A ray of light travelling in air is incident at an angle of incidence $i \approx 90^{o}$ on a long rectangular slab of a transparent medium of thickness $y_{0} .$ The medium has a variable refractive index of $μ (x) = \sqrt{1 + e^{2 x / a}} \forall x \geq 0,$ Where ‘a’ is a positive constant. Which of the following is/are correct?

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If $y_{0} = \frac{a}{2},$ the angle made by light ray with +ve x-axis at the upper surface of slab-air interface, inside the medium is $\tan^{- 1} (2)$

In the above situation if $y_{0} = a / 2,$ the coordinates of the point where the ray intersects the upper surface of the slab- air boundary are $(\frac{a}{2} l n 2, a / 2)$

If $y_{0} = \frac{a}{2},$ the angle made by light ray with +ve x-axis at the upper surface of slab –air interface, inside the medium is $\tan^{- 1} (\frac{1}{2})$

If $y_{0} = a / 2,$ the coordinates of the point where the ray intersects the upper surface of the slab- air boundary are $(a l n 2, a / 2)$

answer is A, C.

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Detailed Solution

$\tan θ = \frac{1}{\sqrt{μ^{2} - 1}} [∵ μ \sin θ = 1]$
$∴ \frac{d y}{d x} = e^{- x / a}$
$∴ y = a (1 - e^{- x / a})$
$∵$ At y = a/2, x = a In 2
$\tan θ = 1 / 2 a t x = a I n 2$
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