To find the change in energy per unit area of the capacitor due to insertion of the conductor, we need to compare the energy stored in the capacitor before and after insertion.
Before insertion, the energy stored in the capacitor is given by the formula:
E1 = (1/2) * ε0 * A * V^2,
where ε0 is the permittivity of free space, A is the area of the capacitor plates, and V is the voltage across the capacitor.
After insertion, the energy stored in the capacitor is given by the formula:
E2 = (1/2) * ε0 * A_eff * V^2,
where A_eff is the effective area of the capacitor plates after insertion.
The change in energy per unit area is then given by the formula:
ΔE/A = (E2 - E1) / A,
where ΔE is the change in energy due to insertion.
To find the effective area A_eff, we need to consider the electric field inside the capacitor when the conductor is inserted. The electric field will be zero inside the conductor because it is a perfect conductor. Therefore, the electric field only exists in the region where the dielectric is not. This region will be reduced in size due to the insertion of the conductor.
The effective area A_eff can be calculated using the formula:
A_eff = (d-d_conductor) * A,
where d is the distance between the capacitor plates and d_conductor is the thickness of the inserted conductor.
Finally, substituting the values into the formula for ΔE/A:
ΔE/A = [(1/2) * ε0 * A_eff * V^2 - (1/2) * ε0 * A * V^2] / A,
ΔE/A = [(1/2) * ε0 * (d-d_conductor) * A * V^2 - (1/2) * ε0 * A * V^2] / A,
Simplifying,
ΔE/A = (1/2) * ε0 * (d-d_conductor) * V^2.
Therefore, the change in energy per unit area of the capacitor due to insertion of the conductor is (1/2) * ε0 * (d-d_conductor) * V^2.
A parallel plate capacitor is connected to a voltage source as shown in Figure 4.76a. The dielectric is free space. A flat perfect conductor of thickness a [m] is inserted as shown in Figure 4.76b: (a) What is the change in energy per unit area of the capacitor due to insertion of the conductor (i.e., what is the change in energy for a 1 m2 section of the capacitor)
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