Animated Solution for Physics - Optics: A double convex lens has power $P$ and same radii of curvature $R$ of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power $1.5P$ is

A Tale of Two Curvatures

Imagine you are an optician tasked with crafting a new lens. You have a double convex lens with a known power, and you need to create a plano-convex lens from the same material but with a significantly higher power. How do you determine the exact curvature required? This problem takes us on a beautiful journey through the Lens Maker's formula.

We start with a double convex lens where both surfaces have the same radius of curvature, $R$. To find its power, we invoke the master equation of lens design: the Lens Maker's formula.

By standard sign convention, the first surface bulges outward towards the incoming light, so $R_1=R$. The second surface curves away, so $R_2=-R$. Substituting these into our formula:

This gives us a clean expression for the power of our original lens. Let's hold onto this as our baseline.

Now, let's look at the new plano-convex lens. It's made of the same material, so the refractive index $\mu$ remains constant. One surface is curved with an unknown radius $R^{\prime }$, and the other is perfectly flat. A flat surface is essentially a sphere with an infinite radius, so $R_2=\infty$.

Applying the Lens Maker's formula again:

Since $\frac{1}{\infty }=0$, the equation simplifies beautifully:

We are given a crucial piece of information: the power of this new lens is $1.5$ times the power of the original lens. This means $P^{\prime }=1.5P$, or equivalently, $\frac{P}{P^{\prime }}=\frac{1}{1.5}=\frac{2}{3}$.

Let's divide our two power equations to eliminate the refractive index term:

The math from here is incredibly satisfying. The factor of $2$ cancels out on both sides, leaving us with a direct relationship between the radii:

This result makes perfect physical sense. If we had simply cut the original double convex lens in half, its power would have been halved ($0.5P$). To achieve a power of $1.5P$ with only one curved surface, that single surface must be curved much more sharply. A smaller radius of curvature ($R/3$) means a sharper curve, which bends light more aggressively, resulting in a higher optical power.