EM-12 · Electrostatics field

Electrostatic Energy

The electrostatic energy stored in an arrangement of charges actually resides in the electric field itself. You see for yourself that strengthening the field raises the energy density as the square of the field, E², and confirm that a capacitor's energy is the field's energy (u = ½εE²).

Strengthen the field and see the energy

A uniform electric field spans the gap between the red plate on the left and the blue plate on the right, and the brightness of the glow filling the gap shows the energy density. Strengthening the field with the slider brightens the field lines, and the glow in the gap brightens even more steeply. The percentage on the indicator lets you check how many times the energy grows when the field doubles.

Field strength EE = 4.0

Drag to rotate; the slider changes the field strength (E).

The energy density in the field

u = ½εE² · energy lives in the electric field

U = ½CV² = ∫u dV · the capacitor energy is the sum of field energy

Energy density (vs maximum) · 16%

Middle

Where the energy is

Charging a capacitor stores energy (EM-11). But where is that energy? It seems to sit on the charges, yet the deeper answer is that it is held in the electric field itself. As long as there is a field in space, the energy is there too, and electrostatic energy is spread throughout the space the field fills.

Energy density u=½εE²

The energy per unit volume at a point in space with an electric field is called the energy density, written u = ½εE². The key point is that it is proportional to the square of the field, E². Doubling the field makes the energy density four times larger, so even a small increase in the field raises the energy steeply. On the first screen, the glow in the gap brightened faster than the field lines because of exactly this relation.

Observeu = ½εE²

The energy density is ½ times ε times the square of the field.

Chooseu ∝ ?

Doubling the field makes the energy density four times larger.

It concentrates where the field is strong

Since the energy density is proportional to E², energy crowds where the field is strong. In a parallel-plate capacitor the field is uniform, so the energy is spread evenly across the gap, but around a point charge the field goes as 1/r², so the energy density goes as 1/r⁴ and piles up right next to the charge. If a point charge is taken as a true point with no size, the total of this energy grows without limit, which is also why classical electromagnetism has trouble with a charge that is a single point. With a map of the electric field, you can read at once where and how much energy is stored.

Capacitor energy = field energy

The two views meet here. The circuit view's capacitor energy U = ½CV² and the field view's energy density integrated over volume, ∫u dV, are exactly the same value. You can check this directly for parallel plates. Put E = V/d and the volume Ad into u = ½εE², and you get ½ε(V/d)²(Ad) = ½(εA/d)V² = ½CV². Count the same energy by charges or by fields, and there is one answer. This view that energy lives in the field leads on to the reason electromagnetic waves carry energy through empty space.

Fill inU = u ?

The total energy is the density added up over the volume.

On your ownU = ?

The energy counted by the field equals ½CV².

Back to the first screen

On the first screen, strengthening the field made the glow filling the gap brighten steeply, and doubling the field made it four times brighter. That is because the energy density follows the square of the field, u = ½εE². That glow was the energy spread through space. What held the energy was not the charges but the electric field, and the ½CV² gathered in the capacitor exactly equals the sum of ½εE² over every bit of volume.

Electrostatic energy is held not in the charges but in the electric field itself. The energy density per unit volume is u = ½εE², so energy crowds where the field is strong (proportional to the square of the field). The total energy is this density integrated over volume, U = ∫u dV, and for a capacitor it exactly equals ½CV². Doubling the field E makes the energy density four times larger.

What comes next

This completes the 12 blocks of electrostatics, the world of charges at rest, from coordinate systems through the electric field, Gauss's law, and potential to electrostatic energy. Now we move on to moving charges, that is, the magnetic field made by currents. The first unit of the next block is the Biot-Savart law (EM-13). The view that energy lives in fields returns in magnetic energy (EM-19) and in the Poynting vector of electromagnetic waves (EM-24).