Animated Solution for Physics - Optics: The refractive index of glass is $1.520$ for red light and $1.525$ for blue light. Let $D_1$ and $D_2$ be angles of minimum deviations for red and blue light respectively in a prism of this glass. Then,

Have you ever looked at a rainbow and wondered how a simple drop of water or a piece of glass can unpack white light into such a brilliant spectrum of colors? This beautiful phenomenon is called dispersion, and it is the heart of the problem we are tackling today. Imagine you are Isaac Newton, sitting in a dark room, letting a single beam of sunlight pass through a glass prism. You would see exactly what our problem describes: different colors bending by different amounts. But why does this happen? Let's dive into the physics of it.

To understand why colors behave differently, we must first understand the refractive index ($\mu$). The refractive index is a measure of how much a medium slows down light. But here is the catch: a medium doesn't slow down all light equally.

According to Cauchy's Equation, the refractive index of a material depends on the wavelength ($\lambda$) of the light passing through it: Where $A$, $B$, and $C$ are constants for a given material.

Because blue light has a shorter wavelength than red light (${\lambda }_{\text{blue}}<{\lambda }_{\text{red}}$), the denominator for blue light is smaller, making the overall fraction larger. This means the glass presents a higher refractive index to blue light than to red light (${\mu }_{\text{blue}}>{\mu }_{\text{red}}$). The glass literally feels "denser" to the blue light!

Now, let's look at how the prism bends the light. When light passes through a prism, it deviates from its original path. The angle by which it bends is called the angle of deviation ($D$).

For a prism, the refractive index is related to the angle of minimum deviation ($D_m$) and the prism angle ($A$) by the famous prism formula:

If we are dealing with a thin prism (where the angle $A$ is very small), this complex trigonometric equation simplifies beautifully to a linear relationship:

Even if the prism isn't thin, the mathematical truth remains: the angle of minimum deviation ($D_m$) is a strictly increasing function of the refractive index ($\mu$). If you increase $\mu$, the light bends more, and $D_m$ increases.

Red vs. Blue

Let's bring the values from our problem into the arena. We are given the exact refractive indices for our specific glass prism: For red light: ${\mu }_{\text{red}}=1.520$ For blue light: ${\mu }_{\text{blue}}=1.525$

As we predicted using Cauchy's equation, the refractive index for blue light is indeed greater than that for red light:

Now, let's connect this back to our deviation formula. Since the deviation is directly proportional to the refractive index (or strictly increasing with it), the color that experiences the higher refractive index will suffer the greater deviation.

The problem defines $D_1$ as the angle of minimum deviation for red light, and $D_2$ as the angle of minimum deviation for blue light.

Since the blue light faces a higher refractive index, it gets bent more sharply by the prism. Therefore, its deviation $D_2$ must be strictly greater than the deviation of the red light $D_1$.

Or, written the other way around to match our options:

And there we have it! By simply understanding that shorter wavelengths experience higher refractive indices and therefore bend more, we've cracked the problem. This isn't just a math trick; it's the fundamental reason why the sky is blue, why rainbows form, and why a prism creates a spectrum. Physics is truly beautiful when you see the connections!