The angle of a prism is `60^(@)` When light is incident at an angle of `60^(@)` on the prism, the angle of emerge is `40^(@)`. The angle of incidence `i` for which the light ray will deviate the least is such that

Options

A. `I lt 40^(@)`
B. `40^(@) lt I lt 50^(@)`
C. `50^(@) lt I lt 60^(@)`
D. `I gt 60^(@)`

Explanation

To find the angle of incidence \( i \) for which the light ray will deviate the least when passing through a prism with an angle \( A = 60^\circ \), we can follow these steps:

### Step 1: Understand the given information
We have:
- Angle of the prism \( A = 60^\circ \)
- Angle of incidence \( i_1 = 60^\circ \)
- Angle of emergence \( e_1 = 40^\circ \)

### Step 2: Use the formula for deviation
The angle of deviation \( D \) can be calculated using the formula:
\[ D = i + e - A \]
Substituting the known values for \( i_1 \) and \( e_1 \):
\[ D_1 = 60^\circ + 40^\circ - 60^\circ = 40^\circ \]

### Step 3: Analyze the symmetry of the angles
From the information given, we can see that if the angle of incidence is \( 60^\circ \) and the angle of emergence is \( 40^\circ \), there is a symmetry in the angles. If we switch the angles, we can say:
- If \( i = 40^\circ \), then \( e = 60^\circ \).

### Step 4: Calculate the deviation for \( i = 40^\circ \)
Using the same formula for deviation:
\[ D_2 = 40^\circ + 60^\circ - 60^\circ = 40^\circ \]
Thus, for both \( i = 60^\circ \) and \( i = 40^\circ \), the deviation is \( 40^\circ \).

### Step 5: Determine the minimum deviation
To find the angle of incidence \( i \) for which the deviation is minimum, we note that the minimum deviation occurs when the angle of incidence equals the angle of emergence:
\[ i = e \]
Thus, we can express the minimum deviation \( D_{min} \) as:
\[ D_{min} = 2i - A \]

### Step 6: Set up the equation for minimum deviation
Since we know \( A = 60^\circ \), we can set up the equation:
\[ D_{min} = 2i - 60^\circ \]
To find the minimum deviation, we need to ensure that \( D_{min} < 40^\circ \):
\[ 2i - 60^\circ < 40^\circ \]
Adding \( 60^\circ \) to both sides:
\[ 2i < 100^\circ \]
Dividing by 2:
\[ i < 50^\circ \]

### Step 7: Determine the range for \( i \)
From our earlier analysis, we also know that:
- \( i > 40^\circ \) (since at \( i = 40^\circ \), the deviation is not minimum)
- \( i < 50^\circ \)

Thus, the angle of incidence \( i \) for which the light ray will deviate the least is:
\[ 40^\circ < i < 50^\circ \]

### Final Answer
The angle of incidence \( i \) for which the light ray will deviate the least is such that:
\[ i \text{ is between } 40^\circ \text{ and } 50^\circ. \]