Energy and Momentum Carried by Waves
~7 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with the vector calculus identities for divergence and curl, and you must have a solid grasp of the structure of monochromatic plane waves in vacuum, specifically the relationship between the electric and magnetic field amplitudes.
Why this matters
Electromagnetic fields are not abstract mathematical constructs; they are physical entities that transport energy and momentum across space.
In the previous lecture, we established that light is a transverse wave propagating at speed \( c \). However, a wave that carries no energy is merely a mathematical curiosity. To understand how antennas transmit power, how solar panels generate electricity, or how radiation pressure pushes spacecraft, we must quantify the flow of energy and the mechanical impulse delivered by the field. This lecture bridges the gap between the wave description and the physical interaction with matter, introducing the Poynting vector and the concept of field momentum.
The Physical Picture: Flow and Pressure
Imagine a river flowing through a pipe. The water has a certain density, and it moves with a certain velocity. The amount of water passing a cross-section per second is the product of the density and the velocity. In electromagnetism, the "substance" flowing is the electromagnetic field itself. The energy is stored locally in the electric and magnetic fields, much like potential and kinetic energy in a mechanical oscillator. When these fields oscillate and propagate, they carry this stored energy away from the source. The rate at which this energy flows through a unit area perpendicular to the direction of propagation is the electromagnetic power flux. Furthermore, because the field exerts forces on charges (the Lorentz force), it must possess momentum. Just as a stream of water hitting a paddle exerts a force, an electromagnetic wave hitting a surface exerts radiation pressure. This lecture formalizes these intuitive ideas into precise conservation laws.
Local Energy Conservation
The fundamental statement of energy conservation in electromagnetism is local, not just global. It asserts that energy cannot simply disappear from one point in space and appear at another; it must flow continuously through the intervening space. This is captured by a continuity equation. The rate of change of the electromagnetic energy density \( u \) at a point, plus the net outward flux of energy from that point, equals the rate at which the field does work on the charges present at that point. If there are no charges, the energy is strictly conserved within the field.
The energy density \( u \) is the sum of the electric and magnetic contributions. For a linear medium in vacuum, this is given by:
The flow of energy is described by the Poynting vector \( \vec{S} \), which points in the direction of energy propagation. Its magnitude represents the power per unit area. The local conservation law is:
Here, \( \vec{J} \cdot \vec{E} \) is the power per unit volume transferred from the field to the matter. The negative sign indicates that if the field does positive work on the charges, the field energy decreases.
The Poynting Vector
To satisfy the continuity equation derived above, the Poynting vector must be defined as the cross product of the electric and magnetic fields, scaled by the permeability of free space:
This vector is always perpendicular to both \( \vec{E} \) and \( \vec{B} \). In a plane wave, where \( \vec{E} \) and \( \vec{B} \) are orthogonal and in phase, \( \vec{S} \) points exactly in the direction of wave propagation \( \vec{k} \). The magnitude of \( \vec{S} \) is proportional to the product of the field amplitudes. Since \( B = E/c \) in vacuum, we can express the instantaneous power flux solely in terms of the electric field:
Because the fields oscillate sinusoidally, the instantaneous Poynting vector oscillates at twice the frequency of the wave. For most practical applications, such as calculating the power of a laser beam, we are interested in the time-averaged value. For a sinusoidal wave with peak electric field amplitude \( E_0 \), the time-averaged Poynting vector is:
This result is crucial for engineering calculations, as it relates the measurable field amplitude directly to the transmitted power.
Momentum and Radiation Pressure
Just as energy is conserved, momentum is also conserved. The electromagnetic field carries momentum. The momentum density \( \vec{g} \) of the field is proportional to the Poynting vector:
This expression mirrors the relativistic relation between energy and momentum for a massless particle, \( p = E/c \). When an electromagnetic wave is absorbed by a surface, it transfers its momentum to that surface, resulting in a force. This is known as radiation pressure. For a perfectly absorbing surface, the pressure \( P \) is equal to the time-averaged intensity \( I \) (which is the magnitude of \( \langle \vec{S} \rangle \)) divided by \( c \):
If the surface is perfectly reflecting, the momentum change is doubled (the wave reverses direction), so the pressure is \( 2I/c \). This effect, while tiny for everyday light intensities, is significant in astrophysics, where it shapes the tails of comets and the structure of nebulae.
The derivations
Poynting's Theorem and Field Energy
Starting from the microscopic Maxwell equations and the Lorentz force law, we derive the local conservation law for electromagnetic energy: the rate of change of the field energy density \( u = \tfrac{1}{2}\varepsilon_0 E^2 + \tfrac{1}{2\mu_0} B^2 \) plus the divergence of the Poynting vector \( \vec{S} = \tfrac{1}{\mu_0}\,\vec{E}\times\vec{B} \) equals minus the rate at which the fields do work on charges per unit volume, \( \partial u/\partial t + \nabla\cdot\vec{S} = -\,\vec{J}\cdot\vec{E} \). Both \(u\) and \(\vec{S}\) emerge from the derivation rather than being postulated.
Structure of Monochromatic Plane Waves
In a source-free, linear, homogeneous, isotropic, non-dispersive medium (vacuum unless stated), every monochromatic plane-wave solution of Maxwell's equations, E(r,t) = E0 ei(k·r − ωt) and B(r,t) = B0 ei(k·r − ωt), is strictly transverse: k·E = k·B = 0. The triad (E, B, k) is mutually orthogonal and right-handed, the two fields oscillate in phase, their amplitudes obey |E| = c|B|, and the complex vector E0 encodes the polarization state — linear, circular, or elliptical.
A common misconception is that the energy density \( u \) is always positive definite in a way that implies the electric and magnetic energies are independent. While \( u \) is indeed positive, the instantaneous values of the electric and magnetic energy densities in a plane wave are equal at every point in space and time. This is a specific feature of plane waves in vacuum. In standing waves, however, the energy sloshes back and forth between electric and magnetic forms, and the Poynting vector oscillates in direction, leading to no net energy flow over a cycle. Do not confuse the time-averaged flow of a traveling wave with the local oscillations of a standing wave. C
Worked example
Problem. A plane electromagnetic wave in vacuum has a peak electric field amplitude of \( E_0 = 10.0\ \mathrm{V/m} \). Calculate the time-averaged intensity of the wave and the radiation pressure exerted on a perfectly absorbing surface perpendicular to the propagation direction. Use \( c = 3.00 \times 10^8\ \mathrm{m/s} \) and \( \varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{F/m} \).
- Calculate the time-averaged Poynting vector magnitude (Intensity) using \( I = \frac{1}{2} \varepsilon_0 c E_0^2 \).
- Calculate the radiation pressure using \( P = I/c \) for a perfectly absorbing surface.
Units check. \( \varepsilon_0 \) is \( \mathrm{C^2/(N \cdot m^2)} \), \( c \) is \( \mathrm{m/s} \), and \( E_0^2 \) is \( \mathrm{V^2/m^2} = \mathrm{J^2/(C^2 \cdot m^2)} \). The product yields \( \mathrm{J/(s \cdot m^2)} = \mathrm{W/m^2} \). Pressure is Force/Area, \( \mathrm{N/m^2} = \mathrm{Pa} \). The units are consistent.
Three quick tests
- What is the direction of the Poynting vector relative to the electric and magnetic fields in a plane wave?
Answer
It is perpendicular to both, pointing in the direction of propagation. - If the electric field amplitude of a wave is doubled, by what factor does the intensity increase?
Answer
It increases by a factor of four, since intensity is proportional to the square of the amplitude. - Why is the radiation pressure on a perfect reflector twice that of a perfect absorber?
Answer
Because the momentum change of the photon is twice as large when it is reflected compared to when it is absorbed.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-166 · D-165 · Threads: force, energy, light