When a glass prism of refracting angle `60^(@)` is immersed in a liquid its angle of minimum deviation is `30^(@)`. The critical angle of glass with respect to the liquid medium is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `50^(@)`

Answer

To solve the problem, we need to determine the critical angle of glass with respect to the liquid medium using the given information about the glass prism.

### Step-by-step Solution:

1. **Identify Given Values:**
- Refracting angle of the prism (A) = 60°
- Angle of minimum deviation (D) = 30°

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

3. **Substitute the Values into the Formula:**
- Calculate \( A + D = 60° + 30° = 90° \)
- Therefore, \( \frac{A + D}{2} = \frac{90°}{2} = 45° \)
- Calculate \( \frac{A}{2} = \frac{60°}{2} = 30° \)

Now substituting these into the formula:
\[ \mu = \frac{\sin(45°)}{\sin(30°)} \]

4. **Calculate the Sine Values:**
- \( \sin(45°) = \frac{1}{\sqrt{2}} \)
- \( \sin(30°) = \frac{1}{2} \)

5. **Substitute the Sine Values:**
\[ \mu = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{1}{\sqrt{2}} \times 2 = \sqrt{2} \]

6. **Determine the Critical Angle (IC):**
The critical angle can be found using the formula:
\[ \sin(IC) = \frac{1}{\mu} \]
Substituting the value of μ:
\[ \sin(IC) = \frac{1}{\sqrt{2}} \]

7. **Find the Critical Angle:**
To find IC, we take the inverse sine:
\[ IC = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
We know that:
\[ \sin(45°) = \frac{1}{\sqrt{2}} \]
Therefore,
\[ IC = 45° \]

### Final Answer:
The critical angle of glass with respect to the liquid medium is **45°**.

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