Animated Solution for Physics - Optics: A hollow double concave lens is made of very thin transparent material. It can be filled with air or either of two liquids $L_1$ or $L_2$ having refracting indices $n_1$ and $n_2$ respectively ($n_2>n_1>1$). The lens will diverge a parallel beam of light if it is filled with

Imagine a hollow double concave lens. It is essentially a shell made of very thin glass. Because the glass is so incredibly thin, its own refractive power is practically negligible. The true optical behavior of this setup is entirely dictated by what we fill inside the hollow space and the medium we immerse the entire lens into. This gives us a fascinating level of control over the lens's properties!

Lens Maker's Formula

To understand exactly how this fluid-filled lens will behave, we turn to our ultimate optical tool: the Lens Maker's Formula. This elegant equation relates the focal length $f$ of a lens to the refractive indices of the lens material ($n_L$) and the surrounding medium ($n_m$), as well as the radii of curvature of its two surfaces ($R_1$ and $R_2$):

Let's carefully analyze the shape of our double concave lens using the standard Cartesian sign convention. Light travels from left to right.

The first surface (on the left) curves inwards, meaning its center of curvature lies to the left of the lens. Therefore, its radius $R_1$ is negative ($R_1<0$).

The second surface (on the right) curves outwards from the center of the lens, meaning its center of curvature lies to the right. Therefore, its radius $R_2$ is positive ($R_2>0$).

Now, let's plug these signs into the geometric term of our formula:

Since we are subtracting a positive number from a negative number, the entire geometric term becomes strictly negative.

The problem specifically asks for the lens to diverge a parallel beam of light. For any lens to act as a diverging lens, its focal length $f$ must be negative. Consequently, $\frac{1}{f}$ must also be negative.

Let's look back at our Lens Maker's Formula. We need the overall product on the right side to be negative. We have already established that the geometric term $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ is negative.

For the final product to be negative, the first term—the refractive index term—must be positive!

This is a profound physical insight: A concave lens will only act as a diverging lens if the material inside it is optically denser than the medium surrounding it.

We are provided with two liquids, $L_1$ and $L_2$, with refractive indices $n_1$ and $n_2$ respectively. We are given the crucial constraint that $n_2>n_1>1$.

Since our mathematical derivation demands that the inside of the lens must have a higher refractive index than the outside ($n_L>n_m$), we have only one logical choice:

We must fill the hollow lens with the denser liquid $L_2$ (so $n_L=n_2$) and immerse the entire setup in the rarer liquid $L_1$ (so $n_m=n_1$).

This perfectly matches option (d).