Energy of Charge Distributions and Field Energy

The Idea of Field Energy

Imagine you have a collection of point charges scattered infinitely far apart. To bring them together to form a specific configuration, you must do work against the electrostatic forces. This work is stored as potential energy in the system. Traditionally, we might think of this energy as residing in the interactions between pairs of charges. However, a more profound view, consistent with the field concept, is that the energy is distributed throughout space wherever the electric field exists. The stronger the field, the more energy is stored per unit volume. This localizes the energy, making it easier to analyze how energy flows and is conserved in dynamic situations.

Assembling a Charge Distribution

Consider a continuous charge distribution with volume charge density \(\rho(\vec{r})\). To calculate the total energy \(U\) required to assemble this distribution from infinity, we can imagine bringing infinitesimal amounts of charge \(dq\) from infinity to their final positions. The work done to bring a small charge \(dq\) to a point where the potential is already \(\varphi\) is \(dW = \varphi \, dq\). Since \(dq = \rho \, d^3r\), the total work is the integral of \(\varphi \rho \, d^3r\). However, because the potential \(\varphi\) at any point is generated by all the other charges in the distribution, we must be careful not to double-count the interaction energy. Each pair of interacting charges is counted twice in the direct integration of \(\rho \varphi\). Therefore, the total electrostatic energy is given by half the integral of the product of charge density and potential over all space.

This expression is known as the charge-potential form of the energy. It is useful for calculating the energy of discrete systems or when the charge distribution is known explicitly. However, it still relies on the source charges. We can eliminate \(\rho\) using Gauss’s law in differential form, \(\nabla \cdot \vec{E} = \rho / \varepsilon_0\), to express the energy purely in terms of the field.

Energy Density in the Field

Substituting \(\rho = \varepsilon_0 \nabla \cdot \vec{E}\) into the energy expression, we get \(U = \frac{\varepsilon_0}{2} \int (\nabla \cdot \vec{E}) \varphi \, d^3r\). Using the vector identity \(\nabla \cdot (\varphi \vec{E}) = \varphi (\nabla \cdot \vec{E}) + \vec{E} \cdot (\nabla \varphi)\), we can rewrite the integrand. Since \(\vec{E} = -\nabla \varphi\), the term \(\vec{E} \cdot (\nabla \varphi)\) becomes \(-E^2\). Applying the divergence theorem to the volume integral of \(\nabla \cdot (\varphi \vec{E})\), the surface integral vanishes at infinity because the potential falls off as \(1/r\) and the field as \(1/r^2\), making the product decay faster than \(1/r^2\) over a surface area growing as \(r^2\). This leaves us with an integral solely over the square of the electric field magnitude.

This result is profound: the total energy is the sum of a local quantity, the energy density \(u\), integrated over all space. The energy density is defined as \(u = \frac{1}{2} \varepsilon_0 E^2\). This confirms that the electric field itself possesses energy, independent of the specific charges that created it. This local density is always non-negative, reflecting the fact that energy is a positive quantity stored in the field configuration.

Implications for Conductors and Capacitors

Although this lecture focuses on vacuum, the concept extends directly to conductors. Inside a conductor in electrostatic equilibrium, the electric field is zero, so the energy density is zero. All the energy is stored in the field outside the conductor. For a parallel plate capacitor, the field is uniform between the plates and zero outside. The energy density is constant between the plates, \(u = \frac{1}{2} \varepsilon_0 E^2\). Multiplying this density by the volume between the plates (\(A \times d\)) gives the total energy \(U = \frac{1}{2} \varepsilon_0 E^2 A d\). Since \(E = V/d\) and \(C = \varepsilon_0 A / d\), this reduces to the familiar formula \(U = \frac{1}{2} C V^2\). This consistency check validates the field energy interpretation and shows that the energy is physically located in the space between the plates, not in the metal plates themselves.