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Q.
The refracting angle of a prism is A and refractive index is $\cot(A/2)$. The angle of minimum deviation is
JIPMERJIPMER 2012Ray Optics and Optical Instruments
Solution:
Here, $u = \cot \frac{A}{2}$
According to prism formula
$\mu=\frac{\sin\left(\frac{A +\delta_{m}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \:\:\: \therefore \:\: \cot \frac{A}{2} =\frac{\sin\left(\frac{A+\delta_{m}}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
or $\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}=\frac{\sin\left(\frac{A+\delta_{m}}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
or $\cos\left(\frac{A}{2}\right) = \sin\left(\frac{A+\delta_{m}}{2}\right)$
or $\sin \left(90^\circ -\frac{A}{2}\right) = \sin\left(\frac{A+\delta_{m}}{2}\right)$
or $90^\circ - \frac{A}{2} =\frac{A+\delta_{m}}{2}$
or $180^\circ = A = A +\delta_{m} $
or $\delta_{m} = 180^\circ - 2A$