Radiation Pressure

Electromagnetic radiation exerts a slight pressure on matter when it is absorbed or reflected: intensity = I, speed of light = c, pressure = I/c when absorbed and 2I/c when reflected. Two derivations will be shown here, one based on the photon picture and the other on the wave aspect of radiation. Nature has this duality: light is both a wave and a particle.

Ditto for the particles—electrons, etc. have wave and particle characteristics.

Photons are pulses of light of energy hf, where h is Planck’s constant and f is frequency. Set hf equal to mc² and divide by c to solve for momentum mc. So momentum of a photon is its energy divided by c. In mechanics we learn that force is rate of change of momentum. If a bunch of photons are absorbed by an object, we see that the total force is the rate of energy absorbed (or power absorbed) divided by c. But intensity I is power/area, so as noted in paragraph 1, we have pressure = I/c. Reflection doubles the rate of change of momentum of the photons, so this doubles the pressure. QED. Problem: what if the radiation is 1/3 absorbed and 2/3 reflected? Solution: simply add the two: pressure = (1/3)I/c + (2/3)2I/c.

Now for the wave stuff: Consider a wave going horizontally in the +x direction, hitting a vertical surface. The electric field shakes the electrons, and the magnetic field applies a horizontal force to the electrons. Consider an electric field E in the + y direction (up) and the magnetic field B in the +z direction (toward you). (Note- the direction light travels is the direction of ExB, the cross product.) Current in the surface will be in the +y direction at the instant being considered (electrons moving down). Verify for yourself that the magnetic force is in the + x direction. The y force on electrons is eE, where e is electron charge. Power absorbed is given by P = F_yv = eEv, where v is electron velocity. F_x = evB, so we have F_x = PB/E. Elsewhere it is shown that in SI units, E = cB, so F_x = P/c. Divide both sides by area, and the left is pressure and the right is I/c. QED. 2I/c makes sense for the reflected wave, but I won’t attempt to derive it explicitly.

The derivation above involved instantaneous power, but I is normally defined as average power per area. Not a problem, valid for instantaneous- valid for average.

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