Question

why the angle of reflection equal angle of incidence. Proof by mathematically​

Answer 1

Look at the below diagram.

Angle of Incidence

The ray I from a light source strikes the mirror MM’ at point O This ray is called as incident ray. ON is the perpendicular to the mirror MM’ at O which is also called the normal to the mirror at O. The angle ‘x’ made by the ray I, with the normal ON is called the ‘angle of incidence’.

The incident ray, is reflected by the mirror when it strikes the mirror at O and the ray after reflection ‘R’ is called as ‘reflected ray’. The angle ’y’ made by the reflected ray with the same ray is called ‘angle of reflection’.

Angle of Incidence and Reflection

As per law of reflection the angle of incidence is always equal to angle of reflection. This can be proved mathematically and also by a simple experiment. There are many mathematical proofs but the simplest proof is based on Fermat’s Principle.

Angle of Incidence Equals Angle of Reflection

Look at the above diagram. Let a right ray AO from a source A strike a plane mirror at O and reflected as ray OB. Let B be a point on ray OB and let ON be the normal to the mirror at O. The angle of incidence is ‘x’ and angle of reflection is ‘y’, as shown. Let ‘t’ be the time taken for the ray to travel the distance AO + OB and ‘c’ be the velocity of light, which is a constant.

The time ‘t’ can be worked out as,

t = AOc + OBc or,

t = a2+p2√c+(l−a)2+q2√c

As per Fermat's principle light takes the path of least time. Therefore, the derivative dtda must be 0.

That is, 0 = 1c[aa2+p2√+−(l−a)(l−a)2+q2√]

or, aa2+p2√ = (l−a)(l−a)2+q2√

But as per trigonometry, this is same as sin(x) = sin(y) or x = y .

Hence,

The Angle of Incidence = Angle of Reflection