It is required to form an achromatic combination of two prisms of dispersive powers 0.04 and 0.06 respectively . If the first prism has a refracting angle of `10^(@)` and mean refractive index of `1.52` , what must be the refracting angle of the second prism , if its mean refractive index is `1.56` ?

Options

A. `5.5 ^(@)`
B. `4.8^(@)`
C. `6.2 ^(@)`
D. `7.4 ^(@)`

Answer

C

Explanation

Given : `A = 10^(@) , omega = 0.04 , omega' = 0.06`,
`n_(y) = 1.52 ,n'_(y)= 1.56`
For achromatic combination of two prisms of angles A and A' , we get deviation without dispersion and the condition is
`(A)/(A') = (n_(v)- n_(R))/(n_(v) - n_(R)) " " ... (1)`
But `omega = (n_(v) - n_(R))/(n_(y)- 1)`
`therefore n_(v) - n_(R) = omega(n_(y) - 1) = 0.04 (1.52 - 1) = 0.04 xx 0.52`
Similarly `n'_(v) - n'_(R) = omega' (n'_(y) - 1) = 0.06(1.56-1) = 0.06 xx 0.56`
Using the values in (1) , we get
`(10)/(A') = (0.06 xx 0.56)/(0.04 xx 0.52) = (3)/(2) xx (14)/(13) = (21)/(13)`
`therefore 21 A' = 130 therefore A' = (130)/(21) = 6.2 ^(@)`