Radiation Pressure — Question 4

Question 4

A short pulse gives a mirror an impulse before appreciable displacement

A pulse of energy U strikes a free perfect mirror of mass M normally in vacuum. Initially the mirror is at rest and U is much smaller than Mc squared. Work to leading nonzero order in this small ratio.

1. Find mirror impulse and recoil speed, comparing with an absorber.

2. Find the leading recoil kinetic energy and its fraction of incident energy.

Question 4 - Solution

A short pulse gives a mirror an impulse before appreciable displacement

(a) Method and calculation

$$J_{\mathrm{mirror}}\simeq2U/c,\quad\boxed{v\simeq2U/(Mc)}.$$$$J_{\mathrm{absorber}}\simeq U/c,\quad J_{\mathrm{mirror}}/J_{\mathrm{absorber}}\simeq2.$$

(b) Interpretation and further calculation

$$K\simeq\frac{J^2}{2M}=\boxed{\frac{2U^2}{Mc^2}}.$$$$\boxed{K/U\simeq2U/(Mc^2)\ll1},\quad U_{\mathrm{reflected}}=U-K.$$

Physical check

Treating reflected energy as U is sufficient for the leading impulse; the small energy difference supplies recoil kinetic energy.