Correct Answer - D
In figure -I, total internal reflection take place at face `AB`.
`/_EMD+/_MDE=/_BEM`
`implies beta(90-alpha)=45^(@)`
`impliessin(beta+45^(@))gt(1/(mu))`
`implies sin beta+cosbeta gt (sqrt(2))/(mu)`……….(1)
According to law of refraction at point `M`, we can write
`1xxsin(45^(@))=musin betaimpliessin betas=1/(musqrt(2))impliescosbeta=sqrt(1-1/(2mu^(2))`
Putting these value in equation (1), we have
`1/(musqrt(2))+sqrt(1- 1/(2mu^(2))) gt (sqrt(2))/(mu)implies sqrt(2mu^(2)-1) gt 1implies mu gt 1`
In figure II: Replace the Dove Prism, by glass cube of side as `AB`
`Deltal=(DeltaL)/(sqrt(2))`.............(2)
From geometry, we can write
`sinbeta=1/(mu_(R)sqrt(2))` and `sin (beta-Deltabeta)=1/((mu_(R)+Deltamu_(R))sqrt(2))`
`DeltaL=a tan beta-a tan (beta-Deltabeta)=1/(sqrt(2mu_(R)^(2)-1))-a/(sqrt(2(mu_(|r)+Deltamu)^(2)-1))`
`impliesDelta=a/(sqrt(2mu_(R)^(2)-1))(1-sqrt((2mu_(R)^(2)-1)/(2(mu_(R)+Deltamu)^(2)-1)))=a/(sqrt(mu_(R)^(2)-1))(1-sqrt((2mu_(R)^(2)-1)/(2(mu_(R)+Deltamu)^(2)-1)))`
`impliesDeltaL=a/(sqrt(2mu_(R)^(2)-1))(1-sqrt(2mu_(R)^(2)-1)/(2mu_(R)^(2)+4mu_(R)Deltamu+2(Deltamu)^(2)-1))`
Neglecting the value `(Deltamu)^(2)`, we have
`impliesDelta=a/(sqrt(2mu_(R)^(2)-1))(1-sqrt(1/(1+(4mu_(R)Deltamu)/(2mu_(R)^(2)-1))))-a/(sqrt(2m_(R)^(2)-1))(1-{1+(4mu_(R)Deltamu)/(2mu_(R)^(2)-1)}^(-1/2))`
Expand binomially, we have
`impliesDeltaL=(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)(a tan beta))xxDeltabeta`
`DeltaL=(asec^(2)beta)xxDeltabeta`
Also `sinbeta=1/(sqrt(2)mu_(R))`
`impliescos beta Deltabeta=1/(sqrt(2)mu_(R)^(2))Deltamu`
`impliesDeltaL=(asec^(2)betaxxDeltamu)/(sqrt(mu_(R)^(2)cosbeta`
`impliesDeltaL=(2amu_(R)Deltamu)/((2mu_(R)^(2)cosbeta)`
Third method:
`Deltal=(DeltaL)/(sqrt(2)), sin alpha = 1/ (mu_(r) sqrt(2)), sin beta=1/(mu_(v)sqrt(2))`
`DeltaL=atan alpha =a tan beta`
`=[1/(sqrt(2mu_(r)^(2)-1))-1/(sqrt(2mu_(v)^(2)-1))]`
`=a[1/2-1/(2sqrt(1+mu_(r)Deltamu))]`
`=a/2 [a-(1+mu_(r)Deltamu)^(-1//2)]`
`=a/2[1-1+1/2mu_(r)Deltamu]`
`=a/4mu_(r)Deltamu`
`Deltal=(DeltaL)/(sqrt(2))=a/(4sqrt(2)) mu_(r)Deltamu=(4cm)/(4sqrt(2))xx(sqrt(5))/(sqrt(2))xx2/100`
`=(sqrt(5))/100 cm = sqrt(5)xx10^(-4)m`
`=2.25xx10^(-4)m`
Since : `Deltal ge d`
`impliesd le 2.25xx 10^(-4)m`
`sqrt(2mu_(v)^(2)-1)=sqrt(2(mu_(r)+Deltamu)^(2)-1)`
`impliessqrt(4+4mu_(r)Deltamu)[(Deltamu)^(2)~~0]`
`=2sqrt(1+mu_(r)Deltamu)`