Refractive index of a prism is `sqrt(7//3)` and the angle of prism is `60^(@)`. The minimum angle of incidence of a ray that will be transmitted through the prism is

Options

A. `30^@`
B. `45^@`
C. `15^@`
D. None of these

Explanation

To find the minimum angle of incidence of a ray that will be transmitted through a prism, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Refractive index of the prism, \( \mu = \sqrt{\frac{7}{3}} \)
- Angle of the prism, \( A = 60^\circ \)

2. **Calculate the Critical Angle (\( \theta_C \)):**
The critical angle can be calculated using the formula:
\[ \theta_C = \sin^{-1}\left(\frac{1}{\mu}\right) \]
Substituting the value of \( \mu \):
\[ \theta_C = \sin^{-1}\left(\frac{1}{\sqrt{\frac{7}{3}}}\right) = \sin^{-1}\left(\sqrt{\frac{3}{7}}\right) \]
Calculating this gives:
\[ \theta_C \approx 40.9^\circ \]

3. **Relate Angles in the Prism:**
For a prism, the relationship between the angles is given by:
\[ A = r_1 + r_2 \]
Where \( r_1 \) is the angle of refraction at the first surface and \( r_2 \) is the angle of refraction at the second surface. Given \( A = 60^\circ \):
\[ r_1 + r_2 = 60^\circ \]

4. **Apply Snell's Law:**
At the first surface, Snell's law states:
\[ n_1 \sin I = \mu \sin r_1 \]
Where \( n_1 = 1 \) (for air). Thus:
\[ \sin I = \mu \sin r_1 \]

5. **Substituting for \( r_1 \):**
From the relationship \( r_1 = 60^\circ - r_2 \):
\[ \sin I = \mu \sin(60^\circ - r_2) \]

6. **Maximize \( r_2 \):**
To find the minimum angle of incidence \( I \), we need to maximize \( r_2 \) to be equal to the critical angle \( \theta_C \):
\[ r_2 = \theta_C \approx 40.9^\circ \]

7. **Calculate \( r_1 \):**
Substitute \( r_2 \) back to find \( r_1 \):
\[ r_1 = 60^\circ - 40.9^\circ = 19.1^\circ \]

8. **Substituting \( r_1 \) into Snell's Law:**
Now substitute \( r_1 \) into Snell's law:
\[ \sin I = \mu \sin(19.1^\circ) \]
Calculate \( \sin(19.1^\circ) \):
\[ \sin I = \sqrt{\frac{7}{3}} \cdot \sin(19.1^\circ) \]

9. **Find Minimum Angle of Incidence \( I \):**
Calculate \( I \):
\[ I \approx 30^\circ \]

### Final Answer:
The minimum angle of incidence \( I \) that will be transmitted through the prism is \( 30^\circ \).

---