56. For the angle of minimum deviation of a prism to be equal to its refracting angle, the prism must be made of a material whose refractive index ?

56. For the angle of minimum deviation of a prism to be equal to its refracting angle, the prism must be made of a material whose refractive index ?

(a) lies between √2 and 1

(b) lies between 2 and √2

(c) is less than 1

(d) is greater than 2

Answer :  (b) lies between 2 and √2

\(\delta_{min} = A \quad \mu = ?\)

\(\frac{\sin\left(\tfrac{\delta_{min}+A}{2}\right)}{\sin\tfrac{A}{2}} = \mu\)

\(\frac{\sin A}{2\sin\tfrac{A}{2}} = \mu\)

\(\frac{2\sin\tfrac{A}{2}\cos\tfrac{A}{2}}{\sin\tfrac{A}{2}} = \mu\)

\(2\cos\tfrac{A}{2} = \mu\)

\(\text{If } \mu = 2\)

\(\cos\tfrac{A}{2} = 1\)

\(A = 0\)

\(\text{Angle of prism cannot be zero.}\)

\(\text{Max angle}\)

\(A = 90^\circ , \quad \mu = \sqrt{2}\)

\(\cos 45^\circ = \tfrac{1}{\sqrt{2}}\)

\(\mu = 1\)

\(\cos\tfrac{A}{2} = \tfrac{1}{2}\)

\(\tfrac{A}{2} = 60^\circ\)

\(A = 120^\circ\)

\(A \to 0^\circ \text{ to } 90^\circ\)

\(\mu \to 2 \text{ to } \sqrt{2}\)