# How to determine which number is $r_1$ or $r_2$ in the given equation?

A physicist is designing a contact lens. The material has an index of refraction of 1.40. It also had:

• an inner radius of curvature (surface that touches the eye) = +2.60 cm
• an outer radius of curvature (surface that first interacts with incoming light)= +2.06 cm

what is the focal length of this contact lens (in cm)?

I got the correct answer for this question by using the formula,

$$\frac 1f = (n-1)\left(\frac{1}{r_1}+\frac{1}{r_2}\right)$$

which was 24.8 cm. But I just did that by randomly stubbing in 2.60 as $$r_1$$ and I don't really understand which number I should take as $$r_1$$ or $$r_2$$. The question did not have a diagram and did not state whether it was converging or diverging lens, so I was wondering if there was any other formula or method to determine which number should be taken as $$r_1$$ or $$r_2$$.

• Are contact lenses converging or diverging lenses? Dec 1 '19 at 1:10
• The question did not mention whether they were converging or diverging lenses
– rsn
Dec 1 '19 at 1:14
• I think you’re supposed to use common sense to make an assumption about that. Dec 1 '19 at 1:16
• I've added the homework-and-exercises tag. In the future, please add this tag to this type of problem. This is one of the things that we ask you to do in our homework policy: physics.meta.stackexchange.com/questions/714/…
– user4552
Dec 1 '19 at 1:57
• Please don't cut and paste on the internet without crediting the author. It's rude. Please reference the source of this homework question. This is one of the things that we ask you to do in our homework policy: physics.meta.stackexchange.com/questions/714/…
– user4552
Dec 1 '19 at 1:57

$$\frac 1f = (n-1)\left( \frac 1{r_1}+\frac 1{r_2} \right),$$
there are eight possible choices for (a) which side of the lens is $$r_1$$, (b) which direction of curvature makes $$r_1$$ positive, and (c) which direction of curvature makes $$r_2$$ positive. I've seen at least five of these possibilities in different textbooks. Note that my $$+$$ where you have a $$-$$ is one of those choices.
The physics result to match is that, if $$n>1$$ (as usual, and as in your problem), a lens with a thick middle () makes light rays converge, while a lens with a thin middle and thick edges )( makes light rays diverge. Every modern text uses $$f>0$$ for converging lenses and $$f<0$$ for diverging lenses; so do opticians.
The lens in your problem is flatter on the eye-touching side than on the air-touching side 👁|), so it is a fat-middled converging lens for a farsighted person and should have positive $$f$$.