Question

A thin prism \( P_1 \) with angle \( 4^\circ \) made of glass having refractive index 1.54 is combined with another thin prism \( P_2 \) made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism \( P_2 \) in degrees is:

• A. 1.5
• B. 3
• C. \( \frac{16}{3} \)
• D. 4

Hint:

For dispersion without deviation, the deviations from each prism must balance out. This requires using the refractive index and angle of each prism.

Answer 1

The Correct Option is B

To solve this problem, we need to find the angle \( A_2 \) of prism \( P_2 \) such that the combination of prism \( P_1 \) with angle \( A_1 = 4^\circ \) and refractive index \( n_1 = 1.54 \) and prism \( P_2 \) with refractive index \( n_2 = 1.72 \) produces dispersion without deviation.

1. Recall the formula for deviation by a thin prism:
The deviation \( \delta \) caused by a thin prism with angle \( A \) and refractive index \( n \) is given by:
\( \delta = (n - 1)A \)

2. Apply the condition for zero net deviation:
For dispersion without deviation, the net deviation must be zero. Therefore, the sum of the deviations caused by the two prisms must be zero:
\( \delta_1 + \delta_2 = 0 \)

3. Substitute the expressions for \(\delta_1\) and \(\delta_2\):
Substitute \( \delta_1 = (n_1 - 1)A_1 \) and \( \delta_2 = (n_2 - 1)A_2 \) into the equation \( \delta_1 + \delta_2 = 0 \):
\( (n_1 - 1)A_1 + (n_2 - 1)A_2 = 0 \)

4. Substitute the given values:
Substitute the given values \( n_1 = 1.54 \), \( A_1 = 4^\circ \), and \( n_2 = 1.72 \) into the equation:
\( (1.54 - 1)(4) + (1.72 - 1)(A_2) = 0 \) \( (0.54)(4) + (0.72)(A_2) = 0 \)

5. Solve for \(A_2\):
Solve for \( A_2 \):
\( 2.16 + 0.72 A_2 = 0 \) \( 0.72 A_2 = -2.16 \) \( A_2 = -\frac{2.16}{0.72} = -3 \)

6. Interpret the result:
The angle of the prism \( P_2 \) is \( A_2 = -3^\circ \). The negative sign indicates that the prism \( P_2 \) is placed in an inverted position relative to the prism \( P_1 \). Since we are asked for the magnitude of the angle, we take the absolute value.

Final Answer:
The angle of prism \( P_2 \) is \( {|A_2|} = {3^\circ} \).

Answer 2

Step 1: Understand the problem.
We are given two thin prisms \( P_1 \) and \( P_2 \) made of different materials. The condition for **dispersion without deviation** means that the **mean deviation produced by both prisms should cancel each other**, while their **dispersive powers should balance** so that the net dispersion remains zero.

Given data:
For prism \( P_1 \): \( A_1 = 4^\circ \), refractive index \( \mu_1 = 1.54 \)
For prism \( P_2 \): \( A_2 = ? \), refractive index \( \mu_2 = 1.72 \)

Step 2: Use the condition for no deviation.
For no deviation, the net angular deviation must be zero:
\[ (\mu_1 - 1)A_1 = (\mu_2 - 1)A_2 \] This ensures that the light emerging from the second prism has no overall deviation but the dispersions of the two prisms cancel each other.

Step 3: Substitute the given values.
\[ (1.54 - 1) \times 4 = (1.72 - 1) \times A_2 \] \[ 0.54 \times 4 = 0.72 \times A_2 \] \[ A_2 = \frac{0.54 \times 4}{0.72} \] \[ A_2 = 3^\circ \]

Step 4: Conclusion.
The angle of the second prism required to obtain dispersion without deviation is:
\[ \boxed{A_2 = 3^\circ} \]

Final Answer:
\( 3^\circ \)