the capacitance changes because the distance between the plates has increased. To find the new capacitance, we substitute the new distance, 2d, into the capacitance formula:

C' = ε * A / 2d.

We can then calculate the new energy using the formula:

U' = (1/2) * Q^2 / C'.

(c) The difference between (a) and (b) is a result of the change in capacitance. When the plates were initially separated by a distance d, the capacitance was C. However, when the distance between the plates increased to 2d, the capacitance became smaller, C'. As a result, the energy stored in the capacitor decreases when the plates are pulled apart while keeping the same charge.

The change in energy is related to the work done in pulling the plates apart against the electrostatic force between them. The work done is converted into potential energy stored in the capacitor. Therefore, when the plates are pulled apart and the capacitance decreases, the energy stored in the capacitor also decreases.