Learning Goal:

To understand that the charge stored by capacitors represents energy; to be able to calculate the stored energy and its changes under different circumstances.

An air-filled parallel-plate capacitor has plate area A and plate separation d. The capacitor is connected to a battery that creates a constant voltage V.

A. Find the energy U0 stored in the capacitor.
Express your answer in terms of A, d, V, and ?0.
B. The capacitor is now disconnected from the battery, and the plates of the capacitor are then slowly pulled apart until the separation reaches 3d. Find the new energy U1 of the capacitor after this process.
Express your answer in terms of A, d, V, and ?0.
C. The capacitor is now reconnected to the battery, and the plate separation is restored to d. A dielectric plate is slowly moved into the capacitor until the entire space between the plates is filled. Find the energy U2 of the dielectric-filled capacitor. The capacitor remains connected to the battery. The dielectric constant is K.
Express your answer in terms of A, d, V, K, and ?0.

The capacitance of a capacitor is its ability to store charges. For parallel-plate capacitors, this ability depends the material between the plates, the common plate area and the plate separation. The relationship is

[tex]C=\dfrac{\epsilon A}{d}[/tex]

[tex]C[/tex] is the capacitance, [tex]A[/tex] is the common plate area, [tex]d[/tex] is the plate separation and [tex]\epsilon[/tex] is the permittivity of the material between the plates.

For air or free space, [tex]\epsilon[/tex] is [tex]\epsilon_0[/tex] called the permittivity of free space. In general, [tex]\epsilon=\epsilon_r \epsilon_0[/tex] where [tex]\epsilon_r[/tex] is the relative permittivity or dielectric constant of the material between the plates. It is a factor that determines the strength of the material compared to air. In fact, for air or vacuum, [tex]\epsilon_r=1[/tex].

The energy stored in a capacitor is the average of the product of its charge and voltage.

[tex]U = \dfrac{QV}{2}[/tex]

Its charge, [tex]Q[/tex], is related to its capacitance by [tex]Q=CV[/tex] (this is the electrical definition of capacitance, a ratio of the charge to its voltage; the previous formula is the geometric definition). Substituting this in the formula for [tex]U[/tex],

[tex]U = \dfrac{CV^2}{2}[/tex]

A. Substituting for [tex]C[/tex] in [tex]U[/tex],

[tex]U_0 = \dfrac{\epsilon_0 A V^2}{2d}[/tex]

B. When the distance is [tex]3d[/tex],

[tex]U_1 = \dfrac{\epsilon_0 A V^2}{2\times3d}[/tex]

[tex]U_1 = \dfrac{\epsilon_0 A V^2}{6d}[/tex]

C. When the distance is restored but with a dielectric material of dielectric constant, [tex]K[/tex], inserted, we have

[tex]U_2 = \dfrac{K\epsilon_0 A V^2}{2d}[/tex]