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The dispersive powers of crown and flint...

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 from 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.

Text Solution

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Suppose, the angle of the crown prism needed is A and that of the filnt prism is A' We have
`omega = (mu_(upsilon) - mu_r)/(mu -1)`
or, `mu_upsilon - mu_r = (mu -1)omega.`
The angular dispersion produced by the crown prism is
`(mu_(upsilon -1)omega A.`
Similarly, the angular dispersion produced by the flint prism is
`(mu' - 1) omega' A'`
For achromatic combination, the net dispersion should be zero. Thus,
(mu - 1)omegaA = (mu' -1)omega'A`
or `(A')/(A) = ((mu -1)omega)/((mu' - 1)omega') = (0.517 xx 0.03)/(0.621 xx 0.05) = 0.50. ... (i)`
The deviation in the yellow ray produced by the crown prism is `delta = (mu -1)A` and by the combination is
`delta - delta' = (mu -1)A - (mu' - 1)A'`
or , `1^@ = 0.517 A - 0.621 A'. .... (ii)`
Solving (i) and (ii), `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^@.`
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