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Question

The dispersive powers of crown and flint glasses are 0.03 and 0.05 respectively. The refractive indices for yellow light for these glasses are 1.517 and 1.621 respectively. It is desired to form an achromatic combination of prisms of crown and flint glasses which can produce a deviation of 1 in the yellow ray. Find the refracting angles of the two prisms needed.

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Solution

Given,
ω=0.03 ; ω=0.05
μy=1.517 ; μy=1.621

Where, ω ,ω=Dispersive powers of crown and flint prisms respectively
μy ,μy=R.I. of the yellow light for crown and flint prisms respectively

Let, the angle of the crown and flint prism needed is A and A respectively.

Dispersive power is,

ω=μvμrμy1

μvμr=(μy1)ω

The angular dispersion produced by the crown prism is,

(μvμr)A=(μy1)ωA

Similarly, the angular dispersion produced by the flint prism is,

(μvμr)A=(μy1)ωA

For achromatic combination, the net dispersion should be zero. Thus,

(μy1)ωA=(μy1)ωA

AA=(μy1)ω(μy1)ω=0.517×0.030.621×0.05

AA=0.5 .......(1)

The deviation in the yellow ray produced by the crown prism is,

δ=(μy1)A

And by the flint prism is,

δ=(μy1)A

The net deviation produced by the combination is,

δδ=(μy1)A(μy1)A

1=0.517A0.621A ......(2)

Solving (1) and (2) we get,

A=4.8 and A=2.4

Thus, the crown prism should have its refracting angle 4.8 and that of the flint prism should be 2.4.

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, (D) is the correct answer.
Why this question ?

Note: Achromatic combination means that colours of the white light are not separated by the combination of prisms. Therefore, achromatic combination refers to the deviation without dispersion.

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