Animated Solution for Physics - Optics: A convex lens is in contact with concave lens. The magnitude of the ratio of their focal length is $\frac{3}{2}$. Their equivalent focal length is $30\text{ cm}$. What are their individual focal lengths ?

The behavior of lens combinations is one of the most elegant applications of geometric optics. When we place two lenses in contact, their powers add up algebraically. This simple principle allows us to design complex optical instruments, from microscopes to camera lenses, by fine-tuning the equivalent focal length.

Imagine you have a convex lens and a concave lens placed perfectly in contact. The convex lens wants to converge the light rays, while the concave lens wants to diverge them. It's a tug-of-war of optical power!

The problem tells us that the equivalent focal length of this combination is $F=+30\text{ cm}$. The positive sign is a massive clue: it means the converging power of the convex lens wins the battle.

Since power is inversely proportional to focal length ($P=\frac{1}{f}$), the lens with the greater power must have the smaller focal length magnitude. Therefore, the convex lens has a smaller focal length than the concave lens.

We are given that the magnitude of the ratio of their focal lengths is $\frac{3}{2}$. Based on our deduction, the ratio of the concave lens's focal length to the convex lens's focal length must be $\frac{3}{2}$.

Let the focal length of the convex lens be $f_1$ (where $f_1>0$) and the concave lens be $f_2$ (where $f_2<0$). We can write:

Now, we bring in the master equation for lenses in contact:

Substitute the known values into the master equation:

To solve this, we take a common denominator of $3f_1$ on the right side:

Cross-multiplying gives us:

Now, we plug this back into our ratio to find $f_2$:

The individual focal lengths are $-15\text{ cm}$ and $10\text{ cm}$. This perfectly aligns with our initial intuition: the convex lens ($10\text{ cm}$) is stronger than the concave lens ($-15\text{ cm}$), resulting in a net converging system!