A light ray starts at the origin and is reflected off a...

A light ray starts at the origin and is reflected off a mirror along the line $y=1$ to the point (2,0) . See Figure 4.49. Fermat's Principle says that light's path minimizes the time of travel. $^{5}$ The speed of light is a constant. (a) Using Fermat's Principle, find the optimal position of $P$ (b) Using your answer to part (a), derive the Law of Reflection, that $\theta_{1}=\theta_{2}$

A light ray starts at the origin and is reflected off a mirror along the line $y=1$ to the point (2,0) . See Figure 4.49. Fermat's Principle says that light's path minimizes the time of travel. $^{5}$ The speed of light is a constant.
(a) Using Fermat's Principle, find the optimal position of $P$
(b) Using your answer to part (a), derive the Law of Reflection, that $\theta_{1}=\theta_{2}$

Key Concepts

Law of Reflection

The Law of Reflection, which states that the angle of incidence equals the angle of reflection, can be derived from the condition of minimizing the optical path length. By applying the method of reflection, one shows that the optimal reflected path makes equal angles with the normal of the reflective surface, ensuring that the time taken for the light to travel along this path is minimal. This law is a cornerstone in geometrical optics and is directly obtained from Fermat's Principle.

Method of Reflection (Mirror Image Method)

The method of reflection, or mirror image method, is a geometric technique where the reflection problem is transformed by reflecting one of the points across the mirror line. This converts the problem of finding the optimal path with a reflection into a problem of finding a straight-line path between the original point and the image point. This method simplifies the analysis and directly shows how the angles of incidence and reflection are related.

Fermat's Principle

Fermat's Principle states that light follows the path that requires the least time when traveling from one point to another. Since the speed of light is constant, minimizing the time is equivalent to minimizing the optical path length. This principle is fundamental in explaining the behavior of light in reflection and refraction phenomena.

Optical Path Length Minimization

Optical path length minimization involves finding the path that gives the smallest cumulative distance (or more generally, the least time if the speed of light is uniform) between two points. In optics, this is used to determine the path taken by light, leading to accurate predictions of its trajectory when conditions change, such as at the interface between different media.

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