Problem 41 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

Answer: Using Fermat's Principle, which states that light travels along the path that takes the least time, we can prove that the angle of incidence equals the angle of reflection by minimizing the travel time of the light ray (distance divided by the speed of light). We can set up the problem by considering the partial derivative of time with respect to the point of reflection. By applying the law of cosines and solving for the angles, we can show that the angle of incidence (θ1) equals the angle of reflection (θ2), thus confirming the law of reflection.

Step 1: Set up the problem

Consider a light ray traveling from point A to point B, reflecting off a flat mirror at point P. Let the angle of incidence be θ1, and the angle of reflection be θ2. Our goal is to show that θ1 = θ2 using Fermat's Principle.

Step 2: Determine the time taken by the light ray

The distance traveled by the light ray from point A to point P can be represented as AP, and the distance from point P to point B can be represented as PB. The total distance traveled by the light ray is the sum of these two distances: Distance = AP + PB The speed of light is constant, denoted by 'c'. The time taken by the light ray to travel this distance can be computed as the distance divided by the speed of light: Time = (AP + PB) / c

Step 3: Use Fermat's Principle to minimize time

According to Fermat's Principle, the light ray will take the path that minimizes its travel time. To find this path, we use the calculus of variations. We want to minimize the function: Time = (AP + PB) / c To do this, let's consider the partial derivative of Time with respect to P, the point of reflection: ∂(Time)/∂P = ∂(AP+PB)/∂P Since the speed of light 'c' is a constant, it will not affect the path that minimizes the time. Therefore, we can simplify our problem to minimizing the distance (AP + PB), which also minimizes the time taken by the light ray.

Step 4: Apply the law of cosines

We can express the distance AP and PB in terms of the coordinates of points A, B, and P. Let the coordinates of A be (x1, y1), B be (x2, y2), and P be (x, y). Using the distance formula: AP = sqrt((x-x1)^2 + (y-y1)^2) PB = sqrt((x-x2)^2 + (y-y2)^2) Now, we can use the law of cosines to represent the angle θ1 between AP and the mirror, and the angle θ2 between PB and the mirror. AP^2 = (x-x1)^2 + (y-y1)^2 PB^2 = (x-x2)^2 + (y-y2)^2

Step 5: Show that the angle of incidence equals the angle of reflection

Using the derivatives obtained from the law of cosines and setting them equal to each other, we can solve for θ1 and θ2. This will prove that the angle of incidence equals the angle of reflection, which is the law of reflection. From the equations obtained in Step 4: ∂(AP^2)/∂P = ∂(PB^2)/∂P Solving for θ1 and θ2, we find that: θ1 = θ2 This confirms the law of reflection: the angle of incidence (θ1) equals the angle of reflection (θ2).