Animated Solution for Physics - Optics: A convex lens of focal length 40 cm is in contact with a concave lens of focal length 25 cm. The power of the combination is

When dealing with lenses, one of the most practical concepts is the power of a lens. While focal length tells us the distance at which parallel rays converge (or appear to diverge), power tells us how strongly the lens bends light. A shorter focal length means a stronger bend, hence a higher power.

Mathematically, the power $P$ of a lens is the reciprocal of its focal length $f$ measured in meters: The unit of power is the Diopter (D), which is equivalent to ${\text{m}}^{-1}$.

In many optical instruments, like microscopes or cameras, a single lens isn't enough. We often combine lenses to minimize aberrations or achieve a specific focal length. When thin lenses are placed in direct contact, their powers simply add up algebraically.

For two lenses with powers $P_1$ and $P_2$, the equivalent power $P_{\text{eq}}$ of the combination is:

In our problem, we have two lenses: 1. A convex lens with a focal length $f_1=+40\text{ cm}$. 2. A concave lens with a focal length $f_2=-25\text{ cm}$.

Step 1: Convert focal lengths to meters. This is a crucial step where many students make silly mistakes. Power must be calculated using meters!

Step 2: Calculate individual powers. For the convex lens: For the concave lens:

Step 3: Find the equivalent power. Now, we simply add the two powers together:

The net power of the combination is $-1.5\text{ D}$.

What does this tell us? The negative sign indicates that the combined system behaves as a diverging lens. Even though we have a converging convex lens in the mix, the concave lens has a shorter focal length (magnitude-wise), meaning it is "stronger." Its diverging effect overpowers the converging effect of the convex lens, resulting in a net diverging system.