A ray of light passes through a medium whose refractive index varies with distance as $n = n_{0} (1 + \frac{x}{2 a})$ .ray enters the medium parallel to x-axis, what w be the time taken for ray to travel between x=0 an x=a ?

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$t = \frac{5 n_{0} a}{4 c}$

$t = \frac{6 n_{0} a}{4 c}$

$t = \frac{5 n_{0} a}{3 c}$

$t = \frac{5 n_{0} a}{7 c}$

answer is D.

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Detailed Solution

Since, we know that $μ = \frac{c}{v}$

$\Rightarrow v = \frac{c}{μ}$

So, if v be the speed at a distance x from y-axis, then

$v = \frac{c}{n_{0} (1 + \frac{x}{2 a})}$

since, $v = \frac{d x}{d t}$ , so we have

$\frac{d x}{d t} = \frac{c}{n_{0} (1 + \frac{x}{2 a})} \Rightarrow \int_{0}^{1} d t = \frac{n_{0}}{c} \int_{0}^{a} (1 + \frac{x}{2 a}) d x t = \frac{5 n_{0} a}{4 c}$

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