Problem 15 - University Physics with Modern Physics

Question

A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58 . The angle of incidence is \(62.0^{\circ},\) and the ray originates in the glass with \(n=1.70 .\) Compute the angle of refraction.

Step-by-Step Solution

Answer

The angle of refraction is approximately \(71.82^{\circ}\).

Step 1: Understand Snell's Law

Snell's Law relates the angle of incidence and angle of refraction for a wave impinging on an interface between two media with different refractive indices, using the formula: \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \). Here, \( n_1 = 1.70 \) and \( n_2 = 1.58 \), and \( \theta_1 = 62.0^{\circ} \). We need to find \( \theta_2 \).

Step 2: Apply Snell's Law

Insert the given values into Snell's Law: \( 1.70 \sin(62.0^{\circ}) = 1.58 \sin(\theta_2) \). This equation will allow us to solve for \( \theta_2 \).

Step 3: Solve for \( \sin(\theta_2) \)

Calculate \( \sin(62.0^{\circ}) \) using a calculator to find \( 0.88295 \). Then substitute back into the equation: \( 1.70 \, \times \, 0.88295 = 1.58 \sin(\theta_2) \). Simplify to find \( \sin(\theta_2) = \frac{1.70 \, \times \, 0.88295}{1.58} \approx 0.949 \).

Step 4: Calculate \( \theta_2 \)

Now, use the inverse sine function to find \( \theta_2 \): \( \theta_2 = \arcsin(0.949) \). Using a calculator, find that \( \theta_2 \approx 71.82^{\circ} \).

Step 5: Double-Check Calculation

Finally, re-check the insertion of values and computation to ensure accuracy, reaffirming that the angle of refraction \( \theta_2 \) is approximately \( 71.82^{\circ} \).

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in the study of light and optics. It describes how light bends, or refracts, when it passes from one medium to another. This law is named after the Dutch mathematician, Willebrord Snellius. Snell's Law is represented by the equation:\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]Here:

• \(n_1\) is the refractive index of the first medium.
• \(\theta_1\) is the angle of incidence, the angle between the incoming ray and the normal to the interface.
• \(n_2\) is the refractive index of the second medium.
• \(\theta_2\) is the angle of refraction, the angle between the refracted ray and the normal.

This relationship tells us how the angle of the light ray changes as it enters a different medium. By understanding this law, we can predict how light will behave at the boundary between two materials. This principle is critical in designing lenses and understanding optical phenomena like rainbows.

Refractive Index

The refractive index of a material indicates how much the speed of light is reduced inside it compared to the speed in a vacuum. It is a dimensionless number typically greater than 1. Light travels slower in materials with higher refractive indices.

• A vacuum has a refractive index of exactly 1.
• In air, the refractive index is slightly more than 1, approximately \(1.0003\).
• Glass has a higher refractive index, often ranging between 1.5 to 1.7, depending on its composition.

When light passes from a medium with a higher refractive index, like glass, into one with a lower refractive index, like air, it speeds up and bends away from the normal. Conversely, light slows and bends towards the normal when moving into a medium with a higher refractive index. This bending of light is a foundational concept in optics.

Angle of Refraction

The angle of refraction is the angle between the refracted light ray and the normal to the surface at the point of refraction. It is crucial in determining how much a light ray bends when passing through mediums with different refractive indices.

• When light enters a denser medium (higher refractive index), it bends towards the normal, resulting in a smaller angle of refraction compared to the angle of incidence.
• However, when light exits a denser medium into a less dense one, the angle of refraction will be larger than the angle of incidence, causing the light to bend away from the normal.

Using Snell's Law, one can calculate the specific angle of refraction when the angle of incidence and both refractive indices are known. This calculation is essential for understanding optical devices like lenses and fiber optics, as they rely on precise control of light refraction.