A ray of light passes through a medium whose refractive index varies with distance as n=n01+xa. If ray enters the medium parallel to x -axis, The time taken for ray to travel between x = 0 and x = a is Pn0a2c, then what is ‘ P ’ ?

We know that μ=cv⇒v=cμIf v be the speed at a distance x from y-axis, thenv=cn01+xadxdt=cn01+xa⇒∫01 dt=n0c∫0a 1+xadx⇒t=3n0a2c

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A ray of light passes through a medium whose refractive index varies with distance as $n = n_{0} (1 + \frac{x}{a})$ . If ray enters the medium parallel to x -axis, The time taken for ray to travel between x = 0 and x = a is $\frac{{Pn}_{0} a}{2 c}$ , then what is ‘ $P$ ’ ?

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Detailed Solution

We know that $μ = \frac{c}{v} \Rightarrow v = \frac{c}{μ}$
If v be the speed at a distance x from y-axis, then
Question Image
$\begin{array}{l} v = \frac{c}{n_{0} (1 + \frac{x}{a})} \\ \frac{dx}{dt} = \frac{c}{n_{0} (1 + \frac{x}{a})} \\ \Rightarrow \int_{0}^{1} dt = \frac{n_{0}}{c} \int_{0}^{a} (1 + \frac{x}{a}) dx \\ \Rightarrow t = \frac{3 n_{0} a}{2 c} \end{array}$