# The push of light

Light carries momentum as well as energy: its momentum is its energy divided by the speed of light. A surface that absorbs one watt of light is pushed with about 3.34 × 10⁻⁹ newtons, and a mirror that reflects it straight back is pushed twice as hard.

Written for this edition, not translated from Einstein.

## How hard does a beam of light push on what it hits?

Sunlight on your hand warms it but does not seem to push it. It does push. Light carries momentum as well as energy, and hands it to whatever absorbs it. The push is tiny: full sunlight, about 1,000 watts on each square metre, presses on a square metre of black cloth with about 3.3 millionths of a newton.

Maxwell's electrodynamics gives light this momentum. For light travelling in one direction, the momentum it carries is its energy divided by the speed of light, c:

$$
p = \frac{E}{c}
$$

Momentum equals energy divided by c.

A force is momentum delivered per second. A beam that delivers power P, energy per second, to a surface that absorbs it therefore pushes with force P/c. A mirror that sends the light straight back reverses its momentum and so receives twice as much each second, 2P/c. The flow of energy in a beam, in watts per square metre, is the Poynting vector, after John Henry Poynting (1884).

The push is tiny, and it was measured before 1905: by Pyotr Lebedev in 1901, and by Ernest Nichols and Gordon Hull between 1901 and 1903. Both reported forces that agreed with Maxwell's value within their experimental error.

Einstein's mass-energy paper of September 1905 does not use this momentum. It compares the energy of two emitted light pulses as reckoned from two frames, using the energy transformation of §8 of the relativity paper. Light's momentum enters his argument in 1906, in his paper on the motion of the centre of gravity, where light crossing a closed box carries momentum from one end to the other.

## Worked example: The push of a one-watt beam, absorbed and reflected

1. A surface absorbs a beam of 1 W: one joule of light energy arrives each second.
2. Each second it receives momentum of 1 J divided by c, which is 1 ÷ 299 792 458, about 3.34 × 10⁻⁹ kilogram metres per second.
3. Momentum received per second is a force, so the push is about 3.34 × 10⁻⁹ newtons.
4. A mirror that reflects the same beam straight back receives twice the momentum each second: about 6.67 × 10⁻⁹ newtons.

$$
F_{\text{absorbed}} = \frac{P}{c}, \qquad F_{\text{reflected}} = \frac{2P}{c}
$$

The force on an absorbing surface is the power divided by c; on a mirror that reflects the light straight back, it is twice the power divided by c.

For scale, 3.34 × 10⁻⁹ newtons is about the weight of a third of a microgram. That is why the push was first measured in 1901, nearly thirty years after Maxwell predicted it in his Treatise of 1873.

## Where this lesson stops

Light's momentum is the push a beam gives when absorbed or reflected. It follows from Maxwell's electrodynamics, with no quantum hypothesis.

## This lesson builds on

- [Functions and graphs](/foundations/functions-graphs/)
- [Rates of change and derivatives](/foundations/derivatives/)