The minimum deviation produced by a hollow prism filled with a certain liquid to be `30^(@)`. The light ray is also found to be refracted at angle `30^(@)`. The refractive index of the liquid is

Options

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

Answer

To find the refractive index of the liquid in the hollow prism, we can follow these steps:

### Step 1: Understand the Given Information
- Minimum deviation (Δm) = 30°
- Angle of refraction (r) = 30°

### Step 2: Relate the Angle of Prism (A) to the Angle of Refraction
In a prism, at minimum deviation, the angle of the prism (A) is related to the angle of refraction (r) by the formula:
\[ A = 2r \]
Substituting the given value of r:
\[ A = 2 \times 30° = 60° \]

### Step 3: Use the Formula for Refractive Index (μ)
The formula for the refractive index (μ) of the prism filled with a liquid is given by:
\[ \mu = \frac{\sin\left(\frac{A + \Delta m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 4: Substitute the Values into the Formula
Now, substituting A = 60° and Δm = 30° into the formula:
\[ \mu = \frac{\sin\left(\frac{60° + 30°}{2}\right)}{\sin\left(\frac{60°}{2}\right)} \]
Calculating the angles:
- \(\frac{60° + 30°}{2} = \frac{90°}{2} = 45°\)
- \(\frac{60°}{2} = 30°\)

Thus, we have:
\[ \mu = \frac{\sin(45°)}{\sin(30°)} \]

### Step 5: Use Known Values of Sine
Now, we can use the known values:
- \(\sin(45°) = \frac{1}{\sqrt{2}}\)
- \(\sin(30°) = \frac{1}{2}\)

Substituting these values:
\[ \mu = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} \]

### Step 6: Simplify the Expression
To simplify:
\[ \mu = \frac{1}{\sqrt{2}} \times 2 = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer
The refractive index of the liquid in the prism is:
\[ \mu = \sqrt{2} \]

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