Physics

Solution:

$sinc=μ1 =1.5241 =0.65orc=40_{∘}30_{_{′}}r=19_{∘}30_{_{′}}=[180_{∘}−120_{∘}−40_{∘}30_{_{′}}]μ=sinrsini orsini=1.524sin19_{∘}30_{_{′}}=1.524(.3340)=0.5ori=30_{∘}$

A ray of light enters at grazing angle of incidence into an assembly of three isosceles right-angled prisms having refractive indices $μ_{1}=2 ,μ_{2}=x $ and $μ_{3}=3 $ . If finally emergent light ray also emerges at grazing angle then calculate x.

Solution:

Apply Snell's law on various surfaces one by one :

1 sin $90_{∘}$ $=μ_{1}$ sin $r_{1}⇒$ sin $r_{1}=2 1 ⇒r_{1}=45_{∘}$

We have the incident angle for the second interface given using the relation sin$i_{2}$=cos$r_{1}$

Thus we get

$μ_{1}$ cos $r_{1}$= $μ_{2}$ sin $r_{2}⇒$ sin $r_{2}$ = $μ_{2}1 $(cos$r_{2}$=$2 1 $)

$μ_{2}$ cos $r_{2}=μ_{3}$ sin $r_{3}$ $⇒$ sin $r_{3}$ $=3 μ_{2}1−sin_{2}r_{2} $

$μ_{3}$ cos $r_{3}=1$ = $3 μ_{2}−1 $

sin $_{2}r_{3}$ + cos $_{2}r_{3}=1$

$⇒3μ_{2}−1 +31 =1⇒μ_{2}_{2}=3⇒μ_{2}=3 $

Thus we get $x=3.$