The energy stored in a pair of point charges and a distance from one another can be thought of in terms of the work-energy theorem and the work required to bring one of the charges from far away.
The energy of each pair of charges in an arrangement is
What is the energy stored in an arrangement of three charges of magnitude arranged in an equilateral triangle with side length
Since there are three charges, there are three pairs. Each pair has energy
So for three identical pairs, the energy is
Storing charge on the isolated conductors of a capacitor requires work to move the charge onto the conductors. By definition of the potential difference, if charge is added to one of the conductors, causing a potential difference , then a work of is required. So the total work required to charge one of the conductors from neutrality up to charge is
By the law of conservation of energy, the work done in charging the capacitor is stored as potential energy in the electric field of the capacitor. Using this can be rewritten several ways:
What is the energy stored on a parallel-plate capacitor?
As computed above, the capacitance of the parallel-plate capacitor (area , plate separation , charge ) is
Plugging into the formula for the potential energy stored in a capacitor,
If a capacitor is composed of two isolated conductors, after charging the oppositely charged plates will experience a Coulombic attraction. Given a spherical capacitor of inner radius and outer radius , find the attractive force exerted on the outer conductor assuming that each conductor holds charge .
Assume the conductors are mechanically held fixed, so the force is constant in time, and let negative forces correspond to attraction and vice versa.
The capacitance of a capacitor and thus the energy stored in a capacitor at fixed voltage can be increased by use of a dielectric. A dielectric is an insulating material that is polarized in an electric field, which can be inserted between the isolated conductors in a capacitor. That is, when an electric field is applied to a dielectric, the positive and negative charges in the insulating material shift slightly from their neutral equilibrium, creating a small electric field opposing the applied field.
Mathematically, the effect of a dielectric is to modify the permittivity of free space by some constant:
The constant is often called the dielectric constant, and takes into account how the presence of the dielectric modifies the strength of the electric field in the insulating material. In vacuum, ; otherwise, since any atoms present may be slightly polarized.
Find the capacitance of a parallel-plate capacitor with a dielectric of constant inserted between the plates.
The capacitance of the parallel-plate capacitor was derived to be
with the area of each plate and the plate separation. With the dielectric inserted, . The resulting capacitance is
Since , inserting the dielectric increases the capacitance and therefore increases the energy stored at a fixed voltage.
where is the amplitude of the electric field and is the permittivity of free space.
What is the electric energy density of an electromagnetic wave with magnetic field amplitude
First, express the magnetic field amplitude in terms of the electric field amplitude.
Then, calculate the energy density.