A prism has a refracting angle 'A'. The secon | vedclass

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(D) For the light ray to retrace its path after reflection from the silvered surface,it must strike the silvered surface normally.In the prism,the ang

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$2 \cos A$

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(D) For the light ray to retrace its path after reflection from the silvered surface,it must strike the silvered surface normally.In the prism,the angle of refraction at the first surface is '$r_1$'. From the geometry of the prism,the angle of incidence at the second surface is '$r_2$'. Since the ray strikes normally,'$r_2 = 0$'.Using the relation '$A = r_1 + r_2$',we get '$A = r_1 + 0$',so '$r_1 = A$'.Applying Snell's law at the first surface: '$\mu = \frac{\sin i}{\sin r_1}$'.Given the angle of incidence '$i = 2A$',we have '$\mu = \frac{\sin 2A}{\sin A}$'.Using the trigonometric identity '$\sin 2A = 2 \sin A \cos A$',we get '$\mu = \frac{2 \sin A \cos A}{\sin A} = 2 \cos A$'.Thus,the refractive index of the material of the prism is '$2 \cos A$'.

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