For a prism kept in air, it is found that for an angle of incidence `60^(@)` , the angle of refraction 'A', angle of deviation `delta`, and angle of emergence 'e' become equal. The minimum angle of incidence of a ray that will be transmitted through the prism is

Options

A. 1.73
B. 1.15
C. 1.5
D. 1.33

Explanation

To solve the problem, we need to determine the minimum angle of incidence for a ray that will be transmitted through a prism, given that for an angle of incidence of \(60^\circ\), the angle of refraction \(A\), angle of deviation \(\delta\), and angle of emergence \(e\) are all equal.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that:
- Angle of incidence \(I = 60^\circ\)
- Angle of refraction \(A = \delta = e\)

2. **Using the Formula for Angle of Deviation**:
The formula for the angle of deviation \(\delta\) in a prism is given by:
\[ \delta = I + e - A \]
Since \(A = \delta = e\), we can substitute these values into the equation.

3. **Substituting the Values**:
Substituting \(e\) and \(A\) with \(\delta\):
\[ \delta = I + \delta - \delta \]
This simplifies to:
\[ \delta = I \]
Therefore, we have:
\[ \delta = 60^\circ \]

4. **Finding the Refractive Index**:
The refractive index \(\mu\) of the prism can be calculated using the formula:
\[ \mu = \frac{\sin\left(\frac{A + \delta}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Since \(A = \delta = 60^\circ\), we can substitute:
\[ \mu = \frac{\sin\left(\frac{60^\circ + 60^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} = \frac{\sin(60^\circ)}{\sin(30^\circ)} \]

5. **Calculating the Sine Values**:
We know:
- \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
- \(\sin(30^\circ) = \frac{1}{2}\)

Substituting these values:
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

6. **Finding the Minimum Angle of Incidence**:
The minimum angle of incidence \(I_{\text{min}}\) can be found using the formula:
\[ I_{\text{min}} = \sin^{-1}\left(\frac{\mu - 1}{\mu}\right) \]
Substituting \(\mu = \sqrt{3}\):
\[ I_{\text{min}} = \sin^{-1}\left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]

7. **Calculating the Minimum Angle of Incidence**:
We can calculate this value:
\[ I_{\text{min}} = \sin^{-1}\left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]
This value can be computed using a calculator or trigonometric tables, which will yield the minimum angle of incidence.

### Final Answer:
The minimum angle of incidence of a ray that will be transmitted through the prism is approximately \(I_{\text{min}} \approx 30^\circ\).