The angular dispersion of a prism depends on

Analysis

The question asks for the factors that determine the angular dispersion of a prism. Angular dispersion is the difference in the angle of deviation between the extreme colors (usually red and violet) of a spectrum produced by a prism.

Concept Explanation

When white light passes through a prism, it splits into its constituent colors because the refractive index ($n$) of the glass varies with the wavelength of light. This phenomenon is called dispersion.

The angular dispersion ($\theta$) is given by the formula:

Where:

• $\delta_v$ is the angle of deviation for violet light.
• $\delta_r$ is the angle of deviation for red light.

For a thin prism, the angle of deviation ($\delta$) is calculated as:

Where:

• $A$ is the refracting angle of the prism.
• $n$ is the refractive index of the material.

However, for a prism in a general state (not necessarily thin), the deviation depends on the angle of incidence ($i$), the refracting angle of the prism ($A$), and the refractive index ($n$) of the material. Since the dispersion is the difference in deviations for different wavelengths, it is fundamentally dependent on how the refractive index changes with wavelength and the geometry of the light's path through the prism, which is dictated by the angle of incidence.

Step-by-Step Solution

1. Refractive Index: The refractive index ($n$) determines how much each color is bent. Since different colors have different refractive indices in the same medium, this is the primary cause of dispersion.
2. Angle of Incidence: The path the light takes through the prism is determined by the angle of incidence. If the angle of incidence changes, the deviation of the light rays changes. Consequently, the difference between the deviations of the extreme colors (the angular dispersion) is also affected by the angle of incidence.
3. Conclusion: Because the deviation of light rays is a function of both the material's properties (refractive index) and the orientation of the light entering the prism (angle of incidence), the angular dispersion depends on both.

Identifying the Answer

This is because the deviation of light through a prism is not constant; it varies with the angle of incidence, and therefore the spread (dispersion) between the colors is also dependent on the angle at which the light enters the prism.

Exam Tip

When dealing with prism questions, always remember the mnemonic "D-I-A": Deviation depends on the Degree of incidence (angle of incidence), the Index of refraction, and the Angle of the prism. If you are asked about dispersion, remember that it is simply the difference in deviation for two different wavelengths, so it inherits these dependencies!