The light ray is incidence at angle of `60^(@)` on a prism of angle `45^(@)` . When the light ray falls on the other surface at `90^(@)` , the refractive index of the material of prism` mu` and the angle of devation `delta` are given by

Options

A. `mu=sqrt(2), delta=30^(@)`
B. `mu=1.5, delta=15^(@)`
C. `mu=(sqrt(3))/(2), delta=30^(@)`
D. `mu=(sqrt(3))/(2), delta=15^(@)`

Explanation

To solve the problem, we need to find the refractive index (μ) of the prism material and the angle of deviation (δ) when a light ray is incident at an angle of 60° on a prism with an angle of 45°.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Angle of incidence (i) = 60°
- Angle of the prism (A) = 45°
- Angle of emergence (R2) = 0° (since the light ray emerges at 90° to the normal)

2. **Determine the Angle of Refraction (R1):**
- According to the prism rules, the angle of refraction (R1) at the first surface is equal to the angle of the prism (A).
- Therefore, R1 = A = 45°.

3. **Apply Snell's Law at the First Surface:**
- Snell's Law states: n1 * sin(i) = n2 * sin(R1)
- Here, n1 (refractive index of air) = 1, n2 (refractive index of the prism) = μ.
- Substituting the values:
\[ 1 * \sin(60°) = μ * \sin(45°) \]
- We know:
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\sin(45°) = \frac{1}{\sqrt{2}}\)

4. **Solve for μ:**
- Rearranging the equation gives:
\[ μ = \frac{\sin(60°)}{\sin(45°)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} \]
- Simplifying:
\[ μ = \frac{\sqrt{3}}{2} * \frac{\sqrt{2}}{1} = \frac{\sqrt{6}}{2} \]

5. **Calculate the Angle of Deviation (δ):**
- The formula for the angle of deviation (δ) is:
\[ δ = (μ - 1) * A \]
- Substituting the values:
\[ δ = \left(\frac{\sqrt{6}}{2} - 1\right) * 45° \]
- To simplify:
- Convert 1 to have a common denominator:
\[ 1 = \frac{2}{2} \Rightarrow δ = \left(\frac{\sqrt{6} - 2}{2}\right) * 45° \]

6. **Final Calculation:**
- Calculate the numerical value of δ:
\[ δ = \frac{(\sqrt{6} - 2) * 45}{2} \]
- This will give the angle of deviation in degrees.

### Final Results:
- Refractive index (μ) = \(\frac{\sqrt{6}}{2}\)
- Angle of deviation (δ) = \(\frac{(\sqrt{6} - 2) * 45}{2}\)