Intensity of a light beam can be defined as amount of light energy incident in unit time on a unit area held normal to the direction of beam. A light beam of intensity $I_{0}$ has a circular cross section of diameter $d_{0}$ . This beam is travelling in a medium of refractive index $μ = \sqrt{2}$ and gets incident on the medium – air boundary at an angle of incidence $i = 30 °$ . Assume that entire light energy gets transmitted into air. The intensity $(I)$ of transmitted light beam is $\frac{x}{2} I_{0} . x is$

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Detailed Solution

$\sqrt{2} \sin 30 ° = 1. \sin r \Rightarrow r = 45 ° \frac{P R}{P Q} = \cos i \Rightarrow P Q = \frac{d_{0}}{\cos 30 °} = \frac{d_{0}}{\sqrt{3}} Q S = d = diameter of refracted light \frac{d}{P Q} = \cos r \Rightarrow P Q = \frac{d}{\cos 45 °} = \sqrt{2} d ∴ \sqrt{2} d = \frac{2 d_{0}}{\sqrt{3}} \Rightarrow d = \sqrt{\frac{2}{3}} d_{0}$
The refracted beam has lesser diameter. Intensity of refracted beam
$I = \frac{I_{0} π {(\frac{d_{0}}{2})}^{2}}{π {(\frac{d}{2})}^{2}} = I_{0} . {(\frac{d_{0}}{d})}^{2}$
$∴ I = \frac{3}{2} I_{0}$
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