Question

If minimum deviation for an equilateral prism is \(30^\circ\), then refractive index of the prism is:

• A. \(\sqrt{2}\)
• B. \(\frac{\sqrt{3}}{2}\)
• C. 2
• D. 4

Hint:

For an equilateral prism, the minimum deviation is used to calculate the refractive index using the formula: \(n = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)}\), where \(A\) is the angle of the prism and \(D\) is the minimum deviation.

Answer 1

The Correct Option is A

For an equilateral prism, the angle of the prism (\(A\)) is \(60^\circ\), and the minimum deviation (\(D\)) is \(30^\circ\). The refractive index (\(n\)) is given by the formula: \[ n = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \] Substitute the given values: \[ n = \frac{\sin\left(\frac{60^\circ + 30^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \] \[ n = \frac{\sin(45^\circ)}{\sin(30^\circ)} \] We know that: \[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \] Substitute these values into the equation: \[ n = \frac{\frac{\sqrt{2}}{2}}{\frac{1}{2}} = \sqrt{2} \] Thus, the refractive index of the prism is \(\sqrt{2}\). Final Answer: Option (A) \(\sqrt{2}\).