Question

Derive an expression for the energy shored in the series arrangement.

Answer 1

Answer:

The expression in Equation 8.4.2 for the energy stored in a parallel-plate capacitor is generally valid for all types of capacitors.

If we know the energy density, the energy can be found as UC=uE(Ad) . We will learn in Electromagnetic Waves (after completing the study of Maxwell’s equations) that the energy density uE in a region of free space occupied by an electrical field E depends only on the magnitude of the field and is

uE=12ϵ0E2.(8.4.1)

If we multiply the energy density by the volume between the plates, we obtain the amount of energy stored between the plates of a parallel-plate capacitor UC=uE(Ad)=12ϵ0E2Ad=12ϵ0V2d2Ad=12V2ϵ0Ad=12V2C .

In this derivation, we used the fact that the electrical field between the plates is uniform so that E=V/d and C=ϵ0A/d . Because C=Q/V , we can express this result in other equivalent forms:

UC=12V2C=12Q2C=12QV.(8.4.2)

The expression in Equation 8.4.2 for the energy stored in a parallel-plate capacitor is generally valid for all types of capacitors.