Problem 47 - Arihant AIEEE Physics

Question

If a crown glass prism of refracting angle \(10^{\circ}\) have refractive indices for red and violet rays \(1.514\) and \(1.523\) respectively, then the dispersion caused by a crown glass prism is : (a) \(0.07^{\circ}\) (b) \(0.08^{\circ}\) (c) \(0.09^{\circ}\) (d) \(0.10^{\circ}\)

Step-by-Step Solution

Answer

The dispersion caused by the crown glass prism is (c) \(0.09^{\circ}\).

Step 1: Understanding Dispersion in Prisms

Dispersion in a prism occurs because different wavelengths of light refract by different amounts. This difference in deviation causes the dispersion. We can calculate this difference using the formula for the difference in deviation angles for the two colors of light, red and violet: \( \Delta D = (n_v - n_r) A \), where \( n_v \) and \( n_r \) are the refractive indices for violet and red rays, respectively, and \( A \) is the angle of the prism.

Step 2: Identifying Given Values

Identify the values given in the problem: the refracting angle \( A = 10^{\circ} \), refractive index for red rays \( n_r = 1.514 \), and refractive index for violet rays \( n_v = 1.523 \).

Step 3: Calculate the Difference in Refractive Indices

Calculate the difference between the refractive indices of violet and red light: \( n_v - n_r = 1.523 - 1.514 = 0.009 \).

Step 4: Calculate the Dispersion Angle

Use the formula \( \Delta D = (n_v - n_r) A \) to find the dispersion angle: \( \Delta D = 0.009 \times 10^{\circ} = 0.09^{\circ} \).

Key Concepts

Refractive Indices

Refractive indices are crucial when it comes to understanding how light interacts with different materials, like a prism. They are essentially a measure of how much light is bent, or refracted, when it passes through a material. This bending occurs because light travels at different speeds in different mediums.

• The refractive index is often denoted by the letter "n."
• In this context, we have two important refractive indices: one for red light ( _r = 1.514") and one for violet light ( _v = 1.523").

The reason these numbers differ is because different colors (or wavelengths) of light bend by different amounts when they enter a new medium. This property is what leads to dispersion, where white light is spread out into its component colors as it passes through a prism. Violet light, having a shorter wavelength, is refracted more than red light, which has a longer wavelength.

Prism Angle

The prism angle is often referred to as the refracting angle of the prism. In this problem, the prism angle is given as 10 degrees, denoted as "A = 10^{\circ}." This angle is significant because it helps determine how much the light will bend when passing through each face of the prism.

• The prism angle affects the path that light takes through the prism, influencing how much it bends at each interface.
• A larger prism angle typically results in more bending and, consequently, more dispersion since the light travels through more of the material.

This angle is also an essential factor in calculating the dispersion caused by the prism. By using the formula \( \Delta D = (n_v - n_r) A \), one can find how much the deviation, and therefore the dispersion, is influenced by both the difference in refractive indices and the prism angle.

Violet and Red Light

Violet and red light are two extremes of the visible spectrum, with violet having the shortest wavelength and red having the longest. This difference in wavelength means that they behave differently when passing through materials like a prism.

• Violet light has the highest refractive index, meaning it bends the most. In our example, \( n_v = 1.523 \).
• Red light has a lower refractive index, so it bends less. Here, \( n_r = 1.514 \).

Because violet light is bent more than red light, when white light (which contains all colors) enters a prism, the light spreads out into a spectrum ranging from red, at one end, to violet, at the other. This spreading is known as dispersion and is why we see a rainbow effect through prisms. Understanding how violet and red light are dispersed differently is key to knowing how various optical elements, such as lenses and prisms, function in applications like cameras and spectacles.