KVPY SX SB Exam 2016 Question Paper

Question : 99 of 120

Marks: +1, -0

A glass prism has a right-triangular cross section ABC, with ∠A = 90º. A ray of light parallel to the hypotenuse BC and incident on the side AB emerges grazing the side AC. Another ray, again parallel to the hypotenuse BC, incident on the side AC suffers total internal reflection at the side AB. Which one of the following must be true about the refractive index µ of the material of the prism ?

$√{3/2} < µ < √2$
$µ > √3$
$µ < √{3/2}$
$√2 < µ √3$

Validate

Solution:

r + θ_C = 90º

…..(1)
1 × sin (90º – α) = µsin r
cos α = µ sin r …..(2)
90º – e >

θ_C

…..(3)
µsin e = 1 × sin α …..(4)
(3) & (4)
90º –

θ_C

> e
cos

θ_C

> sin e

\cosθ_c > {\sin α}/µ

1-\sin^2θ_C > 1/µ^2[1-µ^2\sin^2r]

1-1/µ^2 > 1/µ^2 [1-µ^2\sin^2(90°-θ_C)]

1-1/µ^2 > 1/µ^2-\cos^2θ_C

1-2/µ^2 > -[1-1/µ^2]

2 > 3/µ^2

µ > √{3/2}

(1) & (2)
cos α = µ sin (90º –

θ_C

)
cos α = µ cos

θ_C

cos α < 1
µ cos

θ_C

< 1

√{1-1/µ^2} < 1/µ

1-1/µ^2 < 1/µ^2

1 < 2/µ^2

µ < √2

∴√{3/2} < µ < √2

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