What will be the angle of minimum deviation if the refracting angle of a prism be \( 2\theta \), and the refractive index of the material of the prism be \( \cot\theta \)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( (\pi - 4\theta) \)
The refractive index of a prism in terms of its refracting angle \( A \) and the angle of minimum deviation \( \delta_m \) is
\( \displaystyle \mu = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \)
Put \( A = 2\theta \) and \( \mu = \cot\theta \).
\( \displaystyle \sin\left(\theta + \frac{\delta_m}{2}\right) = \cot\theta\,\sin\theta = \cos\theta = \sin\left(\frac{\pi}{2} - \theta\right) \)
\( \displaystyle \theta + \frac{\delta_m}{2} = \frac{\pi}{2} - \theta \;\Rightarrow\; \delta_m = \pi - 4\theta \)