Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are 2.8 (medium-1) and 6.8 (medium-2), respectively. To satisfy the condition, so that the reflected and refracted rays are perpendicular to each other, the angle of incidence should be $\tan^{- 1} {(1 + \frac{10}{θ})}^{\frac{1}{2}}$ the value of $θ$ is ________

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answer is 7.

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Detailed Solution

We have $μ_{1} = \sqrt{2.8 \times 1} = \sqrt{2.8} [∵ μ = \sqrt{ε_{r} μ_{r}}]$ and $μ_{2} = \sqrt{6.8 \times 1} = \sqrt{6.8}$ $i + r = 90^{o}$

$\Rightarrow r = 90^{0} - i \Rightarrow \sin r = \sin (90 - i) \Rightarrow \sin r = \cos i B y s n e l l s l a w, μ_{1} \sin i = μ_{2} \sin r \Rightarrow μ_{1} \sin i = μ_{2} \cos i \Rightarrow \tan i = \frac{μ_{2}}{μ_{1}} = \frac{\sqrt{6.8}}{2.8} \Rightarrow \tan i = {(\frac{2.8 + 4}{2.8})}^{1 / 2} \Rightarrow i = \tan^{- 1} {(1 + \frac{10}{7})}^{1 / 2}$