Law of reflection by Fermat’s principle | Optics

Law of reflection by Fermat’s principle

Fermat's principle

According to Fermat’s principle of extremum path, a ray of light in passing from one point to another through a set of media by any number of reflections or refractions selects a path for which the time taken is either minimum or maximum.

Law of reflection by Fermat's principle

Since whole path is travelled by light in air.
So, the optical path between A and B is
l = AO + OB
l = √{a² + x²} + √{b² + (c - x)²)}
From Fermat’s principle, the position of point O is such the optical path l should be either maximum or minimum.
dl/dx = 0
Therefore

∴ sin i = sin r
or ∠ i = ∠ r

Conclusion

Since i is angle of incidence and r is angle of reflection
So, for reflection form any surface, the angle of incidence should be equal to the angle of reflection.
The incident ray, reflected ray and normal to the reflecting surface all should lie in the same plane and that plane is plane of incidence.

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