Animated Solution for Physics - Optics: A double convex lens made of glass of refractive index $1.5$ and radii of curvature of the curved surfaces $20\text{ cm}$ each is immersed in a liquid of refractive index $n_L$. The correct plot showing the variation of the power, in the units of diopter (D), as a function of $n_L$ is :

A Lens in Disguise

Imagine you are holding a standard double convex lens made of glass. In the air, it behaves exactly as you'd expect—converging light rays to a neat focal point. But what happens when you plunge this lens into a liquid? The rules of the game change entirely!

The glass has a fixed refractive index of $n_g=1.5$, and both of its curved surfaces have a radius of curvature of $20\text{ cm}$. However, the liquid's refractive index, $n_L$, is our independent variable. We are on a mission to discover how the power of this lens dances as we vary $n_L$.

Lens Maker's Formula

To bridge the gap between the physical shape of the lens and its optical behavior, we call upon the legendary Lens Maker's Formula:

Notice the crucial term $\frac{n_g}{n_L}$. This is the relative refractive index of the glass with respect to the liquid. It dictates how much the light bends at the interface.

Let's substitute our known values. We must be extremely careful with our sign convention! The first surface bulges outward to the right, so $R_1=+0.2\text{ m}$. The second surface bulges outward to the left, so $R_2=-0.2\text{ m}$.

Let's simplify the geometric part of our equation.

Plugging this back into our main equation, we get a beautiful, streamlined expression for the reciprocal of the focal length:

What is Power, Really?

Here is where the problem reveals its true colors. How do we define the power of a lens when it is submerged in a medium?

According to the standard NCERT convention (which is the holy grail for JEE), the power of a lens is strictly defined as the reciprocal of its focal length in meters:

If we adopt this definition, our power equation becomes:

This is the equation of a hyperbola. Let's test a few critical points to visualize it. If the liquid is air ($n_L=1.0$), the power is $5\text{ D}$. If the liquid perfectly matches the glass ($n_L=1.5$), the lens vanishes optically, and the power drops to $0\text{ D}$. If we push further to a denser liquid ($n_L=2.0$), the power becomes $-2.5\text{ D}$. This hyperbolic behavior perfectly matches the curve shown in Option (A).

But wait, there is a twist! In advanced optics, there is an alternative convention. Sometimes, the power of a lens in a medium is defined as the refractive index of the medium divided by the focal length:

If we apply this alternative definition, the $n_L$ in the denominator cancels out beautifully:

This is the equation of a straight line with a negative slope! Testing our points again: at $n_L=1.0$, $P=5\text{ D}$. At $n_L=1.5$, $P=0\text{ D}$. At $n_L=2.0$, $P=-5\text{ D}$. This linear behavior perfectly matches the graph in Option (B).

Because of this dual convention, both Option (A) and Option (B) are technically correct depending on the textbook you follow. However, for strict NCERT compliance, Option (A) is the intended path!