An equilateral prism deviates a ray through `45^(@)` for the two angles of incidence differing by `20^(@)`. The angle of incidence is

Options

A. `60^@ and 40^@`
B. `50^@ and 70^@`
C. `62^@30' and 42^@30'`
D. `60^@30' and 40^@30'`

Answer

To solve the problem, we need to find the angle of incidence (I) for an equilateral prism that deviates a ray through 45 degrees, given that the two angles of incidence differ by 20 degrees.

### Step-by-Step Solution:

1. **Understanding the Prism and Deviation**:
- For an equilateral prism, the angle of the prism (A) is 60 degrees.
- The formula for the deviation (D) of light passing through a prism is given by:
\[ D = I + E - A \]
Where:
- \(D\) = deviation
- \(I\) = angle of incidence
- \(E\) = angle of emergence
- \(A\) = angle of the prism

2. **Setting Up the Equation**:
- We know that the deviation \(D\) is 45 degrees:
\[ 45 = I + E - 60 \]
- Rearranging gives:
\[ I + E = 105 \quad \text{(Equation 1)} \]

3. **Using the Given Information**:
- We are told that the two angles of incidence differ by 20 degrees. Let:
\[ I_1 = I \quad \text{and} \quad I_2 = I - 20 \]
- The corresponding angles of emergence will be:
\[ E_1 \quad \text{and} \quad E_2 \]
- From the symmetry of the prism, we can say:
\[ E_1 = 105 - I_1 \quad \text{and} \quad E_2 = 105 - I_2 \]

4. **Setting Up the Second Equation**:
- Using the angles of incidence and emergence:
\[ I_1 + E_1 = 105 \quad \text{and} \quad I_2 + E_2 = 105 \]
- Substituting \(E_1\) and \(E_2\):
\[ I_1 + (105 - I_1) = 105 \quad \text{(which is always true)} \]
\[ I_2 + (105 - I_2) = 105 \quad \text{(which is also always true)} \]

5. **Finding the Angles**:
- Now, substituting \(I_2 = I_1 - 20\) into the equation:
\[ (I_1 - 20) + (105 - (I_1 - 20)) = 105 \]
- Simplifying gives:
\[ I_1 - 20 + 105 - I_1 + 20 = 105 \]
- This confirms our equations are consistent.

6. **Solving for \(I\)**:
- From Equation 1, we know:
\[ I + E = 105 \]
- We also have:
\[ E = 105 - I \]
- Substituting into the deviation equation:
\[ D = I + (105 - I) - 60 = 45 \]
- This simplifies to:
\[ 105 - 60 = 45 \]
- Which is consistent.

7. **Final Calculation**:
- Now, we can find \(I\) and \(E\) using the difference:
\[ I - E = 20 \]
- Substituting \(E = 105 - I\):
\[ I - (105 - I) = 20 \]
- This gives:
\[ 2I - 105 = 20 \]
- Rearranging:
\[ 2I = 125 \implies I = 62.5 \]

### Conclusion:
The angle of incidence \(I\) is \(62.5^\circ\) or \(62^\circ 30'\).