Radiation Pressure — Question 10

Question 10

A symmetric lossless deflector feels force despite zero heating

A stationary optical device receives vacuum power P along +z and sends P/2 into each of two outgoing beams making angles plus and minus theta from +z in the xz plane. There is no absorption or backward beam.

1. Find the net force using vector momentum balance.

2. Evaluate the longitudinal force at theta=60 degrees and its small-angle approximation.

Question 10 - Solution

A symmetric lossless deflector feels force despite zero heating

(a) Method and calculation

$$\dot{\mathbf p}_{\mathrm{out}}=\frac{P}{2c}[(\sin\theta\hat x+\cos\theta\hat z)+(-\sin\theta\hat x+\cos\theta\hat z)].$$$$\boxed{\mathbf F=\frac{P}{c}(1-\cos\theta)\hat z,\quad F_x=0}.$$

(b) Interpretation and further calculation

$$\boxed{F_z(60^\circ)=P/(2c),\quad\dot Q=0}.$$$$1-\cos\theta\simeq\theta^2/2\Rightarrow\boxed{F_z\simeq P\theta^2/(2c)}\quad(\theta\ll1).$$

Physical check

The output power equals the input power, but its longitudinal momentum is smaller; the device receives the difference.