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The length of deviation for a glass prism is equal to its refracting angle. The refractive index of glass is 1.5. Then the angle of prism is
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The correct answer is:
$2 \cos ^{-1}(3 / 4)$
Here, $\delta_m=A, \mu=1.5$
$\therefore \mu=\frac{\sin \frac{\left(A+\delta_m\right)}{2}}{\sin \frac{A}{2}}=\frac{\sin \frac{(A+A)}{2}}{\sin \frac{A}{2}}$
$=\frac{2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right)}{\sin \left(\frac{A}{2}\right)}$
$\frac{3}{2}=2 \cos \frac{A}{2} \quad\left(\because \mu=1.5=\frac{3}{2}\right)$
$\cos \frac{A}{2}=\frac{3}{4}$
$\frac{A}{2}=\cos ^{-1}\left(\frac{3}{4}\right)$ or $A=2 \cos ^{-1}\left(\frac{3}{4}\right)$
$\therefore \mu=\frac{\sin \frac{\left(A+\delta_m\right)}{2}}{\sin \frac{A}{2}}=\frac{\sin \frac{(A+A)}{2}}{\sin \frac{A}{2}}$
$=\frac{2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right)}{\sin \left(\frac{A}{2}\right)}$
$\frac{3}{2}=2 \cos \frac{A}{2} \quad\left(\because \mu=1.5=\frac{3}{2}\right)$
$\cos \frac{A}{2}=\frac{3}{4}$
$\frac{A}{2}=\cos ^{-1}\left(\frac{3}{4}\right)$ or $A=2 \cos ^{-1}\left(\frac{3}{4}\right)$
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