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When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light and the direction in which the ray is traveling change. (This is why a fish under water is in a different position from where it appears to be.) The changes are given by Snell's law, $$\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$$ where \(c_{1}\) is the speed of light in the first medium, \(c_{2}\) is the speed of light in the second medium, and \(\theta_{1}\) and \(\theta_{2}\) are the angles shown in the figure below. In Exercises assume that \(c_{1}=3 \times 10^{8}\) meters per second. (Figure cant copy) Approximate the speed of light in the second medium. Find \(\theta_{2}\) for the following values of \(\theta_{1}\) and \(c_{2} .\) Round to the nearest degree. \(\theta_{1}=40^{\circ} ; c_{2}=1.5 \times 10^{8}\) meters per second

Step by step solution

01

Understand Snell's Law

Snell's law relates the ratios of the sines of the angles of incidence and refraction to the ratio of the speeds of light in two different media. The formula is \( \frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}} \) where \( c_1 \) is the speed of light in the first medium (air), \( c_2 \) is the speed of light in the second medium, \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

02

Identify Given Values

We are given that \( c_1 = 3 \times 10^8 \) m/s, \( \theta_1 = 40^\circ \), and \( c_2 = 1.5 \times 10^8 \) m/s. Our task is to find \( \theta_2 \).

03

Insert Values into Snell's Law

Insert the given values into the Snell's Law formula: \( \frac{3 \times 10^8}{1.5 \times 10^8} = \frac{\sin 40^\circ}{\sin \theta_2} \). Simplify the left side to get \( 2 = \frac{\sin 40^\circ}{\sin \theta_2} \).

04

Solve for \( \sin \theta_2 \)

To solve for \( \sin \theta_2 \), multiply both sides of the equation by \( \sin \theta_2 \) and divide by 2: \( \sin \theta_2 = \frac{\sin 40^\circ}{2} \). Calculate \( \sin 40^\circ \approx 0.6428 \), so \( \sin \theta_2 = \frac{0.6428}{2} = 0.3214 \).

05

Calculate \( \theta_2 \)

Calculate \( \theta_2 \) by finding the inverse sine (arcsin) of \( 0.3214 \). Using a calculator, \( \theta_2 = \text{arcsin}(0.3214) \approx 18.75^\circ \). Round to the nearest degree to get \( \theta_2 = 19^\circ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Refraction

Refraction is a fascinating optical phenomenon that occurs when light passes from one medium to another. Imagine a ray of sunlight entering water from air. This transition causes the light to change both direction and speed. The bending of light, known as refraction, explains why objects under water may appear to be at different locations than they actually are.
Snell's Law is the mathematical description that defines how refraction works. It tells us that the bending of light depends on the speeds of light in the two media, as well as the angles at which light enters and exits the materials. A critical part of understanding refraction is recognizing that it happens because light travels at different speeds in different materials. This change in speed causes a shift in direction, making Snell's Law essential in predicting how light behaves at interfaces.

Angle of Incidence

The angle of incidence is the angle formed between an incoming light ray and a line perpendicular to the surface it strikes, known as the normal. This angle is crucial because it significantly influences how much the light will bend when it enters a new medium.
In our exercise, the given angle of incidence is \( 40^{\circ} \). Understanding this angle helps us apply Snell's Law appropriately. By using the formula \(\frac{\sin \theta_1}{\sin \theta_2}\), we can relate the angle of incidence to the angle of refraction. This relationship allows us to predict how the light will travel in the second medium.
Knowing the initial angle of incidence enables us to control and manipulate light in practical applications, such as designing lenses and creating effects in optical devices.

Speed of Light

The speed of light varies depending on the medium through which it moves. In a vacuum, light travels at an astonishing speed of \(3 \times 10^8\) meters per second. However, when light enters materials like water or glass, its speed reduces.
In the given problem, we are dealing with two media: air and another medium where the speed of light is \(1.5 \times 10^8\) meters per second. The decrease in the speed of light as it travels from air into another medium significantly impacts how much the light will bend.
This concept of varying light speed is what Snell's Law utilizes. By examining the ratio of the speeds in the two media, we can determine the extent to which light will bend when switching mediums. This principle is not only important for understanding basic physics but also for practical applications in optics and telecommunications.