Problem 51 - Essential Calculus Early Transcendentals

Question

Let \(v_{1}\) be the velocity of light in air and \(v_{2}\) the velocity of light in water. According to Fermat's Principle, a ray of light will travel from a point \(A\) in the air to a point \(B\) in the water by a path \(A C B\) that minimizes the time taken. Show that $$\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{v_{1}}{v_{2}} $$ where \(\theta_{1}\) (the angle of incidence) and \(\theta_{2}\) (the angle of refraction) are as shown. This cquation is known as Snell's Law.

Step-by-Step Solution

Answer

Proof uses Fermat's Principle and calculus to derive Snell's Law: \(\frac{\sin \theta_{1}}{\sin \theta_{2}} = \frac{v_{1}}{v_{2}}\).

Step 1: Understand Fermat's Principle

Fermat's Principle states that light travels between two points along the path that requires the least time. In this context, light is traveling from point \(A\) in the air to point \(B\) in the water via an intermediate point \(C\) on the boundary between air and water.

Step 2: Define Variables and Equations

Define \(d_1\) as the horizontal distance from \(A\) to \(C\) and \(d_2\) as the horizontal distance from \(C\) to \(B\). Let \(h\) be the vertical distance from \(A\) to the surface (boundary line) and from the surface to \(B\). Then the total time \(T\) taken by light is \(T = \frac{d_1}{v_1}\cos\theta_1 + \frac{d_2}{v_2}\cos\theta_2\).

Step 3: Use Calculus to Minimize Time

Using calculus, Fermat’s Principle requires taking the derivative of the time function \(T\) with respect to the variable \(x\) (where \(x\) is the horizontal coordinate of point \(C\)) and setting it to zero. This gives \(\frac{d}{dx}(\frac{x}{v_1}+\frac{\sqrt{d_2^2+h^2-x^2}}{v_2}) = 0\).

Step 4: Apply Trigonometry

Relate trigonometric functions to the path: \(\sin\theta_1 = \frac{h}{d_1}\) and \(\sin\theta_2 = \frac{h}{d_2}\). Given the distances \(d_1 = x\) and \(d_2 = h^2 + (d-x)^2\), use calculus result \(\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}\).

Step 5: Conclude with Snell's Law

From our calculations, \(\frac{\sin \theta_{1}}{\sin \theta_{2}} = \frac{v_{1}}{v_{2}}\). This equation is known as Snell's Law and verifies that the path of light refraction follows this trigonometric relationship.

Key Concepts

Fermat's Principle

Understanding the behavior of light as it travels from one medium to another is crucial in optics. Fermat's Principle provides an elegant explanation for this phenomenon. It states that light will follow the path that takes the least amount of time when traveling between two points. Imagine light traveling from a point in the air, point A, to a point underwater, point B. Rather than taking a direct route, light often bends at the boundary between the two mediums, such as air and water.

The intermediate point C on this boundary becomes crucial. The path from point A to C to B is chosen because it takes less time than any other possible path. This principle is fundamental in deriving Snell's Law, as it introduces the concept of time minimization, which underlies many optical phenomena. Fermat's Principle simplifies the complex nature of light's interaction with different mediums by focusing on the element of time.

Light Refraction

Refraction is the bending of light as it passes from one medium into another where its speed is different. This change in speed causes the light to change direction at the boundary between the two mediums, such as air and water. Imagine a straw appearing bent when placed in a glass of water; this visual effect is due to refraction.

The angle at which light enters a new medium is called the angle of incidence (\(\theta_1\)), while the angle it bends to is called the angle of refraction (\(\theta_2\)). The amount of bending depends on the velocities of light in the two media, which are characterized by \(v_1\) in air and \(v_2\) in water in the given exercise.

Understanding refraction is crucial in many practical applications, like designing lenses for glasses and cameras. This bending not only influences everyday experiences but also forms the basis of many technological advancements.

Trigonometry

Trigonometry provides the tools necessary to quantify the behavior of light during refraction. In the context of Snell's Law, trigonometric functions help relate the angles of incidence and refraction to the speed of light in different media. The trigonometric function sine (\(\sin\)) is particularly important.

For instance, in the exercise's solution, \(\sin\theta_1\) and \(\sin\theta_2\) represent the sine of angles of incidence and refraction, respectively. These relationships allow us to connect various elements of the path taken by light as it transitions from one medium to another. By setting up the equations using these sine values, we can derive that \(\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}\).

Through such calculations, one can better understand not just how light behaves when crossing media boundaries, but also how to apply trigonometric principles to solve real-world problems. This begins with grasping basic trigonometric concepts but extends to utilizing them in practical optical designs and research.