A monochromatic ray of light travels through an equilateral prism such that angle of deviation is `30^(@)`. If the difference between angles of incidence and emergence is `10^(@)`, then I and r respectively are

Options

A. `45^(@),55^(@)`
B. `50^(@),40^(@)`
C. `35^(@),45^(@)`
D. `30^(@),40^(@)`

Answer

To solve the problem, we need to find the angles of incidence (I) and emergence (E) for a monochromatic ray of light passing through an equilateral prism with a given angle of deviation (Δ) and a difference between the angles of incidence and emergence.

### Step-by-Step Solution:

1. **Identify Given Values**:
- The angle of deviation (Δ) = 30 degrees.
- The angle of the prism (A) = 60 degrees (since it is an equilateral prism).
- The difference between the angle of incidence (I) and angle of emergence (E) = 10 degrees.

2. **Use the Formula for Angle of Deviation**:
The formula relating the angles is:
\[ Δ = I + E - A \]
Substituting the known values:
\[ 30 = I + E - 60 \]

3. **Rearrange the Equation**:
Rearranging gives:
\[ I + E = 30 + 60 = 90 \quad \text{(Equation 1)} \]

4. **Set Up the Second Equation**:
From the difference between the angles of incidence and emergence:
\[ I - E = 10 \quad \text{(Equation 2)} \]

5. **Solve the Simultaneous Equations**:
We now have two equations:
- Equation 1: \( I + E = 90 \)
- Equation 2: \( I - E = 10 \)

Adding these two equations:
\[ (I + E) + (I - E) = 90 + 10 \]
This simplifies to:
\[ 2I = 100 \]
Therefore:
\[ I = \frac{100}{2} = 50 \text{ degrees} \]

6. **Find the Angle of Emergence (E)**:
Now substitute \( I \) back into Equation 1:
\[ 50 + E = 90 \]
Solving for \( E \):
\[ E = 90 - 50 = 40 \text{ degrees} \]

### Final Answer:
The angles of incidence (I) and emergence (E) are:
- \( I = 50 \text{ degrees} \)
- \( E = 40 \text{ degrees} \)