Question

The refractive index of the material of a small angled prism is \( 1.6 \). If the angle of minimum deviation is \( 4.2^\circ \), the angle of the prism is?

• A. \( 4.2^\circ \)
• B. \( 7^\circ \)
• C. \( 4.8^\circ \)
• D. \( 9^\circ \)

Hint:

For small-angle prisms, use the approximation \( \sin x \approx x \) (in radians) to simplify calculations in the prism formula.

Answer 1

The Correct Option is B

Step 1: Use the prism formula 
For a small-angled prism, the refractive index (\( n \)) is related to the angle of the prism (\( A \)) and the angle of minimum deviation (\( D_m \)) by the formula: \[ n = \frac{\sin \left(\frac{A + D_m}{2} \right)}{\sin \left(\frac{A}{2} \right)} \] Given: \[ n = 1.6, \quad D_m = 4.2^\circ \] For small angles (in degrees), we approximate: \[ \sin x \approx x \text{ (in radians)} \] Step 2: Solve for \( A \) 
Rewriting the equation: \[ 1.6 = \frac{\left(\frac{A + 4.2}{2} \right)}{\left(\frac{A}{2} \right)} \] \[ 1.6 \times \frac{A}{2} = \frac{A + 4.2}{2} \] Multiplying by 2: \[ 1.6 A = A + 4.2 \] \[ 1.6A - A = 4.2 \] \[ 0.6A = 4.2 \] \[ A = \frac{4.2}{0.6} = 7^\circ \] Thus, the angle of the prism is \( 7^\circ \).