Velocity of light in diamond is $\left(\frac{5}{12}\right)^{\text {th }}$ times that in air. Velocity of…

Velocity of light in diamond is $\left(\frac{5}{12}\right)^{\text {th }}$ times that in air. Velocity of light in water is $\left(\frac{3}{4}\right)^{\text {th }}$ times that in air. The angle of incidence of ray of light travelling from water to diamond is (angle of refraction $\left.(r)=30^{\circ}\right)\left\lceil\right.$ Given $\left.\rightarrow \sin 30^{\circ}=\frac{1}{2}\right\rceil$
  1. $\sin ^{-1}(9 / 10)$
  2. $\sin ^{-1}(3 / 4)$
  3. $\sin ^{-1}(5 / 12)$
  4. $\sin ^{-1}(9 / 5)$

Solution

Given, $\mathrm{v}_{\mathrm{d}}=\frac{5}{12} \mathrm{c} \Rightarrow \frac{\mathrm{v}_{\mathrm{d}}}{\mathrm{c}}=\frac{5}{12}$ and $\mathrm{v}_{\mathrm{w}}=\frac{3}{4} \mathrm{c} \Rightarrow \frac{\mathrm{v}_{\mathrm{w}}}{\mathrm{c}}=\frac{3}{4}$ ${ }_w \mathrm{n}_{\mathrm{d}}=\frac{\mathrm{n}_{\mathrm{d}}}{\mathrm{n}_{\mathrm{w}}}=\frac{\frac{\mathrm{c}}{\mathrm{v}_{\mathrm{d}}}}{\frac{\mathrm{c}}{v_{\mathrm{w}}}}=\frac{\frac{12}{5}}{\frac{4}{3}}=\frac{12}{5} \times \frac{3}{4}=\frac{9}{5}$
Using, ${ }_w \mathrm{n}_{\mathrm{d}}=\frac{\sin \mathrm{i}}{\sin \mathrm{r}}$ $\begin{aligned} & \therefore \quad \sin i={ }_w n_d \times \sin r=\frac{9}{5} \times \sin 30^{\circ}=\frac{9}{5} \times \frac{1}{2}=\frac{9}{10} \\ & \therefore \quad i=\sin ^{-1}\left(\frac{9}{10}\right) \end{aligned}$

Asked in: MHT CET 2024 (09 May Shift 1)

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