• # question_answer A large transparent cube (refractive index$=1.5$) has a small air bubble inside it. When a coin (diameter$2\text{ }cm$) is placed symmetrically above the bubble on the top surface of the cube, the bubble cannot be seen by looking down into the cube at any angle. However, when a smaller coin (diameter$1.5cm$) is placed directly over it, the bubble can be seen by looking down into the cube. What is the range of the possible depths d of the air bubble beneath the top surface?   A) $\frac{3\sqrt{5}}{8}cm<d<\frac{\sqrt{5}}{2}cm$ B) $\frac{2\sqrt{3}}{8}cm<d<\frac{3\sqrt{5}}{2}cm$ C) $\frac{\sqrt{5}}{7}cm<d<\frac{2\sqrt{5}}{7}cm$ D) $\frac{\sqrt{3}}{7}cm<d<\frac{3\sqrt{3}}{7}cm$

 [a] The depth of the bubble shall not be more than that shown in the figure, when the coin of diameter $2\text{ }cm$is placed above it. A larger depth will mean that that the angle of incidence at points not covered by the coin can be lesser than the critical angle [c]. $\mu \,\,\sin C=1$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\operatorname{tanC}=\frac{1}{\sqrt{{{\mu }^{2}}-1}}$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\operatorname{tanC}=\frac{2}{\sqrt{5}}$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{d}=\frac{2}{\sqrt{5}}$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,d=\frac{\sqrt{5}}{2}$ The depth at which the bubble will be just visible if the smaller coin is placed can be similarly calculated as $\tan C=\frac{2}{\sqrt{5}}$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\frac{3/4}{d}=\frac{2}{\sqrt{5}}$ $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,d=\frac{3\sqrt{5}}{8}$ If the bubble happens to be below this depth, it will be certainly visible. Therefore, \[\frac{3\sqrt{5}}{8}