A ray of monochromatic light is incident at angle ‘i’ on the surface of glass slab. If the angle of refraction ‘r’ is twice the angle of incidence, the angle of incidence is_ _ _ _ _ _ _ _ .

a)      i =  sin ^{- 1} ( μ / 2 )

b)      i = 2 cos ^{- 1} ( μ / 2 )

c)       i =  cos ^{- 1} ( μ / 2 )

d)      i = 2 sin ^{- 1} ( μ / 2 )

Solution :

Given :

i = 2 r

i . e . 

r = i / 2

- - - - - - - - - - - - - - - - - - - - - - - - ( 1 )

According to Snell’s law of refraction ‘the refractive index of material is equal the ratio of the sine of incident angle to the sine of angle of refraction’.

i .e .

μ = ( sin i ) / ( sin r )

 - - - - - - - - - - - - - - - - - - - - - - - ( 2 )

From equation ( 1 ) & ( 2 ) ,

μ = ( sin i ) / [ Sin ( I / 2 ) ]

Therefore ,

μ = [sin 2 ( I / 2 ) ] / [Sin ( I / 2 ) ]

 - - - - - - - - - - - - - - - - - - - - - - - (3)

we know that ,

sin 2 θ = 2 sin θ cos θ

Therefore equation ( 3 ) becomes ,

μ = [ 2 sin ( I / 2 ) cos ( I / 2 ) ] / [ Sin ( I / 2 ) ]

μ = [ 2 cos ( I / 2 ) ]

cos ( I / 2 ) = μ / 2

i / 2 = cos ^{- 1} ( μ / 2 )

i = 2 cos ^{- 1} ( μ / 2 )

A ray of monochromatic light is incident at angle ‘i’ on the surface of glass slab. If the angle of refraction ‘r’ is twice the angle of incidence, the angle of incidence is i = 2 cos ^{- 1} ( μ / 2 )