Foundation lesson
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:
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
- A surface absorbs a beam of 1 W: one joule of light energy arrives each second.
- 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.
- Momentum received per second is a force, so the push is about 3.34 × 10⁻⁹ newtons.
- A mirror that reflects the same beam straight back receives twice the momentum each second: about 6.67 × 10⁻⁹ newtons.
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.
Try it: how hard the beam pushes
Set the power of the beam and say what the surface does with it. The push is the momentum the light delivers each second.
A beam of 1 W on a surface that absorbs it pushes with 3.3356 × 10−9 newtons. A surface that sent it straight back instead would feel 6.6713 × 10−9 N, a factor of 2.
What it shows, in words
The push is the power divided by the speed of light, and a mirror receives twice as much because it reverses the light's momentum instead of merely stopping it. At the setting above, 1 watts on a surface that absorbs the light, the push is 3.3356 × 10−9 newtons. Doubling the power doubles the push, and switching between the two surfaces changes it by exactly a factor of 2, at any power.
One watt absorbed gives about 3.34 × 10⁻⁹ newtons, which is the number the lesson prints and why the push went unnoticed for so long.
What it does not show: the surface is at rest and stays at rest. A mirror that is moving changes the light's frequency as well as its direction, which is §8 of the relativity paper and a harder problem than this; nothing here bears on it. Nor does it say anything about how the momentum gets there, only how much arrives each second.
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.
If you came here from a passage, Back returns you to the exact place you left.