Question

Use Fermat's Principle to derive the law of reflection.

Short answer

Answer: Using Fermat's Principle, which states that light travels between two points along the path that takes the least time, we can set up and calculate the time taken for light to travel from one point to another after reflecting off a surface. After calculating the total time taken and applying Fermat's Principle to differentiate and minimize the time, we can substitute Snell's Law condition. This result leads to the law of reflection, stating that the angle of incidence is equal to the angle of reflection when light encounters a reflective surface.

Step-by-step solution

Set up the problem

Consider a light ray that travels from point A to a point B after reflecting off a surface. Let point C be the point on the surface where the light ray reflects, and the separating line between A and C be the angle of incidence (θi). The angle of reflection (θr) is between the separating line of C and B. The light ray covers equal distances in the two mediums, so the time taken is the sum of the time taken in each medium.

Calculate time taken from A to C

The distance from A to C is AC, and the speed of light in the medium is given by c. So, the time taken from A to C, denoted as t_1, will be t_1 = \frac{AC}{c}.

Calculate time taken from C to B

Similarly, the distance from C to B is CB, and the speed of light in the medium is given by c. So, the time taken from C to B, denoted as t_2, will be t_2 = \frac{CB}{c}.

Calculate total time taken

The total time taken, denoted as T, is the sum of the time taken from A to C and C to B. So, T = t_1 + t_2 = \frac{AC}{c} + \frac{CB}{c}.

Apply Fermat's Principle

According to Fermat's Principle, the path taken by the light from A to B should be the one that minimizes the total time T. Therefore, the derivative of T with respect to AC and CB should be equal to zero. In order to minimize T, let's consider an auxiliary variable h, the distance between points A and B. The time T can now be written as: T = \frac{\sqrt{AC^2 + h^2}}{c} + \frac{\sqrt{CB^2 + h^2}}{c}. Now, we differentiate T with respect to h and set it to zero to minimize the time: \frac{d}{dh}\left(\frac{\sqrt{AC^2 + h^2}}{c} + \frac{\sqrt{CB^2 + h^2}}{c}\right) = 0.

Apply the Snell's Law condition

Substituting Snell's Law condition, that is, h = AC * sinθi = CB * sinθr, and simplifying the equation in step 4, it would result in: θi = θr. As a result, Fermat's Principle has been used to derive the law of reflection, which states that the angle of incidence is equal to the angle of reflection when light encounters a reflective surface.