Obtain an equation for the trajectory y(x) of the ray in the medium.

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Solution

The Variation of refractive index is given by
$n (y) = (k y^{\frac{3}{2}} + 1)^{\frac{1}{2}}$
It follows from the relation that the refractive index incraeses in the slab from $y = 0$ to $y = t$ . therefore, the ray $S A$ entering the slab at $A$ at grazing incidence ( $i = 90^{o}$ ) keeps bending towards the normal as it progresses in the slab, following a curved path $A B P$ .
If the angle of incidence at point $B (x, y)$ is $i$ , then the slope $C D$ of the trajectory at $B$ is
$\frac{d y}{d x} = t a n α = t a n (90^{o} - i) = c o t i$

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