Question

A thin prism having refracting angle 10° is made of glass of refractive index 1.42. This prism is combined with another thin prism of glass of refractive index 1.7. This combination produces dispersion without deviation. The refracting angle of second prism should be

• A. 4°
• B. 6°
• C. 8°
• D. 10°

Answer 1

The Correct Option is B

To solve this problem, we need to find the refracting angle of the second prism so that the combination of two prisms produces dispersion without deviation.

When two prisms are combined to neutralize their deviations while still creating dispersion, the net deviation produced by the combination should be zero. This condition is known as producing dispersion without deviation. The deviation formula for a prism is:

\(\delta = (n - 1)A\),

where:

• \(\delta\) is the deviation.
• \(n\) is the refractive index of the prism material.
• \(A\) is the refracting angle of the prism.

For the combination of two prisms, the condition to achieve dispersion without deviation is:

(\delta_1 + \delta_2 = 0)

where:

• \(\delta_1 = (n_1 - 1)A_1\) is the deviation by the first prism,
• \(\delta_2 = (n_2 - 1)A_2\) is the deviation by the second prism.

Given values:

• Refracting angle of the first prism A_1 = 10^\circ
• Refractive index of the first prism n_1 = 1.42
• Refractive index of the second prism n_2 = 1.7

According to the condition for dispersion without deviation:

(n_1 - 1)A_1 + (n_2 - 1)A_2 = 0

Substituting the values:

(1.42 - 1)(10^\circ) + (1.7 - 1)A_2 = 0

0.42 \times 10 + 0.7 A_2 = 0

4.2 + 0.7A_2 = 0

Rearranging gives:

0.7A_2 = -4.2

Solving for \(A_2\):

A_2 = \frac{-4.2}{0.7} = -6^\circ

Since the angle cannot be negative, we take the positive equivalent refracting angle:

The refracting angle of the second prism should be 6°.