Question

A small-angled prism is made of a material of refractive index \( \frac{3}{2} \). The ratio of the angles of minimum deviations when the prism is placed in air and in water of refractive index \( \frac{4}{3} \) is:

• A. \( 4:1 \)
• B. \( 3:4 \)
• C. \( 2:3 \)
• D. \( 1:3 \)

Hint:

Use the formula \( \delta_m = (\mu - 1) A \) for the angle of minimum deviation.
- The refractive index of the prism is relative to the surrounding medium, so adjust accordingly.

Answer 1

The Correct Option is A

The angle of minimum deviation for a prism is given by: \[ \delta_m = (\mu - 1) A \] where: - \( \mu \) is the refractive index of the prism with respect to the surrounding medium. - \( A \) is the prism angle. 1. When the prism is in air (\( \mu_{\text{air}} = 1 \)): \[ \mu_{\text{relative}} = \frac{\mu_{\text{prism}}}{\mu_{\text{air}}} = \frac{\frac{3}{2}}{1} = \frac{3}{2} \] \[ \delta_m^{\text{air}} = \left(\frac{3}{2} - 1\right) A = \frac{1}{2} A \] 2. When the prism is in water (\( \mu_{\text{water}} = \frac{4}{3} \)): \[ \mu_{\text{relative}} = \frac{\mu_{\text{prism}}}{\mu_{\text{water}}} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{9}{8} \] \[ \delta_m^{\text{water}} = \left(\frac{9}{8} - 1\right) A = \frac{1}{8} A \] 3. Ratio of minimum deviations: \[ \frac{\delta_m^{\text{air}}}{\delta_m^{\text{water}}} = \frac{\frac{1}{2} A}{\frac{1}{8} A} = \frac{1}{2} \times \frac{8}{1} = 4:1 \] Thus, the correct answer is \(\boxed{4:1}\).