Question

Why angle of incidence of a ray is always equal to the angle of reflection?

Answer 1

Fermat's principle:

1. According to the principle, the light follows the shortest path or path that requires the least time of travel.
2. Suppose a light from source $A$ reflects from a surface to destination $B$, the shortest distance between $A$ and $B$ is a straight line.

Explanation:

1. Consider reflection of a light ray $AP$ at the point of incidence $P$. $PB$ is the reflected ray.${\theta }_1$and ${\theta }_2$ are angles of incidence and reflection respectively. The distance between source $A$ and destination $B$ is $l$. The distances $A$ between and $B$ from the reflecting surface are $h_1$ and $h_2$ respectively. $c$ is the speed of light used to calculate the time taken by the light ray to travel a certain distance at point $P$.
2. $\mathrm{Total}\mathrm{time}=\mathrm{Time}\mathrm{to}\mathrm{travel}\mathrm{distance}AP+\mathrm{Time}\mathrm{to}\mathrm{travel}\mathrm{distance}PB$

$t=\frac{\sqrt{x^2+{h_1}^2}}{c}+\frac{\sqrt{{\left(1-x\right)}^2+{h_2}^2}}{c}$

1. As the time taken is minimum,
   $$\begin{gathered} \Rightarrow \frac{dt}{dx}=0 \\ \frac{x}{c\sqrt{x^2+{h_1}^2}}+\frac{-(1-x)}{c\sqrt{{\left(1-x\right)}^2+{h_2}^2}}...\left(1\right) \end{gathered}$$
2. With reference to the angles in the diagram equation (1) can be written as,

$$\begin{gathered} \frac{\sin {\theta }_1}{c}-\frac{\sin {\theta }_2}{c}=0 \\ \frac{\sin {\theta }_1}{c}=\frac{\sin {\theta }_2}{c} \\ \sin {\theta }_1=\sin {\theta }_2 \\ \Rightarrow {\theta }_1={\theta }_2 \end{gathered}$$

This proves that the angle of incidence is equal to the angle of reflection at $P$.

Conclusion:

1. According to Fermat's principle of the behavior of light, a light ray always selects the shortest path to reach the destination.
2. The same behavior is observed at the point of reflection. When a ray gets reflected from the plane, the angle of incidence is equal to the angle of reflection.