The angle of minimum deviation produced by an equilateral prism is `46^@` The refractive index of material of the prism.

Options

A. 1.6
B. 1.5
C. 1.4
D. 1.8

Answer

To find the refractive index of an equilateral prism given the angle of minimum deviation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry of the Prism**:
- The prism is equilateral, which means each angle of the prism is \(60^\circ\).
- The angle of minimum deviation \(D\) is given as \(46^\circ\).

2. **Use the Relation for Minimum Deviation**:
- For an equilateral prism, the relation between the angle of incidence \(i\), the angle of minimum deviation \(D\), and the angle of the prism \(A\) can be expressed as:
\[ D = 2i - A \]
- Here, \(A = 60^\circ\) (the angle of the prism).

3. **Substituting the Values**:
- Substitute \(D = 46^\circ\) and \(A = 60^\circ\) into the equation:
\[ 46 = 2i - 60 \]

4. **Solve for the Angle of Incidence \(i\)**:
- Rearranging the equation gives:
\[ 2i = 46 + 60 \]
\[ 2i = 106 \]
\[ i = \frac{106}{2} = 53^\circ \]

5. **Apply Snell's Law**:
- According to Snell's Law at the first surface of the prism:
\[ n_1 \sin(i) = n_2 \sin(r) \]
- Here, \(n_1 = 1\) (the refractive index of air), \(n_2 = \mu\) (the refractive index of the prism), \(i = 53^\circ\), and \(r\) is the angle of refraction at the prism surface.

6. **Determine the Angle of Refraction**:
- For an equilateral prism, at minimum deviation, the angle of refraction \(r\) can be calculated as:
\[ r = \frac{A}{2} = \frac{60}{2} = 30^\circ \]

7. **Substituting into Snell's Law**:
- Now substituting the known values into Snell's Law:
\[ 1 \cdot \sin(53^\circ) = \mu \cdot \sin(30^\circ) \]

8. **Calculate the Sine Values**:
- We know:
\[ \sin(30^\circ) = \frac{1}{2} \]
- We can calculate \(\sin(53^\circ)\) using a calculator or trigonometric tables:
\[ \sin(53^\circ) \approx 0.7986 \]

9. **Rearranging for the Refractive Index \(\mu\)**:
- Now substituting the sine values:
\[ 0.7986 = \mu \cdot \frac{1}{2} \]
- Rearranging gives:
\[ \mu = 0.7986 \cdot 2 = 1.5972 \]

10. **Final Answer**:
- The refractive index of the material of the prism is approximately \(1.6\).