Question

An equilateral prism deviates a ray through \(45^{\circ }\) for the two angles of incidence differing by \(20^{\circ }\). One of the angle of incidence is

• A. \(62.5^{\circ }\)
• B. \(32.5^{\circ }\)
• C. \(52.5^{\circ }\)
• D. \(22.5^{\circ }\)

Hint:

Use i + e = A + deviation and the fact that the two incidence angles for the same deviation are related by reversibility of light.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
For a prism, \(i + e = A + \delta\).
For one deviation there are two possible incidence angles, and the emergence angle for one equals the incidence angle for the other.

Step 2: Sum of the Two Incidence Angles
For an equilateral prism \(A = 60^{\circ}\) and \(\delta = 45^{\circ}\).
So \(i_1 + i_2 = 60^{\circ} + 45^{\circ} = 105^{\circ}\).

Step 3: Use the Given Difference
The two angles differ by \(20^{\circ}\), so \(i_1 - i_2 = 20^{\circ}\).

Step 4: Calculation
Adding the two equations, \(2i_1 = 125^{\circ}\), so \(i_1 = 62.5^{\circ}\).
Then \(i_2 = 42.5^{\circ}\).

Step 5: Compare the options
Only \(62.5^{\circ}\) appears among the options.

Final Answer:
One of the angles of incidence is 62.5 degrees. \[ \boxed{\text{(A) }62.5^{\circ}} \]