Radiation Pressure — Question 8

Question 8

Unequal beam spots exert a radiation torque on a mirror

Two narrow vacuum beams travel along +z and reflect normally from a mirror in the xy plane. Their powers are PL at x=-a and PR at x=+a. Let total power be P. Neglect mirror motion.

1. Find net force and torque about the y axis.

2. Invert the torque relation for the two powers and find the maximum possible torque magnitude for fixed P.

Question 8 - Solution

Unequal beam spots exert a radiation torque on a mirror

(a) Method and calculation

$$\boxed{F_z=2(P_L+P_R)/c=2P/c}.$$$$\mathbf r\times\mathbf F:\quad\boxed{\tau_y=2a(P_L-P_R)/c}.$$

(b) Interpretation and further calculation

$$\boxed{P_L=P/2+c\tau_y/(4a),\quad P_R=P/2-c\tau_y/(4a)}.$$$$P_L,P_R\ge0\Rightarrow\boxed{|\tau_y|\le2aP/c}.$$

Physical check

At the left spot, minus x crossed with plus z points along plus y; equal spot powers cancel torque but retain force.