Animated Solution for Physics - Optics: Two lenses of power $-15\text{ D}$ and $+5\text{ D}$ are in contact with each other. The focal length of the combination is

Imagine two people pushing a heavy box. One pushes forward with a force of 5 units, and the other pushes backward with a force of 15 units. What happens? The box moves backward with a net force of 10 units. Lenses in contact work in a very similar way!

In optics, the power of a lens tells us how strongly it converges or diverges light. A positive power means the lens converges light (like a magnifying glass), while a negative power means it diverges light. The unit of power is the Diopter (D).

In our problem, we have two lenses: 1. A concave lens with a power of $P_1=-15\text{ D}$. 2. A convex lens with a power of $P_2=+5\text{ D}$.

When thin lenses are placed right next to each other (in contact), they act together as a single, new lens. The beautiful part? The power of this new "equivalent" lens is simply the algebraic sum of the individual powers.

Let's plug in our numbers:

The net power is $-10\text{ D}$. The negative sign is a huge clue—it tells us that the combination behaves like a concave (diverging) lens!

Now that we know the equivalent power, finding the focal length is a breeze. Power and focal length are inversely related. The formula is:

Since our options are in centimeters, it's easier to use the modified formula:

Substituting our equivalent power:

And there we have it! The focal length of the combination is $-10\text{ cm}$. It's a straightforward application of the power addition rule, but it's a fundamental concept that appears everywhere in optics.