Energy and Momentum in EM Waves

Light waves are not just mathematical abstractions—they are physical carriers of energy and momentum. The fields themselves store energy, and the propagation of these fields constitutes a flow of power.

1. Energy Flow: The Poynting Vector

The total energy density $u$ is the sum of the electric energy density ($u_e$) and magnetic energy density ($u_m$). Recall from Electrostatics and Deep Dive: Energy in Magnetic Fields:

u = u_e + u_m = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2

For a plane wave in vacuum, the electric and magnetic energy densities are equal. The power per unit area transported by the wave is described by the Poynting Vector $\vec{S}$:

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

$\vec{S}$ points in the direction of wave propagation. Its magnitude fluctuates rapidly; what we usually measure is the average power over time, called the Intensity ($I$).

I = \langle S \rangle = \frac{1}{2}c\epsilon_0 E_0^2

Deep Dive: The Poynting Vector

2. Momentum and Radiation Pressure

Since EM waves carry energy, they also carry momentum. When light hits an object, it transfers this momentum, exerting a tiny but measurable force.

The Radiation Pressure ($P$) depends on whether the surface absorbs or reflects the light:

$P = I/c$ Perfect Absorption. The surface absorbs all incident momentum ($100\%$ transfer).

$P = 2I/c$ Perfect Reflection. The wave bounces back, doubling the total momentum transfer due to the change in direction.

Interactive: Radiation Pressure

Deep Dive: Radiation Pressure