Problem 56 - College Physics

Question

\(\bullet\) The indices of refraction for violet light \((\lambda=400 \mathrm{nm})\) and red light \((\lambda=700 \mathrm{nm})\) in diamond are 2.46 and \(2.41,\) respectively. A ray of light traveling through air strikes the diamond surface at an angle of \(53.5^{\circ}\) to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

Step-by-Step Solution

Answer

The angular separation is \(0.6^\circ\).

Step 1: Understanding Snell's Law

Snell's Law relates the angle of incidence and refraction with the indices of refraction. It is given by:\[n_1 \sin \theta_1 = n_2 \sin \theta_2\]where \(n_1\) is the index of refraction of the first medium (air), \(n_2\) is the index of refraction of the second medium (diamond), \(\theta_1\) is the angle of incidence, and \(\theta_2\) is the angle of refraction.

Step 2: Applying Snell's Law for Violet Light

For violet light, the refractive index in diamond is 2.46. The angle of incidence \(\theta_1 = 53.5^\circ\), and the refractive index in air (\(n_1\)) is approximately 1. Using Snell's Law:\[\sin \theta_v = \frac{n_1 \sin 53.5^\circ}{n_2} = \frac{1 \times \sin 53.5^\circ}{2.46}\]Calculate \(\theta_v\) using a calculator.

Step 3: Calculating the Refracted Angle for Violet Light

Using the previously derived equation:\[\sin \theta_v = \frac{1 \times \sin 53.5^\circ}{2.46} \approx 0.4177\]Therefore:\[\theta_v \approx \sin^{-1}(0.4177) \approx 24.7^\circ\]

Step 4: Applying Snell's Law for Red Light

For red light, the refractive index in diamond is 2.41. Using Snell's Law:\[\sin \theta_r = \frac{n_1 \sin 53.5^\circ}{n_2} = \frac{1 \times \sin 53.5^\circ}{2.41}\]Solve for \(\theta_r\).

Step 5: Calculating the Refracted Angle for Red Light

Using the equation for red light:\[\sin \theta_r = \frac{1 \times \sin 53.5^\circ}{2.41} \approx 0.4273\]Therefore:\[\theta_r \approx \sin^{-1}(0.4273) \approx 25.3^\circ\]

Step 6: Finding the Angular Separation

The angular separation \( \Delta \theta \) between the violet and red light is the difference between their refracted angles:\[\Delta \theta = \theta_r - \theta_v = 25.3^\circ - 24.7^\circ = 0.6^\circ\]

Key Concepts

Index of Refraction

The index of refraction (or refractive index) is a fundamental property that describes how light propagates through a medium. It's represented by the symbol "n" in Snell's Law. When light enters a new medium, its speed changes, which causes the light to bend. The index of refraction is calculated as the ratio of the speed of light in vacuum to the speed of light in the medium.

For example, in this exercise:

• The refractive index for violet light in diamond is 2.46.
• The refractive index for red light in diamond is 2.41.
• The refractive index of air is close to 1, since light travels almost as fast as in vacuum.

A higher refractive index means that light slows down more in that medium.

Angle of Incidence

The angle of incidence is the angle that an incoming ray of light makes with the normal (perpendicular) to the surface at the point of incidence. This angle is crucial in calculating how the light will behave when it enters a different medium.

In this exercise, the angle of incidence is given as 53.5° when light strikes the diamond. The normal is an imaginary line perpendicular to the surface, and this concept is essential for accurate calculations following Snell's Law. The angle of incidence helps determine the angle at which the light travels through the second medium and affects its angle of refraction.

Angle of Refraction

The angle of refraction is the angle that the refracted ray makes with the normal as it passes from one medium into another. It can be calculated by applying Snell's Law:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]where:

• \( n_1 \) and \( n_2 \) are the indices of refraction of the two media,
• \( \theta_1 \) is the angle of incidence,
• \( \theta_2 \) is the angle of refraction.

In this exercise:

• The angle of refraction for violet light is approximately 24.7°.
• The angle of refraction for red light is approximately 25.3°.

These calculations demonstrate how different wavelengths of light (colors) bend at different angles due to their different indices of refraction.

Angular Separation

Angular separation refers to the difference in refraction angles of two rays of light passing through a prism or any refractive medium. It shows how much two different colors of light are separated after passing through a material.

Angular separation is important in phenomena such as dispersion, where different wavelengths of light (each with different refractive indices) spread out, forming a spectrum. In this exercise, the angular separation between the violet and red light is calculated as:\[ \Delta \theta = \theta_r - \theta_v = 25.3° - 24.7° = 0.6° \]This means the refracted rays of violet and red light emerge at angles differing by 0.6°, which is a crucial concept in understanding how prisms and lenses manipulate light.