Animated Solution for Physics - Optics: An eye specialist prescribes spectacles having combination of convex lens of focal length $40\text{ cm}$ in contact with a concave lens of focal length $25\text{ cm}$. The power of this lens combination in diopters is

Imagine you are visiting an eye specialist, and they prescribe a unique pair of spectacles. Instead of a single lens, these spectacles are made by combining two different lenses placed perfectly in contact.

We are given a convex lens with a focal length of $f_1=+40\text{ cm}$ and a concave lens with a focal length of $f_2=-25\text{ cm}$. Notice the sign convention here: convex lenses always have a positive focal length because they converge light, while concave lenses have a negative focal length because they diverge light. Our goal is to find the overall power of this lens combination in diopters.

When two thin lenses are placed in contact, their individual abilities to bend light simply add up. The equivalent focal length $F$ of the combination is given by the reciprocal sum of their individual focal lengths:

Let's carefully substitute our given values into this equation. It is crucial to keep the negative sign for the concave lens!

Now, we solve this fraction. The least common multiple for $40$ and $25$ is $200$.

We have the reciprocal of the equivalent focal length, but we need the power of the combination. The power $P$ of a lens in diopters (D) is defined as the reciprocal of its focal length measured in meters.

Since our current value of $\frac{1}{F}$ is in inverse centimeters (${\text{cm}}^{-1}$), we can easily convert it to inverse meters by multiplying by $100$:

Let's plug in our calculated value:

The hundreds cancel out beautifully, leaving us with:

Conclusion: The power of this lens combination is $-1.5\text{ D}$. The negative sign indicates that the diverging power of the concave lens dominates, making the entire combination behave like a net diverging (concave) lens.