A certain prism is found to produce a minimum of`38^(@)`.It produces a deviation of `44^(@)`when the angle of incident is either`42^(@)`or`62^(@)` .What is the angle of incidence when it is undergoing minimum deviation?

Options

A. `45^(@)`
B. `49^(@)`
C. `40^(@)`
D. `55^(@)`

Explanation

To solve the problem step by step, we will use the principles of prism optics.

### Step 1: Understand the given information
We are given:
- Minimum deviation (D_min) = 38°
- Deviation (D) = 44° when the angle of incidence (I) is either 42° or 62°.

### Step 2: Use the formula for angle of deviation
The angle of deviation (D) can be expressed in terms of the angle of incidence (I), angle of emergence (E), and angle of the prism (A) as follows:
\[ D = I + E - A \]

### Step 3: Calculate the angle of the prism (A)
We can use the information about the deviation when the angles of incidence are 42° and 62° to find the angle of the prism (A).

Using the formula for deviation:
1. For I = 42° and E = 62°:
\[ 44 = 42 + 62 - A \]
\[ 44 = 104 - A \]
\[ A = 104 - 44 = 60° \]

### Step 4: Use the minimum deviation formula
The minimum deviation (D_min) is related to the angle of incidence (I_min) and the angle of the prism (A) as follows:
\[ D_{min} = 2I_{min} - A \]

### Step 5: Rearranging the formula to find I_min
We can rearrange the formula to solve for the angle of incidence at minimum deviation (I_min):
\[ I_{min} = \frac{D_{min} + A}{2} \]

### Step 6: Substitute the known values
Now we can substitute the known values into the equation:
- D_min = 38°
- A = 60°

So,
\[ I_{min} = \frac{38 + 60}{2} \]
\[ I_{min} = \frac{98}{2} \]
\[ I_{min} = 49° \]

### Final Answer
The angle of incidence when the prism is undergoing minimum deviation is **49°**.

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