9.1 Electromagnetic energy density

Electromagnetic energy density

Electromagnetic energy density quantifies how much energy is stored in electric and magnetic fields per unit volume. This concept connects the abstract field quantities from Maxwell's equations to something physically tangible: the energy content of a region of space. It also sets the stage for understanding energy flow (via the Poynting vector) and energy conservation in electromagnetic systems.

Electric field energy density

The energy stored per unit volume in an electric field is

$u_e = \frac{1}{2}\varepsilon_0 E^2$

where $\varepsilon_0$ is the permittivity of free space and $E$ is the electric field magnitude. The quadratic dependence on $E$ means that doubling the field strength quadruples the stored energy density. Systems with high electric field energy density include parallel-plate capacitors (where the field is concentrated between the plates) and the regions around high-voltage transmission lines.

Capacitors and energy storage

A capacitor stores energy in the electric field between its conducting plates. The total stored energy is

$U = \frac{1}{2}CV^2$

where $C$ is the capacitance and $V$ is the voltage across the plates. You can connect this to the energy density picture: for an ideal parallel-plate capacitor with plate area $A$ and separation $d$, the uniform field between the plates is $E = V/d$, so integrating $u_e$ over the volume $Ad$ recovers the same expression. The dielectric material between the plates matters because a higher permittivity increases $C$, allowing more energy to be stored at the same voltage.

Energy in magnetic fields

Magnetic fields store energy just as electric fields do. The magnetic contribution to energy density is central to understanding inductors, transformers, and any system carrying substantial currents.

Magnetic field energy density

The energy stored per unit volume in a magnetic field is

$u_m = \frac{1}{2\mu_0} B^2$

where $\mu_0$ is the permeability of free space and $B$ is the magnetic field magnitude. Like its electric counterpart, this scales as the square of the field. Superconducting magnets and high-current inductors are examples of systems with large magnetic energy densities.

Inductors and energy storage

An inductor stores energy in the magnetic field produced by current flowing through its coil. The total stored energy is

$U = \frac{1}{2}LI^2$

where $L$ is the inductance and $I$ is the current. Again, you can verify this by integrating $u_m$ over the volume where the field exists. A core material with higher magnetic permeability increases $L$, concentrating more field energy for a given current.

Electromagnetic energy in materials

When fields exist inside matter rather than vacuum, the energy density expressions pick up factors that account for the material's response. Getting these right is essential for designing real devices.

Electric energy density in dielectrics

Inside a linear dielectric, the electric energy density becomes

$u_e = \frac{1}{2}\varepsilon_0 \varepsilon_r E^2$

where $\varepsilon_r$ is the relative permittivity of the material. A high-$\varepsilon_r$ dielectric (e.g., barium titanate in ceramic capacitors, with $\varepsilon_r$ in the thousands) stores significantly more energy per unit volume at the same field strength. Physically, the polarization of bound charges in the dielectric does work against the applied field, and that work is captured in the increased energy density.

Magnetic energy density in materials

Inside a linear magnetic material, the magnetic energy density is

$u_m = \frac{B^2}{2\mu_0 \mu_r}$

where $\mu_r$ is the relative permeability. Be careful with the interpretation here: for a given $B$, a higher $\mu_r$ actually decreases $u_m$, because the material's magnetization helps sustain the field with less energy input. Equivalently, you can write $u_m = \frac{1}{2}\mu_0 \mu_r H^2$, which shows that for a given $H$, higher $\mu_r$ increases the energy density. Which form you use depends on whether $B$ or $H$ is the controlled quantity in your problem. Ferromagnetic materials like iron, nickel, and cobalt have very large $\mu_r$ values, which is why they're used in transformer and inductor cores.

Poynting vector

The Poynting vector describes the directional energy flux (power per unit area) of an electromagnetic field. It tells you where energy is going and how fast.

Definition and properties

The Poynting vector is defined as

$\vec{S} = \vec{E} \times \vec{H}$

• Its magnitude gives the power density: energy per unit area per unit time (units of $\text{W/m}^2$).
• Its direction points along the energy flow.
• Because it's a cross product, $\vec{S}$ is perpendicular to both $\vec{E}$ and $\vec{H}$.

Energy flow in electromagnetic fields

For a plane electromagnetic wave, $\vec{S}$ points in the direction of propagation, confirming that the wave carries energy forward. The time-averaged Poynting vector for a sinusoidal wave is

$\langle \vec{S} \rangle = \frac{1}{2} \text{Re}(\vec{E} \times \vec{H}^*)$

This quantity is what you'd measure as the intensity of the wave. The Poynting vector is also used to calculate power transmitted through waveguides and antennas, as well as power dissipated in lossy materials through resistive (Joule) heating.

Conservation of electromagnetic energy

Energy conservation in electrodynamics is expressed through Poynting's theorem, which is the electromagnetic analog of a continuity equation for energy.

Continuity equation

Poynting's theorem in differential form is

$\frac{\partial u}{\partial t} + \nabla \cdot \vec{S} = -\vec{J} \cdot \vec{E}$

where $u = u_e + u_m = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0}B^2$ is the total electromagnetic energy density, $\vec{S}$ is the Poynting vector, and $\vec{J}$ is the current density. Reading this term by term:

1. $\frac{\partial u}{\partial t}$ is the rate of change of stored electromagnetic energy density.
2. $\nabla \cdot \vec{S}$ is the net outward energy flux from a point (energy leaving per unit volume).
3. $-\vec{J} \cdot \vec{E}$ is the rate at which the field does work on charges (power delivered to matter, e.g., Joule heating).

The equation says: the rate at which field energy decreases equals the energy flowing out plus the energy transferred to charges. Integrating over a volume gives the integral form, which is often more intuitive for specific problems.

Energy transfer and dissipation

Energy transfer in electromagnetic systems happens through field propagation, described by $\vec{S}$. Energy dissipation occurs when field energy converts to other forms, most commonly heat via $\vec{J} \cdot \vec{E}$ (Joule heating in resistive materials). Poynting's theorem guarantees a strict accounting: every joule of electromagnetic energy that disappears from a region either flowed out through the boundary or was absorbed by charges inside.

Applications of electromagnetic energy density

Electromagnetic wave propagation

For a plane wave in free space, the electric and magnetic energy densities are equal on average: $\langle u_e \rangle = \langle u_m \rangle$. The total time-averaged energy density is $\langle u \rangle = \varepsilon_0 E_0^2 / 2$, and the intensity (time-averaged Poynting magnitude) relates to it by $\langle S \rangle = c \langle u \rangle$. These relationships are used in antenna design, wireless link budgets, and remote sensing to connect field amplitudes to measurable power levels.

Energy density in waveguides

Waveguides confine electromagnetic energy along a specific path. The spatial distribution of energy density inside a waveguide depends on the propagation mode (TE, TM, or TEM) and the guide geometry (rectangular, circular, etc.). For example, in the $\text{TE}_{10}$ mode of a rectangular waveguide, the energy density peaks at the center of the broad wall and falls to zero at the side walls. Understanding these distributions is critical for avoiding dielectric breakdown and for optimizing coupling into and out of the guide.

Energy density in resonators

Resonators trap electromagnetic energy at discrete resonant frequencies. Inside a resonant cavity, standing wave patterns form, and the energy density is highest at the antinodes. At any instant, energy oscillates between electric and magnetic forms (analogous to a mechanical oscillator swapping kinetic and potential energy). Applications include microwave cavities for particle accelerators, optical cavities in lasers, and RF filters in communication systems.

Measurement techniques

Electric field energy density measurement

Electric field probes (dipole antennas, electro-optic sensors) measure the local field strength $E$, from which you calculate $u_e = \frac{1}{2}\varepsilon_0 E^2$. Practical challenges include calibrating the probe response, achieving sufficient spatial resolution (especially at high frequencies), and accounting for field distortion caused by nearby conductive objects or the probe itself.

Magnetic field energy density measurement

Magnetic field probes (small loop antennas, Hall effect sensors) measure $B$, and you compute $u_m = \frac{1}{2\mu_0}B^2$. Similar challenges arise: calibration, spatial resolution, and distortion from ferromagnetic materials in the vicinity. At microwave frequencies, loop probes must be kept electrically small to avoid picking up electric field components.

Poynting vector measurement

Measuring $\vec{S}$ requires simultaneous measurement of both $\vec{E}$ and $\vec{H}$ at the same point. The cross product is then computed from the measured components. Key difficulties include aligning the probes correctly, maintaining phase coherence between the electric and magnetic measurements, and dealing with near-field effects close to sources where the simple far-field relationship between $E$ and $H$ breaks down.