Question

Using Fermat's principle, prove that the angle of incidence is equal to the angle of reflection.

Answer 1

Fermat’s principle states that light travels between two points along the path that requires the least time. Using this Fermat’s principle, we can derive the law of reflection [i.e., the angle of incidence is equal to the angle of reflection].

Consider a light ray shown in the figure. A ray of light starting at point A reflects off the surface at point P before arriving at point B, a horizontal distance I from point A. We calculate the length of each path and divide the length by the speed of light to determine the time required for the light to travel between the two points.

$t=\frac{\sqrt{x^2+h_1^2}}{c}+\frac{\sqrt{(l-x{)}^2+h_2^2}}{c}$
To minimize the time, we set the derivative of the time with respect to x equal to zero.
$0=\frac{dt}{dx}=\frac{x}{c\sqrt{x^2+h_1^2}}+\frac{-(l-x)}{c\sqrt{(l-x{)}^2+h_2^2}}\Rightarrow \frac{x}{\sqrt{x^2+h_1^2}}=\frac{(l-x)}{\sqrt{(l-x{)}^2+h_2^2}}\Rightarrow sin{\theta }_1=sin{\theta }_2\Rightarrow {\theta }_1={\theta }_2$