The angle of prism is `60^(@)` and its refractive index is 1.5. There will be no emergent light if the angle of incidence on the first face is

Options

A. equal to `30^(@)`
B. less than `27^(@)`
C. more than `30^(@)`
D. equal to `60^(@)`

Answer

To determine the angle of incidence on the first face of a prism with an angle of 60 degrees and a refractive index of 1.5, such that there will be no emergent light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Condition for No Emergent Light**:
- For no emergent light to occur, the light must undergo total internal reflection (TIR) at the second face of the prism. This happens when the angle of refraction at the second face is 90 degrees (grazing condition).

2. **Identify Angles**:
- Let the angle of incidence on the first face be \( i \).
- The angle of the prism is given as \( A = 60^\circ \).
- Let \( R_1 \) be the angle of refraction at the first face and \( R_2 \) be the angle of refraction at the second face.

3. **Using the Prism Formula**:
- The relationship between the angles in the prism is given by:
\[ R_1 + R_2 = A \]
- Since we want \( R_2 = 90^\circ \) for TIR, we can substitute:
\[ R_1 + 90^\circ = 60^\circ \]
- This gives us:
\[ R_1 = 60^\circ - 90^\circ = -30^\circ \]
- This indicates that for \( R_2 \) to be 90 degrees, \( R_1 \) must be \( 60^\circ \).

4. **Applying Snell's Law**:
- At the first face, Snell's law states:
\[ n_1 \sin(i) = n_2 \sin(R_1) \]
- Here, \( n_1 = 1 \) (air), \( n_2 = 1.5 \) (glass), and \( R_1 = 60^\circ \).
- Thus, we have:
\[ \sin(i) = 1.5 \sin(60^\circ) \]

5. **Calculating \( \sin(60^\circ) \)**:
- We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \).
- Therefore:
\[ \sin(i) = 1.5 \cdot \frac{\sqrt{3}}{2} = \frac{1.5\sqrt{3}}{2} \]

6. **Finding the Angle of Incidence \( i \)**:
- To find \( i \), we take the inverse sine:
\[ i = \sin^{-1}\left(\frac{1.5\sqrt{3}}{2}\right) \]
- This value will give us the angle of incidence at which TIR occurs.

7. **Conclusion**:
- The angle of incidence \( i \) must be less than the calculated value for there to be no emergent light.

### Final Answer:
The angle of incidence on the first face must be less than approximately \( 27^\circ \) for there to be no emergent light.