Electric energy density

The relation $u = \tfrac{1}{2}\varepsilon_0 E^2$ would deserve first place in the interpretation of electrostatics: it tells us that the energy is not on the charges, but in the space where the field exists. Wherever there is a field $E$, there is an energy density per unit volume proportional to the square of its magnitude. It is a profound change of perspective: we stop thinking of energy as “accumulated by objects” and start thinking of it as spread through space, following the field point by point.

Principle — Electric energy density

At every point in space where there is an electric field of magnitude $E$, the energy per unit volume equals $\ev{u_E = \tfrac{1}{2}\,\varepsilon_0\,E^2 \qquad [\text{J/m}^3]}$ The total energy stored in a region of volume $V$ is obtained by integrating: $U = \int_V u_E\,\dd V$.

The result was derived for the special case of the parallel-plate capacitor, but it holds generally. Just imagine slowly charging any distribution of charges: the work done against the electrostatic forces is transferred into the field that arises, with the same density $u_E$ at every point in space.

Example — Field energy of a charged sphere

A conducting sphere of radius $R$ carries charge $Q$. Outside, the field is that of a point charge; inside, it is zero: $E(r) = \frac{kQ}{r^2} \quad (r > R), \qquad E(r) = 0 \quad (r < R)$ The energy density is $u_E(r) = \tfrac12\varepsilon_0 E^2 = \dfrac{kQ^2}{8\pi r^4}$ (using $\varepsilon_0 = 1/(4\pi k)$). Integrating over spherical shells $\dd V = 4\pi r^2\,\dd r$ from $R$ to infinity: $U_{\text{tot}} = \int_R^{\infty}\! u_E(r)\,4\pi r^2\,\dd r = \frac{kQ^2}{2R}$ The same result follows from the formula $U = \tfrac12 Q V$ applied to a conducting sphere at potential $V = kQ/R$: this confirms that electrostatic energy can equivalently be “counted” on the charges ($\tfrac12 QV$) or on the field ($\int u_E\,\dd V$).

Curiosity — Where does the energy really live?

The question is not trivial: for stationary charges the two descriptions are equivalent. But as soon as the field becomes time-dependent (electromagnetic waves, induction), the local description $u = \tfrac12 \varepsilon_0 E^2 + \tfrac{1}{2\mu_0} B^2$ is the only one that holds up: the energy travels through space together with the field, at speed $c$. We will see this in the chapter on electromagnetic waves.

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Topics: Electric field and potential Concepts: Field energy density · Electric field

Related exercises: Fermi estimate — energy of a cloud · Energy density in a thunderstorm · Shielding and charge in a hollow conductor