Animated Solution for Physics - Optics: Which one of the following spherical lenses does not exhibit dispersion? The radii of curvature of the surfaces of the lenses are as given in the diagrams

Have you ever wondered why a simple magnifying glass sometimes creates colorful fringes around the edges of an image? This beautiful but often unwanted effect is called chromatic aberration, and it happens because of dispersion.

Dispersion occurs because the refractive index of a material, $\mu$, is not a constant. It actually changes depending on the wavelength (or color) of the light passing through it. Because of this, a lens will bend blue light slightly more than red light, causing them to focus at different points.

To understand how to eliminate this effect, we need to look at the Lens Maker's Formula, which tells us how the focal length $f$ of a thin lens depends on its shape and material:

Here, $R_1$ and $R_2$ are the radii of curvature of the two surfaces of the lens.

If we want a lens that does not exhibit dispersion, its focal length must remain exactly the same for all colors. Mathematically, this means the focal length should not change when the refractive index $\mu$ changes.

We can express this condition by setting the derivative of the lens power with respect to $\mu$ to zero:

Let's differentiate the Lens Maker's Formula. The derivative of $(\mu -1)$ with respect to $\mu$ is simply $1$. This leaves us with:

Solving this simple equation gives us the magic condition:

So, for a lens to have zero dispersion, both of its surfaces must have the exact same radius of curvature and the same sign.

If we look at the given options, option (c) shows a meniscus lens where both surfaces curve in the same direction with the same radius $R$.

But wait, if $R_1=R_2$, what happens to the focal length? If we plug this back into the Lens Maker's Formula, we get $\frac{1}{f}=0$, which means the focal length is infinite.

Such a "lens" has zero optical power. It acts just like a curved glass slab (often called a watch glass). Because it doesn't converge or diverge light rays, it doesn't split them into different colors either. Thus, it perfectly satisfies the condition of exhibiting no dispersion!