10.03.05 · em-sr / electrodynamics

Energy and momentum in the electromagnetic field: the Poynting vector and radiation pressure

shipped3 tiersLean: none

Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.7-6.8; Landau & Lifshitz, Classical Theory of Fields, 4th ed. (1975), Ch. 4.6-4.8

Intuition Beginner

Electromagnetic fields carry energy. When sunlight warms your skin, energy has travelled from the sun to you through empty space as an electromagnetic wave. The quantity that measures this energy flow is the Poynting vector $S$ , named after John Henry Poynting.

The Poynting vector points in the direction the energy is flowing, and its magnitude is the power per unit area. For an electromagnetic wave in vacuum, $∣ S ∣$ is the intensity: the number of watts passing through each square metre.

Fields also carry momentum. When light hits a surface, it exerts a tiny force — this is radiation pressure. For a beam of light hitting a perfect mirror, the pressure is $2 I / c$ where $I$ is the intensity and $c$ is the speed of light. This pressure is tiny for everyday light sources (about $1 0^{- 5}$ Pa for bright sunlight) but becomes significant for intense lasers or in astrophysical settings like the interior of stars.

The energy conservation law for electromagnetism is Poynting's theorem: the rate of energy leaving a volume equals the rate at which the field energy inside decreases, minus the rate at which the field does work on charges. This is the electromagnetic version of the first law of thermodynamics.

Visual Beginner

Quantity	Symbol	Meaning	Units
Energy density	$u$	Energy stored per cubic metre	J/m $^{3}$
Poynting vector	$S$	Energy flow per unit area	W/m $^{2}$
Momentum density	$g$	Momentum per cubic metre	kg/(m $^{2}$ s)
Radiation pressure	$P_{rad}$	Force per unit area from light	Pa

Worked example Beginner

The intensity of sunlight at Earth's surface is approximately $I = 1000$ W/m $^{2}$ . The Poynting vector magnitude is $∣ S ∣ = 1000$ W/m $^{2}$ .

The radiation pressure on a perfectly absorbing surface is $P_{rad} = I / c = 1000/ (3 \times 1 0^{8}) \approx 3.3 \times 1 0^{- 6}$ Pa. This is about $3 \times 1 0^{- 11}$ atmospheres — far too small to feel, but measurable with sensitive instruments.

For a perfect mirror (reflecting all light back), the pressure doubles to $2 I / c \approx 6.7 \times 1 0^{- 6}$ Pa because the momentum change is twice as large. This principle is used in solar sails: a large reflective sheet in space can be pushed by sunlight alone, requiring no fuel.

Check your understanding Beginner

Formal definition Intermediate+

Electromagnetic energy density. In vacuum, the energy stored in the electromagnetic field per unit volume is:

u = \frac{1}{2} (ϵ_{0} E^{2} + \frac{B ^{2}}{μ _{0}})

The first term is the electric energy density and the second is the magnetic energy density.

Poynting vector. The energy flux (power per unit area) carried by the electromagnetic field is:

S = \frac{1}{μ _{0}} E \times B

Poynting's theorem. The electromagnetic energy conservation law is:

\frac{d u}{d t} + \nabla \cdot S = - J \cdot E

In integral form, for a volume $V$ bounded by surface $S$ :

\frac{d}{d t} \int_{V} u d τ + \oint_{S} S \cdot d a = - \int_{V} J \cdot E d τ

The left side is the rate of change of stored energy plus the energy flowing out through the boundary. The right side is the rate at which the field does work on the charges (the power delivered to the charges, with a minus sign because it reduces the stored field energy).

Electromagnetic momentum. The momentum density of the electromagnetic field is:

g = ϵ_{0} E \times B = \frac{S}{c ^{2}}

For a plane wave, $∣ g ∣ = u / c$ and the momentum is in the direction of propagation.

Maxwell stress tensor. The electromagnetic force per unit area on a surface is described by the Maxwell stress tensor 10.03.03:

T_{ij} = ϵ_{0} (E_{i} E_{j} - \frac{1}{2} δ_{ij} E^{2}) + \frac{1}{μ _{0}} (B_{i} B_{j} - \frac{1}{2} δ_{ij} B^{2})

The total electromagnetic force on the charges in volume $V$ is:

F = \oint_{S} T \cdot d a - ϵ_{0} \frac{d}{d t} \int_{V} (E \times B) d τ

The first term is the stress integrated over the boundary (the "field pressure"); the second is the rate of change of electromagnetic momentum inside $V$ .

Radiation pressure. For a plane wave of intensity $I$ incident on a surface:

Perfect absorber: $P_{rad} = I / c = u$ (the energy density at the surface)
Perfect reflector (normal incidence): $P_{rad} = 2 I / c = 2 u$

For oblique incidence at angle $θ$ : $P_{rad} = (2 I cos^{2} θ) / c$ for a reflector.

Key derivation Intermediate+

Derivation (Poynting's theorem from Maxwell's equations).

Theorem. Maxwell's equations in vacuum imply the energy conservation law $\partial u / \partial t + \nabla \cdot S = - J \cdot E$ where $u = (1/2) (ϵ_{0} E^{2} + B^{2} / μ_{0})$ and $S = E \times B / μ_{0}$ .

Proof. Start with the Lorentz force law: the rate of work done by the field on a charge distribution is $d W / d t = \int J \cdot E d τ$ . We want to express this in terms of the fields alone.

Use the Maxwell-Ampere law to eliminate $J$ : $J = (1/ μ_{0}) \nabla \times B - ϵ_{0} \partial E / \partial t$ . Then:

J \cdot E = \frac{1}{μ _{0}} (\nabla \times B) \cdot E - ϵ_{0} \frac{\partial E}{\partial t} \cdot E

The second term is $- (\partial / \partial t) (ϵ_{0} E^{2} /2)$ . For the first term, use the vector identity $(\nabla \times B) \cdot E = B \cdot (\nabla \times E) - \nabla \cdot (E \times B)$ . By Faraday's law, $\nabla \times E = - \partial B / \partial t$ , so:

\frac{1}{μ _{0}} (\nabla \times B) \cdot E = - \frac{1}{μ _{0}} B \cdot \frac{\partial B}{\partial t} - \frac{1}{μ _{0}} \nabla \cdot (E \times B) = - \frac{\partial}{\partial t} \frac{B ^{2}}{2 μ _{0}} - \nabla \cdot S

Combining:

J \cdot E = - \frac{\partial}{\partial t} (\frac{ϵ _{0} E ^{2}}{2} + \frac{B ^{2}}{2 μ _{0}}) - \nabla \cdot S = - \frac{\partial u}{\partial t} - \nabla \cdot S □

Bridge. Poynting's theorem builds toward the covariant energy-momentum conservation law of relativistic electrodynamics. The foundational insight is that the electromagnetic field carries energy and momentum as a continuum, with the Poynting vector playing the role of the energy current. The central message is that energy conservation in electromagnetism is not an additional postulate but a consequence of Maxwell's equations. This is exactly what one expects from Noether's theorem applied to time-translation symmetry. Putting these together, the Poynting vector determines the radiated power of any antenna or radiating system 10.07.01, the Maxwell stress tensor generalises to the covariant stress-energy tensor 10.06.03, and the momentum density provides the basis for understanding radiation pressure in astrophysical and engineering contexts.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

A coaxial cable of inner radius $a$ and outer radius $b$ carries a steady current $I$ (out on the inner, back on the outer) at voltage $V$ . Compute the Poynting vector between the conductors and show that $\int S \cdot d a = V I$ (the power transmitted).

Hint

Between the conductors: $E = V / (r ln (b / a)) \hat{r}$ (radial) and $B = μ_{0} I / (2 π r) \hat{ϕ}$ (azimuthal). Compute $S = E \times B / μ_{0}$ .

Answer

The fields between the conductors: $E = \frac{V}{r l n ( b / a )} \hat{r}$ and $B = \frac{μ _{0} I}{2 π r} \hat{ϕ}$ . The Poynting vector is:

S = \frac{1}{μ _{0}} E \times B = \frac{1}{μ _{0}} \frac{V}{r ln ( b / a )} \frac{μ _{0} I}{2 π r} \hat{r} \times \hat{ϕ} = \frac{V I}{2 π r ^{2} ln ( b / a )} \hat{z}

The power flowing through a cross-section at fixed $z$ is:

P = \int_{a}^{b} S \cdot \hat{z} 2 π r d r = \frac{V I}{ln ( b / a )} \int_{a}^{b} \frac{d r}{r} = \frac{V I}{ln ( b / a )} ln \frac{b}{a} = V I

The power $P = V I$ is carried entirely by the electromagnetic field in the space between the conductors, not through the wires themselves. The wires guide the field; the energy flows in the gap.

Exercise 5 (medium, symbolic).

Compute the radiation pressure at the surface of the sun. The sun radiates $P = 3.8 \times 1 0^{26}$ W from a surface of radius $R = 7 \times 1 0^{8}$ m. Assume blackbody absorption.

Hint

$I = P / (4 π R^{2})$ . Then $P_{rad} = I / c$ for absorption at the surface (actually the radiation emerging from inside).

Answer

The intensity at the sun's surface: $I = P / (4 π R^{2}) = 3.8 \times 1 0^{26} / (4 π \times 4.9 \times 1 0^{17}) \approx 6.2 \times 1 0^{7}$ W/m $^{2}$ .

The radiation pressure at the surface: $P_{rad} = I / c = 6.2 \times 1 0^{7} / (3 \times 1 0^{8}) \approx 0.21$ Pa.

This is small compared to the gas pressure inside the sun ( $\sim 1 0^{14}$ Pa), but radiation pressure dominates in very massive stars ( $M > 50 M_{⊙}$ ), where it can exceed the gas pressure and drives the stellar wind. In stars above the Eddington limit ( $L > 1.3 \times 1 0^{31} (M / M_{⊙})$ W), radiation pressure overcomes gravity and the star blows itself apart.

Exercise 6 (hard, symbolic).

Derive the momentum conservation law for the electromagnetic field: $\partial g / \partial t = \nabla \cdot T - ρ E - J \times B$ , where $T$ is the Maxwell stress tensor.

Hint

Start from the Lorentz force density $f = ρ E + J \times B$ , eliminate $ρ$ and $J$ using Maxwell's equations, and rearrange into a conservation law.

Answer

The force density on the charges is $f = ρ E + J \times B$ . Use Gauss's law ( $ρ = ϵ_{0} \nabla \cdot E$ ) and Ampere's law ( $J = \nabla \times B / μ_{0} - ϵ_{0} \partial E / \partial t$ ) to eliminate $ρ$ and $J$ :

f = ϵ_{0} (\nabla \cdot E) E + \frac{1}{μ _{0}} (\nabla \times B) \times B - ϵ_{0} \frac{\partial E}{\partial t} \times B

The last term is $- \partial g / \partial t$ plus a term from $\partial B / \partial t = - \nabla \times E$ :

- ϵ_{0} \frac{\partial E}{\partial t} \times B = - \frac{\partial}{\partial t} (ϵ_{0} E \times B) + ϵ_{0} E \times \frac{\partial B}{\partial t} = - \frac{\partial g}{\partial t} - ϵ_{0} E \times (\nabla \times E)

After expanding all terms and using vector identities, the result is:

f = - \frac{\partial g}{\partial t} + \nabla \cdot T

where $T_{ij} = ϵ_{0} (E_{i} E_{j} - δ_{ij} E^{2} /2) + (1/ μ_{0}) (B_{i} B_{j} - δ_{ij} B^{2} /2)$ . Rearranging: $\partial g / \partial t = \nabla \cdot T - f$ , which is the stated result. The field momentum changes due to the stress at the boundary minus the momentum transferred to the charges.

Exercise 7 (hard, symbolic).

The Abraham-Minkowski controversy concerns the correct expression for the momentum of light inside a material medium. The Minkowski momentum is $p_{M} = n p_{vacuum}$ and the Abraham momentum is $p_{A} = p_{vacuum} / n$ , where $n$ is the refractive index. Explain why both are correct in different senses.

Hint

The total momentum includes both the electromagnetic field momentum and the mechanical momentum of the medium. Different divisions between field and medium give different "field momenta."

Answer

The total momentum of the system (field + medium) is conserved and unambiguous. The controversy is about how to divide this total between the "field part" and the "medium part."

Minkowski momentum $p_{M} = n E / c$ (for a photon): this is the canonical (crystal) momentum — it is the momentum that enters the conservation law for the photon momentum in a periodic medium (the analogue of crystal momentum in solid state physics). It is correct for describing refraction and diffraction.

Abraham momentum $p_{A} = E / (n c)$ : this is the kinetic momentum — the momentum that produces the centre-of-mass motion of the system. It is correct for describing the recoil of a material object when light passes through it.

The resolution (established experimentally in the 2000s): both are correct, they answer different questions. The Minkowski momentum governs the photon's wavevector $k = nω / c$ inside the medium. The Abraham momentum governs the mechanical force on the medium. The total is conserved; the split depends on what you measure.

Exercise 8 (hard, symbolic).

A spherical charge distribution of total charge $q$ and radius $R$ has a uniform electric field outside. Compute the total electromagnetic energy stored outside the sphere and show that it diverges for a point charge ( $R \to 0$ ). Find the classical electron radius $r_{e} = e^{2} / (4 π ϵ_{0} m c^{2})$ by setting the electrostatic energy equal to the rest energy $m c^{2}$ .

Hint

The field outside is $E = q / (4 π ϵ_{0} r^{2})$ . Integrate the energy density $u = ϵ_{0} E^{2} /2$ from $R$ to infinity.

Answer

The energy density outside the sphere: $u = ϵ_{0} E^{2} /2 = q^{2} / (32 π^{2} ϵ_{0} r^{4})$ . The total energy outside:

U = \int_{R}^{\infty} u 4 π r^{2} d r = \frac{q ^{2}}{8 π ϵ _{0}} \int_{R}^{\infty} \frac{d r}{r ^{2}} = \frac{q ^{2}}{8 π ϵ _{0} R}

For $R \to 0$ : $U \to \infty$ . The self-energy of a point charge is infinite in classical electrodynamics — this is the self-energy problem 10.07.04.

Setting $U = m c^{2}$ : $q^{2} / (8 π ϵ_{0} r_{e}) = m c^{2}$ , giving $r_{e} = q^{2} / (8 π ϵ_{0} m c^{2})$ . For the electron ( $q = e$ ): $r_{e} = e^{2} / (8 π ϵ_{0} m_{e} c^{2}) \approx 1.4 \times 1 0^{- 15}$ m (using the convention where the self-energy includes a factor of $3/5$ for a uniform sphere; the standard value is $r_{e} = e^{2} / (4 π ϵ_{0} m_{e} c^{2}) \approx 2.8 \times 1 0^{- 15}$ m). This is the size the electron would need to be for its electrostatic self-energy to equal its rest mass. Quantum mechanics renders this classical picture obsolete, but the classical electron radius appears in Thomson scattering 10.07.06.

Lean formalization Intermediate+

Mathlib has the divergence theorem but does not contain the Poynting vector, Poynting's theorem as a derived identity from Maxwell's equations, the Maxwell stress tensor, or the radiation pressure formula. Formalising these would require systematic development of vector calculus identities applied to the Maxwell system. lean_status: none.

Advanced results Master

Poynting's theorem in matter. In a linear, isotropic medium with permittivity $ϵ$ and permeability $μ$ :

\frac{\partial}{\partial t} \frac{1}{2} (ϵ E^{2} + B^{2} / μ) + \nabla \cdot (E \times H) = - J \cdot E

For dispersive media (where $ϵ$ and $μ$ depend on frequency), the energy density has additional terms involving $\partial (ω ϵ) / \partial ω$ and $\partial (ω μ) / \partial ω$ . This is the Brillouin formula for the energy density in a dispersive medium, which is essential for understanding the group velocity and energy transport in waveguides 10.04.04.

The relativistic energy-momentum tensor. The 16 components of $T^{μν}$ encode the energy density ( $T^{00} = u$ ), the energy flux ( $T^{0 i} = c (S / c)_{i}$ ), the momentum density ( $T^{i 0} = c g_{i}$ ), and the stress ( $T^{ij} = - T_{ij}$ ). The covariant conservation law $\partial_{μ} T^{μν} = 0$ is the 4-dimensional version of Poynting's theorem and the momentum conservation law combined. This is developed in unit 10.06.03.

Radiation pressure in astrophysics. The Eddington luminosity $L_{Edd} = 4 π GM m c / σ_{T}$ is the maximum luminosity of a star of mass $M$ before radiation pressure overcomes gravity ( $σ_{T}$ is the Thomson cross-section). For the sun: $L_{Edd} \approx 3.3 \times 1 0^{31}$ W, about 100 times the actual solar luminosity. For massive stars ( $M > 50 M_{⊙}$ ), radiation pressure drives strong stellar winds that strip away the outer layers.

Synthesis. The energy-momentum formalism for the electromagnetic field provides the bridge between Maxwell's equations and the mechanical conservation laws. The foundational insight is that the field is not merely a mathematical device for computing forces but a physical entity that carries energy and momentum through space. The central message is that Poynting's theorem is a consequence of Maxwell's equations, not an additional law. Putting these together, the Poynting vector quantifies radiation power 10.07.01, the Maxwell stress tensor leads to the covariant stress-energy tensor 10.06.03, and the momentum density underlies radiation pressure in everything from solar sails to stellar interiors.

Full proof set Master

Proposition (Poynting vector of a plane wave). A monochromatic plane wave in vacuum with fields $E = E_{0} cos (k z - ω t) \hat{x}$ and $B = (E_{0} / c) cos (k z - ω t) \hat{y}$ has time-averaged Poynting vector $⟨ S ⟩ = (E_{0}^{2} /2 μ_{0} c) \hat{z}$ and time-averaged energy density $⟨ u ⟩ = ϵ_{0} E_{0}^{2} /2$ .

Proof. $S = E \times B / μ_{0} = (E_{0}^{2} / μ_{0} c) cos^{2} (k z - ω t) \hat{z}$ . The time average of $cos^{2}$ is $1/2$ , giving $⟨ S ⟩ = E_{0}^{2} / (2 μ_{0} c) \hat{z}$ . The instantaneous energy density is $u = (1/2) (ϵ_{0} E^{2} + B^{2} / μ_{0}) = ϵ_{0} E_{0}^{2} cos^{2} (k z - ω t)$ (using $B^{2} / μ_{0} = ϵ_{0} E^{2}$ for a plane wave). The time average is $⟨ u ⟩ = ϵ_{0} E_{0}^{2} /2$ . The relation $⟨ S ⟩ = ⟨ u ⟩ c$ holds: $E_{0}^{2} / (2 μ_{0} c) = (ϵ_{0} E_{0}^{2} /2) c$ , confirmed by $μ_{0} ϵ_{0} c^{2} = 1$ . $□$

Connections Master

Poynting-Maxwell stress tensor 10.03.03 was introduced earlier; this unit develops the energy and momentum conservation laws that follow from it.
Larmor formula 10.07.01 computes radiated power by integrating the Poynting vector over a large sphere.
Covariant stress-energy tensor 10.06.03 is the relativistic version of $u$ , $S$ , and $T$ combined into a single 4-tensor.
Maxwell equations in matter 10.03.04 modify Poynting's theorem for material media with $ϵ$ and $μ$ .
Plane waves 10.04.02 are the simplest system for which the Poynting vector can be computed explicitly.

Historical & philosophical context Master

Poynting's theorem was derived by John Henry Poynting in 1884, the same year Heaviside independently derived the same result. The theorem resolved a long-standing question: where does the energy carried by electromagnetic waves reside? Poynting showed that energy is stored in the field itself and flows through space as described by $S$ .

The concept of electromagnetic momentum was more controversial. J. J. Thomson (1893) showed that a beam of light carries momentum, and Lebedev (1901) and Nichols and Hull (1903) independently measured radiation pressure experimentally, confirming the prediction.

The Abraham-Minkowski controversy (1908-1910) about the correct momentum of light in a medium persisted for a century. The modern resolution (clarified experimentally by She, Chen, and others in the 2000s) is that both expressions are correct in different contexts: Minkowski for canonical (wavevector) momentum and Abraham for kinetic (centre-of-mass) momentum.

Bibliography Master

Poynting, J. H., "On the Transfer of Energy in the Electromagnetic Field," Phil. Trans. Roy. Soc. 175, 343-361 (1884).
Lebedev, P., "Untersuchungen ueber die Druckkraefte des Lichtes," Ann. Phys. 6, 433-458 (1901).
Abraham, M., "Zur Elektrodynamik bewegter Koerper," Rend. Circ. Mat. Palermo 28, 1-28 (1909).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975).

Prerequisites

10.03.03
10.04.01

Used in

10.06.03
10.07.01

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 8.1-8.2
intermediate: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 8.1-8.2; Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.7
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.7-6.8; Landau & Lifshitz, Classical Theory of Fields, 4th ed. (1975), Ch. 4.6-4.8

References

Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 8.1 Energy conservation: the Poynting vector and electromagnetic energy density; Ch. 8.2 Momentum conservation: the Maxwell stress tensor
Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 6.7 Conservation laws for the electromagnetic field; the Poynting theorem; the Maxwell stress tensor and electromagnetic momentum
Landau & Lifshitz — The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975) · Ch. 4.6 The energy-momentum tensor of the electromagnetic field; Ch. 4.7 The virial theorem; Ch. 4.8 The energy-momentum tensor in a medium

Estimated time

beginner: 15m
intermediate: 35m
master: 55m