Question

The refractive index of the material of a glass prism is \( 3 \). The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

• A. 50°
• B. 60°
• C. 58°
• D. 48°

Hint:

For prisms, when the angle of minimum deviation equals the angle of the prism, use the relation involving the refractive index to calculate the angle of the prism.

Answer 1

The Correct Option is B

To find the angle of a prism when the refractive index of its material is given, and the angle of minimum deviation is equal to the angle of the prism, we utilize the formula for the angle of deviation in a prism.

The formula relating the refractive index (\( n \)), the angle of the prism (\( A \)), and the angle of minimum deviation (\( \delta_m \)) is given by:

\(n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}\) 

Given:

• Refractive index, \( n = 3 \)
• Angle of minimum deviation, \( \delta_m = A \)

Substitute \( \delta_m = A \) in the formula:

\(n = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)}\)

Given \( n = 3 \), the equation becomes:

\(3 = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)}\)

We need to find the angle \( A \) such that the above equation holds.

By solving this equation using trigonometric identities and known values, we find:

\(A = 60^\circ\)

This solution matches with the provided options, confirming that the angle of the prism is indeed 60°.

Therefore, the correct answer is 60°.

Answer 2

For a prism, the angle of minimum deviation \( D_{\text{min}} \) is related to the refractive index \( n \) and the angle of the prism \( A \) by the following equation: \[ n = \frac{\sin\left( \frac{A + D_{\text{min}}}{2} \right)}{\sin\left( \frac{A}{2} \right)}. \] Given that the angle of minimum deviation is equal to the angle of the prism, i.e., \( D_{\text{min}} = A \), we can simplify the equation: \[ n = \frac{\sin\left( \frac{A + A}{2} \right)}{\sin\left( \frac{A}{2} \right)} = \frac{\sin(A)}{\sin\left( \frac{A}{2} \right)}. \] Substitute \( n = 3 \): \[ 3 = \frac{\sin(A)}{\sin\left( \frac{A}{2} \right)}. \] Solving for \( A \), we find that the angle of the prism is \( A = 60^\circ \).