Physics Asked by Ambica Govind on May 23, 2021
This is a combination of three lenses whose equivalent focal length is to be found. Solving for the focal length of the first lens was not a problem, it was obtained by applying Lens-Maker’s Formula directly. But I am not so sure about the water lens in the middle.
$$P_{lens}=frac{(n_{lens}-n_0)}{n_0}left(frac{1}{R_1}-frac{1}{R_2}right)$$
I took $n_0$ to be the refractive index of the glass lens, as my textbook calls $n_0$ to be the refractive index of the medium from which rays enter the lens. However, the solution states that $n_0$ is to be taken as 1. It’s as if they have disintegrated the whole system into the air, calculated the individual focal lengths and added the powers. How can it be so (disregarding the fact that rays in the water lens come from the glass lens)?
Consider the focal length of the simplest lens:
You can see that if you send parallel rays from 2 to 1, the focal length is $f_1=frac{n_1R}{n_2-n_1}$. But if you send rays in opposite direction, the focal length is $f_2=frac{n_2R}{n_2-n_1}$. That's why it's more beneficial to talk about surface power (optical power which doesn't depend on on what side of the lens we are): $$ S = frac{n}{f_n} = frac{n_2-n_1}R = nP_n. $$ For scenarios when the media before and after the lens is the same, the Lens Maker's formula of the sum of surface powers becomes the sum of optical powers: $$ P = frac{S}{n} = frac{S_1}n+frac{S_2}{n} = P_1 + P_2 $$
One way to avoid it is to decompose the lenses to have the same outside medium, usually air. This what is suggested in your problem: you have a glass lens in the air, a water lens in the air, and one more glass lens in the air. The sum of their optical powers: $$ P = 2frac{n_g-n_0}{Rn_0} + 2frac{n_w-n_0}{-Rn_0} + 2frac{n_g-n_0}{Rn_0} = frac{4n_g-2n_w-2n_0}{Rn_0}. $$
However, if you know about surface powers, you can avoid virtual air between water and glass: $$ n_0P = S = frac{n_g-n_0}{R} + frac{n_w-n_g}{-R} + frac{n_g-n_w}{R} + frac{n_0-n_g}{-R} = frac{4n_g-2n_w-2n_0}{R} $$
Answered by Vasily Mitch on May 23, 2021
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