Animated Solution for Physics - Optics: The variation of refractive index of a crown glass thin prism with wavelength of the incident light is shown. Which of the following graphs is the correct one, if $D_m$ is the angle of minimum deviation ?

The problem presents us with a fascinating visual: a graph showing how the refractive index ($n$) of a crown glass thin prism changes with the wavelength ($\lambda$) of incident light. As we observe the curve, a clear trend emerges. As the wavelength increases, the refractive index decreases. But it doesn't just drop linearly; it forms a gentle, downward-sloping curve that is concave upwards. Our mission is to determine how the angle of minimum deviation ($D_m$) behaves under these same conditions.

To bridge the gap between refractive index and minimum deviation, we need to recall the fundamental physics of a thin prism. For a prism with a very small refracting angle ($A$), the angle of minimum deviation is given by the elegant formula:

This equation is our master key. Let's break it down. The prism angle $A$ is a geometric property of the glass piece; it is a constant. Therefore, the entire variation in $D_m$ is driven solely by the term $(n-1)$.

Because $A$ is constant, we can establish a direct proportionality:

This linear relationship is crucial. It tells us that whatever mathematical "dance" the refractive index $n$ performs, the minimum deviation $D_m$ must follow the exact same choreography. If $n$ decreases as a concave-upward curve, then $D_m$ must also decrease as a concave-upward curve. There is no squaring, no inversion, and no complex transformation—just a direct, linear scaling.

When we examine the given options, we are looking for a graph that perfectly mimics the shape of our original $n$ vs $\lambda$ graph. Option (a) is a parabola, option (b) is a straight line, and option (d) is an increasing straight line. Only Option (c) presents the correct decreasing, concave-upward curve.

Cauchy's Formula

You might wonder, why does the refractive index curve in this specific way? The answer lies in Cauchy's Formula, an empirical relationship that describes how light interacts with transparent materials:

For most practical purposes, we approximate this to $n\approx A+\frac{B}{{\lambda }^2}$. This inverse-square dependence on wavelength perfectly explains the concave-upward shape of our graph. Shorter wavelengths (like blue light) experience a higher refractive index and bend more, while longer wavelengths (like red light) experience a lower refractive index and bend less. This beautiful interplay of optics is what creates the dispersion of light, and understanding it is key to mastering JEE Physics!