Photons Have Momentum — Question 3

Question 3

Oblique reflection introduces separate momentum and projected-area factors

A beam of intensity $\mathcal I$ measured per area perpendicular to the beam strikes a perfect stationary mirror at angle $\theta$ from the surface normal. A mirror patch has area A. Reflection is specular.

1. Derive the normal force and pressure on the patch.

2. Evaluate their factors relative to normal incidence at $\theta=60^\circ$.

Question 3 - Solution

Oblique reflection introduces separate momentum and projected-area factors

(a) Method and calculation

$$P_{\mathrm{hit}}=\mathcal I A\cos\theta,\qquad\Delta p_n=2(E_\gamma/c)\cos\theta.$$$$\boxed{F_n=2\mathcal I A\cos^2\theta/c},\qquad\boxed{\mathcal P_n=2\mathcal I\cos^2\theta/c}.$$

(b) Interpretation and further calculation

$$\boxed{F_n(60^\circ)/F_n(0)=\cos^260^\circ=1/4}.$$$$\boxed{\mathcal P_n(60^\circ)=\mathcal I/(2c)}.$$

Physical check

One cosine comes from photon momentum normal to the mirror; the other comes from the illuminated projected area.