Question

If the angle of minimum deviation produced by an equilateral prism is equal to the angle of the prism, then the refractive index of the material of the prism is nearly

• A. 1.515
• B. 1.414
• C. 1.732
• D. 1.625

Hint:

The condition for minimum deviation is a special case where the angle of incidence equals the angle of emergence, and the light ray passes symmetrically through the prism. The prism formula is derived directly from applying Snell's law at both faces under this symmetric condition.

Answer 1

The Correct Option is C

The formula for the refractive index (\(\mu\)) of a prism is given by:
\( \mu = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \), where A is the angle of the prism and \(D_m\) is the angle of minimum deviation.
We are given that the prism is equilateral, so its angle is \( A = 60^\circ \).
We are also given that the angle of minimum deviation is equal to the angle of the prism, so \( D_m = A = 60^\circ \).
Now, substitute these values into the formula for the refractive index.
\( \mu = \frac{\sin\left(\frac{60^\circ + 60^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \).
\( \mu = \frac{\sin\left(\frac{120^\circ}{2}\right)}{\sin(30^\circ)} = \frac{\sin(60^\circ)}{\sin(30^\circ)} \).
We know the values of these trigonometric functions:
\( \sin(60^\circ) = \frac{\sqrt{3}}{2} \).
\( \sin(30^\circ) = \frac{1}{2} \).
Substitute these values to find \(\mu\).
\( \mu = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \).
The numerical value of \( \sqrt{3} \) is approximately 1.732.