The refractive index of the medium within a certain region, $x > 0, y > 0,$ changes with y. A thin light ray travelling in the x-direction strikes the medium at right angles and moves through the medium along a circular arc.
How does the refractive index depend on y?
[R : Radius of path & $η_{0}$ : Initial refractive index]
(Light is entering at origin)

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$\frac{R}{R - y} \frac{η_{0}}{2}$

$\frac{R + y}{R} η_{0}$

$\frac{R}{R - y} η_{0}$

$\frac{2 R}{R - y} η_{0}$

answer is D.

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Detailed Solution

Equation of circle is
${(x - 0)}^{2} + {(y - R)}^{2} = R^{2}$ …………..(1)
$\frac{d y}{d x} = \tan θ$ …………..(2)
By Snell’s law $\sin (90 - θ) \times η = 1 \times η_{0}$ …………..(3)
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