A ray of light goes from medium 1 to medium 2. Velocity of light in the two media are c₁ and c₂ respectively. For an angle of incidence 0 in medium 1, the corresponding angle of refraction in medium 2 is ө/2.
(i) Which of the two media is optically denser and why?
(ii) Establish the relationship between ө, c₁ and c₂.

Solution

Given,

i) Angle of incidence in medium 1 = $θ$
Angle of refraction in medium 2 = $\frac{θ}{2}$

Angle of refraction in medium 2 is less than angle of incidence in medium 1. That is, the ray is bending towards the normal. Therefore, medium 2 is optically denser than medium 1.

ii) Using Snell's law,
$μ = \frac{\sin i}{\sin r}$ = $\frac{\sin θ}{\sin \frac{θ}{2}} = \frac{2 \sin \frac{θ}{2 . \cos \frac{θ}{2}}}{\sin \frac{θ}{2}}$
= 2 cos $\frac{θ}{2}$

Also, refractive index $μ = \frac{c_{1}}{c_{2}}$
This implies,
$μ = 2 \cos \frac{θ}{2} = \frac{c_{1}}{c_{2}}$

i.e., $θ = \frac{2 \cos^{- 1} c_{1}}{2 c_{2}}$
which is the required relationship between ө, c₁ and c₂ .

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