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 the material of an equilateral prism given the angle of minimum deviation, we can use the following steps:

### Step 1: Understand the relationship between the angles
For an equilateral prism, the angle of the prism \( A \) is \( 60^\circ \). The angle of minimum deviation \( D_m \) is given as \( 46^\circ \).

### Step 2: Use the formula for minimum deviation
The relationship between the angle of minimum deviation \( D_m \), angle of incidence \( i \), and angle of emergence \( e \) can be expressed as:
\[ D_m = i + e - A \]
Since \( i = e \) at minimum deviation, we can rewrite this as:
\[ D_m = 2i - A \]

### Step 3: Solve for the angle of incidence \( i \)
Substituting the known values into the equation:
\[ 46^\circ = 2i - 60^\circ \]
Rearranging gives:
\[ 2i = 46^\circ + 60^\circ \]
\[ 2i = 106^\circ \]
\[ i = \frac{106^\circ}{2} = 53^\circ \]

### Step 4: Use Snell's Law
According to Snell's Law:
\[ \mu = \frac{\sin i}{\sin r} \]
Where \( r \) is the angle of refraction. For an equilateral prism, at minimum deviation, the angle of refraction \( r \) can be calculated as:
\[ r = \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ \]

### Step 5: Substitute values into Snell's Law
Now substituting \( i \) and \( r \) into Snell's Law:
\[ \mu = \frac{\sin 53^\circ}{\sin 30^\circ} \]
We know that \( \sin 30^\circ = \frac{1}{2} \).

### Step 6: Calculate the refractive index
Using a calculator or trigonometric tables:
\[ \sin 53^\circ \approx 0.7986 \]
Thus:
\[ \mu = \frac{0.7986}{0.5} = 1.5972 \approx 1.6 \]

### Conclusion
The refractive index of the material of the prism is approximately \( 1.6 \).