The angle of minimum deviation for a `75^(@)` prism of dense glass is found to be `45^(@)` when in air and `15^(@)` when immersed in certain liquid. The refractive index of the liquid is

Options

A. `sqrt(2/3)`
B. `3/2`
C. `sqrt(3/2)`
D. `sqrt3`

Answer

To find the refractive index of the liquid using the given data about the prism, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Angle of the prism (A) = 75°
- Angle of minimum deviation in air (D_a) = 45°
- Angle of minimum deviation in liquid (D_l) = 15°

2. **Use the Formula for Refractive Index:**
The refractive index of the material (glass) can be calculated using the formula:
\[ \mu_{glass} = \frac{\sin\left(\frac{A + D_a}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Calculate the Refractive Index of Glass:**
- Calculate \( \frac{A + D_a}{2} \):
\[ \frac{75° + 45°}{2} = \frac{120°}{2} = 60° \]
- Calculate \( \frac{A}{2} \):
\[ \frac{75°}{2} = 37.5° \]
- Now, substituting into the formula:
\[ \mu_{glass} = \frac{\sin(60°)}{\sin(37.5°)} \]

4. **Calculate the Refractive Index of the Liquid:**
Using the angle of minimum deviation in the liquid:
\[ \mu_{liquid} = \frac{\sin\left(\frac{A + D_l}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
- Calculate \( \frac{A + D_l}{2} \):
\[ \frac{75° + 15°}{2} = \frac{90°}{2} = 45° \]
- Now substituting into the formula:
\[ \mu_{liquid} = \frac{\sin(45°)}{\sin(37.5°)} \]

5. **Relate the Two Refractive Indices:**
Since we know the refractive index of glass (\(\mu_{glass}\)) and the formula for the refractive index of the liquid (\(\mu_{liquid}\)), we can set up the equation:
\[ \mu_{liquid} = \frac{\sin(45°)}{\sin(37.5°)} \cdot \frac{\sin(37.5°)}{\sin(60°)} = \frac{\sin(45°)}{\sin(60°)} \]

6. **Calculate the Final Value:**
- Using known values:
- \(\sin(45°) = \frac{\sqrt{2}}{2}\)
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- Thus,
\[ \mu_{liquid} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}} = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3} \]

### Final Answer:
The refractive index of the liquid is \(\frac{\sqrt{6}}{3}\).