Animated Solution for Physics - Optics: Four combinations of two thin lenses are given in Column I. The radius of curvature of all curved surfaces is $r$ and the refractive index of all the lenses is $1.5$. Match lens combinations in Column I with their focal length in Column II and select the correct answer using the codes given below the lists.

A Journey Through the Lens Maker's Formula

Imagine you are an optical engineer tasked with designing a complex camera system. You have a box full of different thin lenses—biconvex, plano-convex, plano-concave—and you need to know exactly how they behave when stacked together. This problem is a perfect simulation of that scenario. It tests your fundamental grasp of the Lens Maker's Formula and the principle of combination of thin lenses.

Let's break down the physics before we dive into the math.

The core tool we need is the Lens Maker's Formula, which relates the focal length $f$ of a lens to its refractive index $\mu$ and the radii of curvature of its two surfaces, $R_1$ and $R_2$:

When we place two thin lenses in contact, their ability to bend light (their optical power) simply adds up. Since power is the reciprocal of focal length, the equivalent focal length $f_{eq}$ is given by:

Now, let's systematically analyze each combination.

Two Biconvex Lenses

We start with two identical biconvex lenses. For a standard biconvex lens, the first surface bulges outward towards the incident light, meaning its center of curvature is on the right (positive). The second surface bulges outward away from the light, meaning its center is on the left (negative).

So, $R_1=r$ and $R_2=-r$. Substituting these into our master equation with $\mu =1.5$:

This tells us that the focal length of a single biconvex lens here is exactly $r$. When we combine two of them, their powers add up:

This perfectly matches with option (q).

Two Plano-Convex Lenses

Next, we have two plano-convex lenses. A plane surface is essentially a sphere with an infinite radius, so $R_1=\infty$. The curved surface is convex, so $R_2=-r$ (assuming light hits the plane surface first; the result is the same either way).

So, each plano-convex lens has a focal length of $2r$. Combining two of them:

This matches with option (s).

Two Plano-Concave Lenses

Now we encounter diverging lenses. For a plano-concave lens, the plane surface has $R_1=\infty$, and the concave surface curves inward, meaning its center is on the right (positive). So, $R_2=r$.

Each plano-concave lens has a focal length of $-2r$. Combining them:

This matches with option (r).

Biconvex and Plano-Concave

Finally, we have a mixed bag: one biconvex lens and one plano-concave lens. The beauty of physics is that we don't need to recalculate everything. We already found their individual focal lengths!

For the biconvex lens, $f_1=r$. For the plano-concave lens, $f_2=-2r$.

Let's add their powers:

This matches with option (p).

By systematically applying the sign convention and the principle of superposition of powers, we've elegantly decoded the entire matrix. This is the hallmark of a strong foundation in optics!