The refracting angle of a prism is \( A \), and refractive index of the material of the prism is \( \cot(A/2) \). The angle of minimum deviation is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 180^\circ - 2A \)
At minimum deviation the ray passes symmetrically through the prism, and the refractive index is given by the prism formula:
\( \displaystyle \mu = \frac{\sin\left(\frac{A + \delta_{m}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \)
Putting \( \mu = \cot(A/2) = \cos(A/2)/\sin(A/2) \):
\( \displaystyle \frac{\sin\left(\frac{A + \delta_{m}}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \)
\( \displaystyle \sin\left(\frac{A + \delta_{m}}{2}\right) = \cos\left(\frac{A}{2}\right) = \sin\left(90^\circ - \frac{A}{2}\right) \)
\( \displaystyle \frac{A + \delta_{m}}{2} = 90^\circ - \frac{A}{2} \)
\( \delta_{m} = 180^\circ - 2A \)