Momentum and radiation pressure

Light pushes

When you stand in sunlight you do not feel a push. Yet light does push. Maxwell predicted it in 1873 from his equations, and it was measured about 1900 by Pyotr Lebedev, and by Ernest Nichols and Gordon Hull a year later. The push is tiny because $c$ is huge.

Momentum of a wave

A wave that carries energy $U$ in the direction of travel also carries momentum. For a wave in vacuum,

This agrees with the photon picture: a photon has $E = hf$ and $p = h/\lambda = E/c$.

Suppose a beam of intensity $I$ falls straight on an area $A$ for a time $t$ and is absorbed. The energy is $U = IAt$, so the momentum gained by the surface is $p = IAt/c$. The force is the momentum each second:

Here $p_{rad}$ is the radiation pressure. It is the same quantity as the average energy density $\bar{u}$, since $I = \bar{u}c$.

If the light is reflected straight back by a perfect mirror, the momentum changes from $+p$ to $-p$. The change is $2p$, so the force and the pressure are twice as large:

For a surface that reflects a fraction $\rho$ and absorbs the rest, $p_{rad} = (1 + \rho)\,I/c$.

A worked example. Sunlight with $I = 1361$ W/m² falls straight on a black sheet and on a mirror.

1. The black sheet. $p = \dfrac{I}{c} = \dfrac{1361\ \text{W/m}^2}{3.00 \times 10^{8}\ \text{m/s}} = 4.54\ \mu\text{Pa}$.
2. The mirror. $p = \dfrac{2I}{c} = 9.07\ \mu\text{Pa}$.
3. Compare. Atmospheric pressure is about $10^5$ Pa, so the pressure of sunlight is about $10^{-10}$ of it.

Set the reflecting fraction to $0$ and then to $1$. By what factor does the force change? Then raise the angle of incidence. At $60^\circ$, what fraction of the straight-on force is left? It is $\cos^2 60^\circ = 0.25$.

A solar sail

A solar sail is a very large, thin mirror. Light bounces off it and pushes it. There is no fuel to carry, so the push goes on for months.

A worked example. A sail has an area of $2500$ m² and a total mass of $12$ kg. It is a perfect mirror and faces the Sun straight on, near the Earth's distance.

1. The force. $F = \dfrac{2IA}{c} = \dfrac{2 \times 1361\ \text{W/m}^2 \times 2500\ \text{m}^2}{3.00 \times 10^{8}\ \text{m/s}} = 22.68\ \text{mN}$.
2. The acceleration. We want $a$, so $a = \dfrac{F}{m} = \dfrac{0.02268\ \text{N}}{12\ \text{kg}} = 1.89\ \text{mm/s}^2$.
3. After 30 days. The time is $2.592 \times 10^{6}$ s. Then $v = at = 4.9$ km/s, and the sail has moved $\tfrac{1}{2}at^2 = 6.3 \times 10^{6}$ km.

A speed of a few kilometres a second after a month, with no fuel. (We took the push as constant. In fact it falls as $1/d^2$ when the sail goes away from the Sun.)

A laser pointer

A small laser pointer sends out $5.0$ mW. If a black target absorbs all of it, $F = \dfrac{P}{c} = \dfrac{0.005\ \text{W}}{3.00 \times 10^{8}\ \text{m/s}} = 16.7\ \text{pN}$. This is far too small to feel. A strong pulse is a different matter, as the next lesson shows.

An oblique beam

Now let a beam of power $P$ strike a flat surface at an angle of incidence $\theta$, measured from the normal. The beam brings momentum $P/c$ each second along its direction. We split it into a part along the normal, $(P/c)\cos\theta$, and a part along the surface, $(P/c)\sin\theta$.

• A perfect mirror. The part along the surface does not change. The part along the normal reverses. The force is normal to the mirror, and $F = \dfrac{2P\cos\theta}{c}$.
• A black surface. All of the momentum is taken in. There is a normal force $\dfrac{P\cos\theta}{c}$ and a force along the surface $\dfrac{P\sin\theta}{c}$.

A worked example. A beam of power $2.4$ W meets a surface at $50^\circ$ to the normal.

1. A mirror. $F = \dfrac{2P\cos\theta}{c} = \dfrac{2 \times 2.4\ \text{W} \times \cos 50^\circ}{3.00 \times 10^{8}\ \text{m/s}} = 10.28\ \text{nN}$, along the normal.
2. A black surface. The normal part is $5.14$ nN and the part along the surface is $6.13$ nN. The two make a force of $8.00$ nN along the beam, which is $P/c$.

If you write the result with the intensity $I$ of the beam and the area $A$ of the surface, the beam covers an area $A\cos\theta$ across its width. Then the normal force on a mirror is $\dfrac{2IA\cos^2\theta}{c}$, which is the formula of the simulation.

Where radiation pressure matters

• Comet tails. A comet's dust tail points away from the Sun, because sunlight pushes the small dust grains. The gas (ion) tail is pushed by the solar wind.
• Small dust. For very small grains the push of light is larger than the pull of gravity, so they are blown out of the Solar System. The two forces both fall as $1/r^2$, so the winner is the same at every distance.
• Stars. In large, hot stars the pressure of the light helps to hold the star up against its own weight.
• Laser cooling and tweezers. Light can slow atoms and hold tiny beads. These uses depend on the momentum of photons.

Practise with new numbers

Try it in Python

The program finds the force on a solar sail for four areas, then the acceleration, the speed after 30 days and the distance covered, with the push taken as constant.

Common mistakes

• Using $I/c$ for a mirror. A perfect mirror reverses the momentum, so the pressure is $2I/c$.
• Forgetting the angle. For a beam at an angle, the normal force on a mirror carries $\cos^2\theta$ when written with the intensity of the beam.
• Mixing the force and the pressure. $p_{rad}$ is in pascal. The force is $p_{rad}A$ in newton.
• Using the pressure of sunlight at any distance. The intensity of sunlight falls as $1/d^2$ with the distance $d$ from the Sun, and so does the push.

Remember these

• A wave of energy $U$ carries momentum $p = U/c$.
• Absorbed: $F = IA/c$ and pressure $I/c$. Reflected straight back: twice that. Partly reflected: $(1 + \rho)I/c$.
• For an oblique beam of power $P$ on a mirror: $F = 2P\cos\theta/c$, along the normal.
• Sunlight at the Earth: $4.54\ \mu\text{Pa}$ on a black surface, $9.07\ \mu\text{Pa}$ on a mirror.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Light of intensity $2000$ W/m² falls straight on a perfect mirror. What is the radiation pressure?

2. A beam of power $40$ W is absorbed by a target. What force does it exert?

3. Why does a perfect mirror feel twice the force of a black surface in the same light?

4. The sail of the example is moved to twice the distance from the Sun. What happens to its acceleration?

References

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves): momentum and radiation pressure.
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves): the momentum carried by a wave.