A certain prism is that to produce minimum deviation of 3 8°. It produces a deviation of 44° when the angle ofincidence is either 42° or 62° . The refractive index of material of prism is

Options

A. 1.51
B. 1.33
C. 1.62
D. 1.732

Explanation

To find the refractive index of the prism, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The minimum deviation (D_min) = 38°
- The deviation (D) = 44° for angles of incidence (i) = 42° and 62°.

### Step 2: Use the relation for deviation in a prism
When the deviation is the same for two angles of incidence, we can use the formula:
\[ i + e - A = D \]
where:
- \( i \) = angle of incidence
- \( e \) = angle of emergence
- \( A \) = angle of the prism
- \( D \) = deviation

Since the angles of incidence and emergence can be interchanged, we can write:
\[ 42 + 62 - A = 44 \]

### Step 3: Solve for the angle of the prism (A)
Substituting the values:
\[ 104 - A = 44 \]
\[ A = 104 - 44 \]
\[ A = 60° \]

### Step 4: Use the formula for refractive index (n)
The formula for the refractive index of the prism in terms of the angle of prism and the minimum deviation is:
\[ n = \frac{\sin\left(\frac{A + D_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 5: Substitute the known values
Substituting \( A = 60° \) and \( D_{min} = 38° \):
\[ n = \frac{\sin\left(\frac{60 + 38}{2}\right)}{\sin\left(\frac{60}{2}\right)} \]
\[ n = \frac{\sin\left(49°\right)}{\sin\left(30°\right)} \]

### Step 6: Calculate the sine values
We know:
- \( \sin(30°) = 0.5 \)
- \( \sin(49°) \approx 0.754 \)

### Step 7: Calculate the refractive index
Now substituting the sine values:
\[ n = \frac{0.754}{0.5} \]
\[ n = 1.508 \]

### Step 8: Round off the answer
Rounding off gives us:
\[ n \approx 1.51 \]

Thus, the refractive index of the material of the prism is approximately **1.51**.

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