3.3 Energy and Momentum of Electromagnetic Waves

Electromagnetic waves carry energy and momentum, and quantifying these properties is central to understanding how light interacts with matter. This section covers energy density, the Poynting vector, intensity-field relationships, and how conservation laws apply to electromagnetic wave phenomena.

Energy density and Poynting vector

Energy Density in Electromagnetic Waves

Energy density is the amount of energy stored per unit volume in the electric and magnetic fields of a wave. The total energy density is the sum of the electric and magnetic contributions:

• Electric field energy density: $u_e = \frac{1}{2}\epsilon_0 E^2$
• Magnetic field energy density: $u_m = \frac{1}{2\mu_0} B^2$

In vacuum, these two contributions are always equal for an electromagnetic wave. That means the total energy density simplifies to:

$u_{total} = u_e + u_m = \epsilon_0 E^2$

Because energy density scales with the square of the field amplitude, doubling the field strength quadruples the energy density.

Poynting Vector and Energy Flow

The Poynting vector tells you both the direction and the rate of energy flow (power per unit area) in an electromagnetic wave:

$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$

Its direction is perpendicular to both $\vec{E}$ and $\vec{B}$, following the right-hand rule: curl your fingers from $\vec{E}$ toward $\vec{B}$, and your thumb points along $\vec{S}$.

Since $\vec{E}$ and $\vec{B}$ oscillate, the Poynting vector oscillates too. For most applications you care about the time-averaged value, which gives the wave's intensity:

$S_{avg} = \frac{1}{2\mu_0} E_0 B_0 = \frac{1}{2}\epsilon_0 c E_0^2$

Practical examples include calculating the energy flux of solar radiation arriving at Earth (~1361 W/m²) and mapping the radiation patterns of antennas.

Electromagnetic wave energy and momentum

Energy of Electromagnetic Waves

There are two complementary ways to think about the energy carried by electromagnetic waves, depending on whether you're treating the wave classically or as quantized photons.

Classical picture: The total energy delivered by a wave is found by integrating the Poynting vector over the area it illuminates and the time it shines:

$E_{total} = \int_A \int_t \vec{S} \cdot d\vec{A}\, dt$

Photon picture: Energy is carried in discrete packets (photons), each with energy

$E = hf$

where $h = 6.626 \times 10^{-34}$ J·s is Planck's constant and $f$ is the wave frequency. Energy is directly proportional to frequency, so gamma-ray photons carry far more energy per photon than radio-wave photons. This quantized energy is what governs photon-matter interactions like the photoelectric effect and Compton scattering.

Momentum of Electromagnetic Waves

Even though photons are massless, electromagnetic waves carry momentum. The momentum of a photon is related to its energy by:

$p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$

where $c = 3 \times 10^8$ m/s. This momentum is what produces radiation pressure. When a wave of intensity $I$ is completely absorbed by a surface:

$P_{abs} = \frac{I}{c}$

For a perfectly reflecting surface, the momentum change is doubled (the photon bounces back), so the pressure doubles: $P_{ref} = \frac{2I}{c}$.

Circularly polarized photons also carry angular momentum of $\pm\hbar$ per photon (+ for right-circular, − for left-circular polarization).

Real-world consequences of radiation momentum:

• Solar sails use radiation pressure from sunlight to propel spacecraft without fuel.
• Comet tails point away from the Sun partly because solar radiation pressure pushes dust particles outward.

Intensity, Electric, and Magnetic Fields

Intensity and Field Amplitude Relationships

Intensity is the time-averaged power per unit area delivered by the wave. You can express it in terms of either field amplitude:

• Using the electric field: $I = \frac{1}{2}\epsilon_0 c E_0^2$
• Using the magnetic field: $I = \frac{c}{2\mu_0} B_0^2$

The electric and magnetic field amplitudes in vacuum are linked by $E_0 = cB_0$, so these two expressions are equivalent.

For AC-style calculations, root-mean-square (RMS) field values are often more convenient:

• $E_{rms} = \frac{E_0}{\sqrt{2}}$
• $B_{rms} = \frac{B_0}{\sqrt{2}}$

Using RMS values, intensity becomes simply $I = \epsilon_0 c E_{rms}^2$ (no factor of 1/2 needed).

Intensity Variations and Media Effects

For a point source radiating equally in all directions, intensity falls off with the inverse square law:

$I \propto \frac{1}{r^2}$

This is purely geometric: the same total power spreads over a sphere of area $4\pi r^2$, so doubling your distance from the source cuts the intensity to one quarter.

In a material medium (not vacuum), you replace the vacuum constants with the material's permittivity $\epsilon$ and permeability $\mu$:

• $\epsilon_0 \rightarrow \epsilon$
• $\mu_0 \rightarrow \mu$

The wave speed also changes to $v = \frac{1}{\sqrt{\epsilon\mu}}$, which modifies the intensity-field relationships accordingly.

Energy and Momentum Conservation for Waves

Conservation of Energy in Electromagnetic Systems

Energy conservation applies to every interaction between electromagnetic waves and matter. The total energy of an isolated system stays constant; it just shifts between forms:

• Electromagnetic → kinetic: In the photoelectric effect, a photon's energy converts into an electron's kinetic energy (plus the work function).
• Electromagnetic → thermal: Microwave ovens transfer wave energy into molecular rotational energy, which becomes heat.
• Absorption and re-emission: In fluorescence, a material absorbs a high-energy photon and re-emits one or more lower-energy photons. The total emitted energy cannot exceed the absorbed energy.

Conservation of Momentum in Wave-Matter Interactions

Momentum conservation governs how electromagnetic waves push and scatter off matter.

Radiation pressure on a perfectly reflecting surface is:

$P = \frac{2I}{c}$

The factor of 2 appears because the photon's momentum reverses direction, so the total momentum transfer is twice the incoming momentum. For perfect absorption, the factor is just 1 (as noted above).

Compton scattering is a clean demonstration of both energy and momentum conservation. When a photon scatters off a free electron, the photon loses energy and its wavelength increases by:

$\Delta \lambda = \frac{h}{m_e c}(1 - \cos\theta)$

where $\theta$ is the scattering angle and $m_e$ is the electron mass. The quantity $\frac{h}{m_e c} \approx 2.43 \times 10^{-12}$ m is called the Compton wavelength of the electron. At $\theta = 180°$ (backscatter), the wavelength shift is maximized at twice this value.

Photon recoil occurs whenever an atom absorbs or emits a photon; the atom kicks back to conserve momentum. This effect is the basis for laser cooling, where carefully tuned lasers slow atoms down to extremely low temperatures.

Pair production and annihilation also obey both conservation laws. A sufficiently energetic photon ($E \geq 2m_e c^2$) near a nucleus can produce an electron-positron pair, and when a particle meets its antiparticle, they annihilate into photons whose total energy and momentum match those of the original particles.