Problem 134 - A Graphical Approach to Precalculus with Limits

Question

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. \(\theta_{1}=62^{\circ} ; c_{2}=2.6 \times 10^{8}\) meters per second

Step-by-Step Solution

Answer

\( \theta_{2} \approx 50^{\circ} \).

Step 1: Understand Snell's Law

Snell's Law relates the angles and speeds of light as it passes between two mediums. We have \( \frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}} \), where \( c_{1} \) and \( c_{2} \) are the speeds of light in the two media, and \( \theta_{1} \) and \( \theta_{2} \) are the angles of the light ray with respect to the normal in the first and second mediums, respectively.

Step 2: Substitute Known Values into Snell's Law

We know that \( c_{1} = 3 \times 10^{8} \) m/s, \( c_{2} = 2.6 \times 10^8 \) m/s, and \( \theta_{1} = 62^{\circ} \). Substitute these values into Snell's Law: \[\frac{3 \times 10^8}{2.6 \times 10^8} = \frac{\sin 62^{\circ}}{\sin \theta_{2}}\]

Step 3: Simplify the Speed Ratio

Calculate the ratio of speeds: \[\frac{3 \times 10^8}{2.6 \times 10^8} = \frac{3}{2.6} = 1.1538 \]

Step 4: Solve for \( \sin \theta_{2} \)

Using the simplified speed ratio, solve for \( \sin \theta_{2} \):\[1.1538 = \frac{\sin 62^{\circ}}{\sin \theta_{2}}\]\[\sin \theta_{2} = \frac{\sin 62^{\circ}}{1.1538}\]

Step 5: Calculate \( \sin 62^{\circ} \) and \( \sin \theta_{2} \)

Compute \( \sin 62^{\circ} \):\[\sin 62^{\circ} \approx 0.8829\]Substitute to find \( \sin \theta_{2} \):\[\sin \theta_{2} = \frac{0.8829}{1.1538} \approx 0.7651\]

Step 6: Find \( \theta_{2} \)

Calculate the angle \( \theta_{2} \) using the inverse sine function:\[\theta_{2} = \arcsin(0.7651)\]This yields \( \theta_{2} \approx 50^{\circ} \).

Key Concepts

Light Refraction

When light enters a new medium, such as moving from air into water or glass, its path bends. This bending effect is known as light refraction. The change in direction occurs because light travels at different speeds in different materials. Refraction is the reason why objects submerged in water appear to be in a different position. The bending can be toward or away from the normal line, depending on whether the light is speeding up or slowing down as it transitions from one medium to another.

Angle of Refraction

The angle of refraction refers to the angle between the refracted light ray and the normal, which is an imaginary line perpendicular to the boundary of the two media. This angle tells us how much the light path has been altered.
- **Snell's Law** allows us to calculate this angle, using the known speeds of light in each medium and the initial angle of incidence.
- In Snell's scenario, the angle of refraction helps us determine the path light will take when transitioning between differing refractive indices, such as air and water.
Calculating the angle of refraction is crucial for applications like designing lenses and optical instruments.

Media Transition

Media transition is the process of light moving from one medium to another, such as air to water. Each time light transitions, it can change direction and speed. This change occurs due to the difference in optical density between the two materials.
- **Optical Density Influence**: A medium with higher optical density will slow down light more, leading to a greater bending effect.
- **Boundary Surface**: The surface between two media is where refraction takes place, causing light to bend.
Knowing how light behaves across different media transitions is essential in understanding how lenses focus light, how glasses correct vision, and even how rainbows are formed.

Speed of Light in Different Media

The speed of light varies depending on the medium through which it travels. In a vacuum, light travels at approximately 3 x 10^8 meters per second, the fastest possible speed. However, when light enters other materials like glass or water, it slows down. This reduction in speed is crucial in refraction calculation.
- **Examples of Speed Changes**: In this exercise, light slows down from 3 x 10^8 m/s in air to 2.6 x 10^8 m/s in water.
- **Index of Refraction**: This is a measure of how much light slows down in a medium compared to a vacuum. It is calculated using the ratio of the speed of light in a vacuum to the speed in the medium.
Understanding how light speed changes in different media can help explain phenomena like underwater mirages and the focusing power of lenses.