Animated Solution for Physics - Optics: A concave lens of glass, refractive index 1.5 has both surfaces of same radius of curvature $R$. On immersion in a medium of refractive index 1.75, it will behave as a

The behavior of a lens is often thought to be an intrinsic property of its shape. A convex lens converges light, and a concave lens diverges light, right? Well, not always! The magic of optics lies in the interaction between the lens material and its surrounding environment. Let's embark on a thrilling journey to understand how a simple change of scenery can completely flip the identity of a lens.

Imagine you are holding a biconcave lens made of glass. The refractive index of this glass is ${\mu }_g=1.5$. Both surfaces of this lens have the exact same radius of curvature, $R$. In the familiar environment of air, this lens acts as a diverging lens, spreading out parallel rays of light.

But what happens if we plunge this lens into a mysterious liquid medium with a refractive index of ${\mu }_m=1.75$? To predict its new behavior, we must consult the master equation of optics.

The Lens Maker's Formula is the ultimate tool for determining the focal length of a lens in any given medium. It beautifully links the geometry of the lens with the optical densities of the materials involved:

Before we rush into substituting numbers, we must establish our sign convention. This is where many students fall into a trap! For a biconcave lens, the first surface (the one the light hits first) curves inwards. This means its center of curvature lies against the direction of the incident light, making $R_1=-R$. The second surface curves outwards, so its center of curvature lies in the direction of the light, making $R_2=+R$.

Now, let's carefully substitute our known values into the Lens Maker's Formula:

Let's break this down into atomic steps. First, we tackle the relative refractive index term:

Notice that this term is negative! This is the critical moment. Because the surrounding medium is optically denser than the lens material, the relative refractive index is less than 1.

Next, we simplify the geometric term:

Now, we multiply these two results together. The negative signs beautifully cancel each other out:

Taking the reciprocal, we find the new focal length of the lens:

The focal length is positive! In the realm of optics, a positive focal length is the hallmark of a convergent lens.

Our initially diverging concave lens has miraculously transformed into a converging lens simply because it was immersed in a denser medium. This profound physical insight teaches us that the nature of a lens is not absolute; it is entirely relative to the world around it.