Reflection and Refraction — Question 2

Question 2

Rotating a plane mirror doubles the reflected-ray rotation

In a fixed two-dimensional laboratory plane, a ray arrives with propagation direction angle alpha. A plane mirror has tangent direction angle beta. Angles are signed counterclockwise from a fixed axis; the ray stays on the same reflecting face during a small mirror rotation.

1. Derive the reflected propagation direction angle in terms of alpha and beta.

2. If the mirror rotates counterclockwise by 3 degrees while the incident direction is fixed, find the outgoing rotation. Find its sign if instead only the incident direction rotates by 3 degrees.

Question 2 - Solution

Rotating a plane mirror doubles the reflected-ray rotation

(a) Method and calculation

$$\mathbf k_r=\mathbf k_i-2(\mathbf k_i\cdot\hat n)\hat n,\quad\boxed{\alpha_r=2\beta-\alpha\pmod{2\pi}}.$$$$d\alpha_r=2\,d\beta-d\alpha.$$

(b) Interpretation and further calculation

$$d\alpha=0,\ d\beta=3^\circ\Rightarrow\boxed{d\alpha_r=6^\circ}.$$$$d\beta=0,\ d\alpha=3^\circ\Rightarrow\boxed{d\alpha_r=-3^\circ}.$$

Physical check

The factor of two refers to mirror rotation with the incoming ray fixed, not to a changing incident ray at a fixed mirror.