Electromagnetic Waves, Antennas, and RF: Electromagnetic Momentum, Radiation Pressure, and Photon Momentum

EM17 established that electromagnetic fields carry energy. In vacuum the energy flux is described by the Poynting vector

(1)

and for a plane wave the time-averaged magnitude is the intensity

(2)

Energy transport is only part of the story. An electromagnetic wave can push on matter. Light can exert force on a mirror, radio waves transfer momentum when absorbed by an antenna or lossy material, and sunlight can accelerate a reflective sail in empty space. These effects require the electromagnetic field itself to carry momentum.

The central results of this article are

(3)

where g is electromagnetic momentum density, and for a vacuum plane wave

(4)

where u is total electromagnetic energy density. The resulting radiation pressure at normal incidence is

(5)

and

(6)

At the quantum level the same momentum transfer is described in terms of photons. A photon of frequency ν and wavelength λ has

(7)

The classical field and photon descriptions therefore agree on the momentum delivered per unit electromagnetic energy [1, 2, 3, 4, 5].

1 Why a wave must carry momentum

Momentum is the quantity whose transfer produces force. For ordinary matter,

(8)

If an electromagnetic wave changes the mechanical momentum of a material object, then total momentum conservation requires an equal and opposite change somewhere else. The missing momentum is carried by the electromagnetic field.

This is not merely a quantum effect. Classical Maxwell theory already contains a local momentum density and a momentum-balance law. Photon momentum will appear later as the quantum counterpart of the same physical conservation law.

2 From energy flow to electromagnetic momentum density

The Poynting vector has units

(9)

Dividing by c² gives

	=	(10)
	= .	(11)

Momentum per volume has units

(12)

which is the same dimension. Maxwell’s equations make this identification exact in vacuum:

(13)

Using

(14)

and

(15)

we obtain

g	= E × B	(16)
	= μ0𝜖0E × B.	(17)

Therefore

(18)

The total electromagnetic momentum inside a volume V is

(19)

3 A more rigorous Maxwell-equation origin of g

The expression for field momentum is not chosen only by dimensional analysis. It follows from combining Maxwell’s equations with the Lorentz force density.

The electromagnetic force per unit volume on matter is

(20)

Use Gauss’s Law,

(21)

and the Ampere–Maxwell law,

(22)

Substitution gives

f

= 𝜖0(∇⋅ E)E + (∇× B) × B − 𝜖0 × B.

(23)

Now differentiate the cross product E × B:

(24)

Therefore

(25)

Faraday’s law gives

(26)

so

(27)

Using the Vector Identities

(28)

and

(29)

with ∇⋅ B = 0, the force density can be collected into the form

(30)

where

(31)

and σ is the Maxwell stress tensor,

(32)

This is the electromagnetic momentum-balance law. The term ∂g∕∂t represents local change of field momentum, while the stress tensor describes momentum transfer through surfaces. The force density f is the rate at which field momentum is transferred to matter [1, 2].

4 Integral momentum balance

Integrate the local relation over a fixed volume V :

(33)

Applying the divergence theorem component by component gives

(34)

This equation is the momentum analogue of Poynting’s theorem. Poynting’s theorem tracks electromagnetic energy; the Maxwell stress tensor and field momentum density track electromagnetic momentum.

For many radiation-pressure calculations, we do not need to evaluate the full tensor. Once the wave is locally plane and propagating at speed c, momentum flux can be obtained directly from the energy flux.

5 Plane-wave relation between energy and momentum density

For a vacuum plane wave, EM17 showed that

(35)

where u is the instantaneous total electromagnetic energy density. Since

(36)

we immediately obtain

(37)

Vectorially,

(38)

where k points in the direction of propagation.

If a localized wave packet has total energy

(39)

then its field momentum is

PEM	= ∫ V gdV	(40)
	= ∫ V udV k.	(41)

Therefore

(42)

This is a completely classical result for a plane-wave packet in vacuum.

6 Momentum flux and radiation pressure

Suppose a normally incident plane wave delivers energy dU to a surface during time dt. If the wave is completely absorbed, the corresponding incident momentum is

(43)

The force on the surface is the momentum delivered per unit time:

(44)

If the illuminated area is A, then

(45)

Therefore

	=	(46)
	= .	(47)

Pressure is force per area, so for complete absorption,

(48)

The pressure is extremely small for ordinary terrestrial intensities because c is so large, but it is nonzero and directly measurable.

7 Why a mirror receives twice the pressure

For ideal reflection at normal incidence, the electromagnetic momentum reverses direction.

Take the incident momentum of an energy packet to be

(49)

After reflection,

(50)

The change in field momentum is

Δpfield	= pf − pi	(51)
	= − −	(52)
	= −.	(53)

The material receives the opposite momentum,

(54)

Hence

(55)

Reflection doubles the pressure because the normal component of electromagnetic momentum is reversed rather than merely removed from the wave.

8 Example 1: pressure from a 1000 W/m² beam

Consider normal incidence with

(56)

For complete absorption,

pabs	=	(57)
	≈ 3.34 × 10−6 Pa.	(58)

Thus

(59)

For ideal reflection,

(60)

The small magnitude explains why radiation pressure is easy to overlook in everyday mechanics, even though it becomes important in precision optical systems, microscopic particles, and spacecraft propulsion.

9 Force in terms of incident optical or RF power

For a beam normally incident on a surface and completely intercepted by it,

(61)

Therefore an absorbing surface experiences

(62)

while an ideal reflecting surface experiences

(63)

Notice that the force depends on total intercepted power, not directly on beam area. Beam area determines the pressure because pressure is force divided by area.

10 Example 2: force from a 5 W laser on a perfect mirror

For a perfectly reflecting mirror at normal incidence,

F	=	(64)
	=	(65)
	≈ 3.34 × 10−8 N.	(66)

Hence

(67)

The force is tiny, but modern force sensors can measure forces in this range and far below it.

11 Partially absorbing and partially reflecting surfaces

Let fractions A, R, and T of the incident power be absorbed, reflected, and transmitted, with

(68)

At normal incidence, assume the incident, reflected, and transmitted beams are all in vacuum and that the transmitted radiation continues in the original propagation direction. This avoids the additional momentum-partition issues that arise inside material media. The incoming momentum flux is

(69)

The outgoing momentum flux is

(70)

Therefore the momentum delivered to the material per unit area per unit time is

prad	= −	(71)
	= .	(72)

Since A = 1 − R − T,

(73)

This formula recovers both limiting cases:

(74)

and

(75)

12 Oblique incidence and the role of projected area

Suppose a plane wave strikes a flat surface at angle 𝜃 measured from the surface normal. If the actual surface area is A, the projected area normal to the beam is

(76)

Hence the incident power is

(77)

For complete absorption, the incident momentum delivered per unit time has magnitude

(78)

in the beam direction. Its normal component is

(79)

Thus the normal pressure on the actual surface area is

(80)

For ideal specular reflection, the tangential momentum component is unchanged while the normal component reverses, so

(81)

13 Electromagnetic momentum and the Maxwell stress tensor

Radiation pressure can also be read directly from the electromagnetic stress carried by the fields. For a plane wave propagating in the +z direction with

(82)

there are no z components of either field. The zz component of the Maxwell stress tensor is

σzz	= 𝜖0 +	(83)
	= −𝜖0E2 −	(84)
	= −u.	(85)

The negative sign indicates a compressive normal stress in this sign convention. Its magnitude is

(86)

For a plane wave,

(87)

so

(88)

After time averaging,

(89)

This is the same momentum-flux scale that appeared in the absorption argument. Reflection changes the boundary condition and doubles the momentum change of the radiation.

14 Radiation pressure as momentum flux

It is useful to compare energy flux and momentum flux side by side.

Energy flux:

(90)

Momentum flux for a vacuum beam:

(91)

Because force is momentum per time, momentum flux has the same dimensions as pressure:

(92)

This is why radiation pressure is naturally interpreted as electromagnetic momentum flux.

15 From classical field momentum to photon momentum

Maxwell’s equations describe electromagnetic waves classically and do not by themselves quantize electromagnetic energy. Quantum physics adds the Planck–Einstein relation

(93)

where h is Planck’s constant.

For a photon in vacuum, relativity gives the energy–momentum relation for a massless particle,

(94)

Therefore

(95)

Since

(96)

we have

(97)

This result is the quantum version of the classical plane-wave relation

(98)

One photon has

(99)

and a classical beam made of many photons has the same momentum-to-energy ratio.

16 Example 3: momentum of a 532 nm photon

For

(100)

and

(101)

photon momentum is

pγ	=	(102)
	=	(103)
	≈ 1.25 × 10−27 kg m/s.	(104)

Thus

(105)

The energy of the same photon is

Eγ	=	(106)
	≈ 3.73 × 10−19 J.	(107)

As required,

(108)

17 Photon flux reproduces classical radiation pressure

Suppose a monochromatic beam has intensity I. The number of photons crossing unit area per unit time is the photon flux

(109)

Each absorbed photon delivers momentum

(110)

Therefore the momentum delivered per unit area per unit time is

ṄApγ	=	(111)
	= .	(112)

Hence

(113)

For ideal reflection, each photon’s normal momentum changes by twice as much, giving

(114)

Thus the photon-counting and classical-field calculations give the same radiation pressure.

18 Example 4: photon rate in a 1 W, 532 nm laser

The photon energy is approximately

(115)

A 1 W beam carries 1 J each second, so the photon rate is

Ṅ	=	(116)
	=	(117)
	≈ 2.68 × 1018 s−1.	(118)

Each photon carries only about 1.25 × 10⁻²⁷ kg m/s, but the enormous photon rate produces the macroscopic force

(119)

19 Frequency and wavelength dependence of single-photon momentum

Because

(120)

higher-frequency photons carry more momentum individually. Equivalently,

(121)

so shorter-wavelength photons carry more momentum individually.

However, for a beam of fixed total power P, the radiation force does not depend on photon frequency:

(122)

and therefore

Fabs	= Ṅ	(123)
	= .	(124)

A higher-frequency beam has fewer photons per second at fixed power, but each photon carries proportionally more momentum. The two effects cancel.

20 Solar sailing: a direct application of radiation pressure

A reflective spacecraft sail uses radiation pressure to create continuous thrust without expelling propellant. For a normally illuminated ideal mirror of area A exposed to intensity I,

(125)

For spacecraft mass M, the acceleration is

(126)

Even when the acceleration is tiny, it acts continuously and can accumulate a substantial velocity change over long times.

21 Example 5: idealized solar-sail acceleration

Take

(127)

For ideal normal reflection,

F	=	(128)
	=	(129)
	≈ 9.08 × 10−4 N.	(130)

Thus

(131)

The corresponding acceleration is

a	=	(132)
	≈ 9.08 × 10−5 m/s2.	(133)

This is only about 9.3 × 10⁻⁶ of standard terrestrial gravitational acceleration, but unlike a brief impulse, solar radiation pressure can act for months or years.

22 Radiation pressure and antennas

The same momentum physics applies at radio frequencies. An antenna interacting with an electromagnetic field absorbs, scatters, and reradiates energy and momentum. In many RF engineering calculations the mechanical force is negligibly small, so the momentum aspect is omitted. Nevertheless, the underlying field still carries momentum density

(134)

This is useful conceptually because energy flow, momentum flow, scattering, antenna force, and radiation pressure are not separate phenomena. They are different consequences of the same Maxwell fields.

23 A useful hierarchy of electromagnetic transport quantities

The quantities developed in EM17 and EM18 form a clean chain.

Field energy density:

(135)

Energy flux:

(136)

Time-averaged energy flux:

(137)

Momentum density:

(138)

Momentum flux and radiation-pressure scale:

(139)

For a plane wave in vacuum,

(140)

These equations make energy transport and momentum transport two parts of the same wave description.

24 Important qualification: momentum in material media

The vacuum relation

(141)

is unambiguous for electromagnetic fields in vacuum. In material media, separating total momentum into electromagnetic and material pieces is more subtle. Different useful momentum forms arise depending on how field and material momentum are partitioned, leading to the historical Abraham–Minkowski discussion.

The total momentum of the complete field-plus-matter system remains conserved. This article therefore keeps the derivations in vacuum, where the physics needed for radiation pressure and photon momentum is cleanest [6].

25 Common mistakes

• Confusing electromagnetic momentum density g with the Poynting vector S. They differ by a factor c² in vacuum.
• Writing g = u∕c² for a plane wave. The correct relation is g = u∕c; it is g = S∕c².
• Forgetting that radiation pressure is momentum flux, not energy density.
• Using p_rad = 2I∕c for an absorbing black surface. The factor of two belongs to ideal reflection.
• Forgetting that an oblique beam illuminates projected area A cos 𝜃.
• Using only one factor of cos 𝜃 for the normal pressure at oblique incidence. The second factor comes from taking the normal component of momentum.
• Treating the Maxwell stress tensor as a scalar pressure in arbitrary field configurations. It is a tensor because electromagnetic stresses can be directional and include shear components.
• Claiming that Maxwell’s classical equations derive energy quantization. The photon energy E_γ = hν is a quantum postulate/result, not a consequence of classical Maxwell theory alone.
• Confusing photon energy hν with photon momentum h∕λ.
• Assuming that higher photon energy means greater radiation force at fixed beam power. At fixed power, fewer higher-energy photons arrive per second, leaving F = P∕c for absorption.
• Applying vacuum momentum formulas inside material media without specifying how field and material momentum are being partitioned.

26 What EM18 adds to the series

EM17 established electromagnetic energy density, energy flux, intensity, and inverse-square spreading. EM18 adds the momentum carried by those same fields.

The vacuum electromagnetic momentum density is

(142)

For a plane wave,

(143)

Momentum flux produces radiation pressure:

(144)

At the quantum level,

(145)

These results connect Maxwell’s field theory to optical forces, radiation pressure, solar sailing, photon momentum, and eventually the microscopic interaction of electromagnetic radiation with matter.

References