InterviewSolution
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InterviewSolution
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ABC is an isosceles right angled triangular reflecting prism of average refractive index mu . When incident rays on face AB are parallel to face BC , then they emerges from Face AC . which are also parallel to Face BC as shown in figure-l. The prism capable to do so , known as Dove Prism. In figure-ll, the Dove Prism is used for dispersion of incident light containing red colour and violet colour only. The red colour and violet colour lights are separated( displaced ) on screen by a distance Af. In reality , each ray of any colour has some width, which can be designated as d. It is clear that an observer can distinguish the red and violet rays that emerges from prism only if Delta/varphige d .Otherwise the bundles of rays will overlap and mix. Q . Choose the INCORRECT option. |
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Answer» As per figure-l, the average refractive index of Dove Prism may be greater than 1. `/_EMD + /_ MDE = /_ BEM` `beta + (90^@ - alpha) = 45^@ implies alpha = beta+ 45^@` `implies sin(beta + 45^@)> (1/mu)` `sin beta + cos beta gtsqrt(2)/mu`…..(1) According to law of refraction at point M, we can WRITE `1xxsin45^@= musn beta implies sin beta = 1/(musqrt(2)) implies cos beta= sqrt(1 - 1/(2mu^(2)))` Putting these value in equation (1), we have `1/(musqrt(2)) + sqrt(1 - 1/(2mu^(2))) > sqrt(2)/mu implies sqrt(2mu^(2) -1) > implies mu gt 1` In figure-II : REPLACE the Dove Prism by glass cube of side as AB. `Delta l =(DeltaL)/sqrt(2)` ....(2) From geometry, we can write `sin beta = 1/ (mu_(r)sqrt(2))` and `sin(beta - Deltabeta)= 1/((mu_(R) + Deltamu_(R))sqrt(2))` `Delta L = atan beta - a tan (beta- Deltabeta) = a/sqrt(2mu_(g)^(2)-1) - a/sqrt(2(mu_(R) + Deltamu)^(2) -1)` `DeltaL = a/sqrt(2mu_(R)^(2) -1 )(1- sqrt((2mu_(R)^(2)-1)/(2mu_(g)^(2) + 4mu_(R)Deltamu + 2(Deltamu)^(2) -1)))` Neglecting the value `(Deltamu)^(2)`, we have, `implies DeltaL = a/(sqrt(2mu_(R)^(2)-1)) (1 -sqrt(1/(1+ (4mu_(R)Deltamu)/(2mu_(R)^(2)-1)))) = a/sqrt(2mu_(R)^(2) -1) ( 1-{1 + (4mu_(R)Deltamu)/(2mu_(R)^(2)-1)}^(1/2))` Expand binomically, we have `implies Delta L = (2amu_(R)Deltamu)/((2mu_(R)^(2)-1)^(3/2))` Putting this value in equation (2), we have `Deltal = (sqrt(2)amu_(R)Deltamu)/((2mu_(R)^(2)-1)^(3/2))` Second method `DeltaL = (d/(dbeta)(atanbeta))xxDeltabeta` `DeltaL=(asec^(2)beta)xxDeltabeta` Also `sin beta= 1/(sqrt(2)mu_(R))` `implies cos betaDeltabeta= 1/(sqrt(2)mu_(R)^(2)) Deltamu` `implies Delta L = (asec^(2)betaxxDeltamu)/(sqrt(mu_(R)^(2)cosbeta)` `DeltaL = (2amu_(R)Deltamu)/((2mu_(R)^(2)-1)^(3/2))` 
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