Plane Mirrors — Question 9

Question 9

The shortest reflected path obeys the image construction

A source is at (-a,0) and a receiver at (-b,H), with positive a and b on the same side of the mirror x=0. A candidate reflection point is (0,y).

1. Minimize the path length with respect to y and find the reflection height.

2. Find the minimum path length and check the equal-angle relation.

Question 9 - Solution

The shortest reflected path obeys the image construction

(a) Method and calculation

$$\ell(y)=\sqrt{a^2+y^2}+\sqrt{b^2+(H-y)^2},\quad\ell^\prime=\frac{y}{\sqrt{a^2+y^2}}-\frac{H-y}{\sqrt{b^2+(H-y)^2}}.$$$$\ell^\prime=0\Rightarrow y/a=(H-y)/b\Rightarrow\boxed{y_*=aH/(a+b)}.$$

(b) Interpretation and further calculation

$$\boxed{\ell_{\min}=\sqrt{(a+b)^2+H^2}}.$$$$\boxed{\tan\theta_i=y_*/a=H/(a+b)=(H-y_*)/b=\tan\theta_r}.$$

Physical check

Both square-root terms have positive second derivative, so this stationary path is the unique minimum.