EM-12 · Electrostatics field

Electrostatic Energy

The electrostatic energy stored in an arrangement of charge in fact resides in the electric field itself. See the energy density grow as E² as you strengthen the field, and learn that a capacitor’s energy is the field’s energy (u = ½εE²).

Strengthen the field and watch the energy

Use the slider to strengthen the field between the capacitor plates. How does the glow filling the gap (the energy density) brighten? When you double the field, by how much does the energy grow?

Field strength EE = 4.0

Drag to orbit. The slider sets 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 resides in the electric field itself. Even if you carry the charges away, as long as a field remains in space the energy stays there too. Electrostatic energy is spread through every part of space the field fills.

Energy density u=½εE²

The energy per unit volume at a point where a field exists is the energy density u: u = ½εE². The key is that it scales with the square of the field. Double the field and the energy density quadruples, so even a modest increase in field makes the energy climb steeply. The glow in the gap brightening as E² on the first screen is exactly this relation.

Observeu = ½εE²

Energy density is one half times ε times the field squared.

Chooseu ∝ ?

Double the field and the energy density quadruples.

It concentrates where the field is strong

Since the density goes as E², energy concentrates where the field is strong. A parallel-plate capacitor has a uniform field, so it spreads evenly across the gap; but around a point charge E goes as 1/r², so the density goes as 1/r⁴ — overwhelmingly concentrated right next to the charge. Given just the field map, you can read off where and how much energy is stored.

Capacitor energy = field energy

Two views meet. The circuit view’s capacitor energy U = ½CV² and the field view’s ∫u dV (the density integrated over volume) are exactly equal. The parallel plate confirms it directly: putting E = V/d and the volume Ad into u = ½εE² gives ½ε(V/d)²(Ad) = ½(εA/d)V² = ½CV². Counting the same energy through charges or through the field gives one answer. This view that energy lives in the field is why electromagnetic waves can carry energy through empty space.

Fill inU = u ?

Total energy is density times volume.

On your ownU = ?

The energy counted from the field equals ½CV².

Back to the first screen

On the first screen, strengthening the field made the glow in the gap brighten steeply: double the field, four times the brightness. That is because the energy density follows the square of the field, u = ½εE². The glow was the energy spread through space — the field, not the charges, held it. The ½CV² gathered in the capacitor was exactly the sum of ½εE² over every bit of volume.

Electrostatic energy resides not in the charges but in the electric field itself. The energy density per unit volume is u = ½εE², so energy concentrates where the field is strong (it scales with the square of the field). The total energy is this density integrated over volume, U = ∫u dV, exactly equal to ½CV² for a capacitor. Doubling the field E quadruples the energy density.

What comes next

With this, the twelve electrostatics units are complete — from coordinate systems through the electric field, Gauss’s law, potential, and electrostatic energy: the world of charges at rest. Now we cross to the magnetic field made by moving charge, that is, current. The next block opens with the Biot–Savart law (EM-13). The view that energy lives in fields returns with magnetic energy (EM-19) and the Poynting vector of electromagnetic waves (EM-24).