Animated Solution for Physics - Optics: Two thin lenses, when in contact, produce a combination of power + 10 D. When they are 0.25 m apart, the power reduces to + 6 D. The focal length of the lenses are ...... m and ...... m.

Imagine a team of two workers. When they work side-by-side, their combined effort is simply the sum of their individual strengths. But what happens if you separate them? Communication gaps arise, and their combined efficiency drops. Lenses behave in a remarkably similar way!

Their equivalent power is simply the algebraic sum of their individual powers: In our problem, this combined power is given as $+10\text{ D}$. So, we have our first equation:

When separated by a distance $d$, the lenses don't cooperate as perfectly. The equivalent power decreases by a factor proportional to the distance and the product of their powers. The new formula becomes: We are told that when the distance $d=0.25\text{ m}$, the power drops to $+6\text{ D}$. Let's plug this into our equation:

Let's substitute this into our second equation: Rearranging the terms to isolate the product, we get: Multiplying both sides by 4 gives us:

Now we face a classic algebra puzzle: find two numbers that add up to 10 and multiply to 16. A quick mental check reveals the numbers are 8 and 2. Therefore, the powers of the individual lenses are $8\text{ D}$ and $2\text{ D}$.

Focal Lengths The question asks for the focal lengths of the lenses. Remember the fundamental relationship between power and focal length: $f=\frac{1}{P}$, where $f$ is in meters and $P$ is in diopters.

For the first lens:

For the second lens:

And there we have it! The focal lengths of the two lenses are $0.5\text{ m}$ and $0.125\text{ m}$.