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

Options

A. `62.5^(@)`
B. `42.5^(@)`
C. Both are correct
D. Both are wrong

Explanation

To solve the problem, we need to find the angle of incidence for an equilateral prism that deviates a ray through \(45^\circ\) for two angles of incidence differing by \(20^\circ\).

### Step-by-Step Solution:

1. **Define the Angles of Incidence:**
Let the first angle of incidence be \(I_1\) and the second angle of incidence be \(I_2\). According to the problem, we know that:
\[ I_1 - I_2 = 20^\circ \quad \text{(Equation 1)} \]

2. **Use the Deviation Formula:**
The deviation \(D\) for a prism can be expressed as:
\[ D = I_1 + I_2 - A \]
where \(A\) is the angle of the prism. For an equilateral prism, \(A = 60^\circ\). Given that the total deviation \(D = 45^\circ\), we can write:
\[ I_1 + I_2 - 60^\circ = 45^\circ \]
Rearranging this gives us:
\[ I_1 + I_2 = 105^\circ \quad \text{(Equation 2)} \]

3. **Add the Two Equations:**
Now, we can add Equation 1 and Equation 2:
\[ (I_1 - I_2) + (I_1 + I_2) = 20^\circ + 105^\circ \]
This simplifies to:
\[ 2I_1 = 125^\circ \]
Thus, we find:
\[ I_1 = \frac{125^\circ}{2} = 62.5^\circ \]

4. **Calculate the Second Angle of Incidence:**
Now, we can find \(I_2\) using Equation 1:
\[ I_2 = I_1 - 20^\circ = 62.5^\circ - 20^\circ = 42.5^\circ \]

5. **Conclusion:**
The two angles of incidence are \(I_1 = 62.5^\circ\) and \(I_2 = 42.5^\circ\).

### Final Answer:
The angles of incidence are \(62.5^\circ\) and \(42.5^\circ\).

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