Question

Two immiscible liquids of refractive indices \( \frac{8}{5} \) and \( \frac{3}{2} \) respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is \( \frac{\alpha}{4} \) cm. The value of \( \alpha \) is ______.

Answer 1

Correct Answer: 31

For layered media, the apparent depth \( d_{\text{app}} \) is given by:

\[ d_{\text{app}} = \frac{h_1}{\mu_1} + \frac{h_2}{\mu_2} \]

where \( h_1 = h_2 = 6 \, \text{cm} \), \( \mu_1 = \frac{8}{5} \), and \( \mu_2 = \frac{3}{2} \).

Calculating:

\[ d_{\text{app}} = \frac{6}{8/5} + \frac{6}{3/2} = \frac{6 \times 5}{8} + \frac{6 \times 2}{3} = \frac{30}{8} + 4 = \frac{31}{4} \, \text{cm} \]

Thus, \( \alpha = 31 \).