# Can focal length of convex lens be negative?

Is focal length of convex lens negative when it is surrounded by a medium with greater refractive index than that of lens?

• I would argue that you're talking about a concave lens of the other medium with a positive focal length then. Dec 29, 2016 at 12:17

A convex lens in a medium with index larger than that of the lens will act like a divergent lens, yes. This is the physical fact. If you consider the focal length positive or negative will depend on the convention used. By usual convention, the focal lens of divergent lenses is taken with minus sign. If you use the lens-maker formula (in the thin lens approximation) $$\frac{1}{f}=(n_{lens} -n_{medium})(\frac{1}{R_1}- \frac{1}{R_2})$$ you can see that if $\ n_{lens} <n_{medium}$ the focal length will be negative. Note that the formula is written with the convention that $R_2$ is negative for convex lenses so that the content of the second parenthesis on the RHS is always positive for convex lenses.

If the lens is in a medium of greater refractive index, then it will act as a concave mirror.

• There is going to be some reflection whenever light goes into another medium with a different refractive index (larger or smaller) but why do you say it would act as a concave mirror rather then convex? Dec 30, 2016 at 3:04

If you are talking about a biconvex lens then it can not have a negative focal length if the surrounding medium is denser than the lens.

As we know the formula for focal length of a thin lens(refractive index=$n_{lens}$) surrounded by a medium(refractive index = $n_o$) is $$\frac{1}{f}=(\frac{n_{lens}}{n_o}-1) (\frac{1}{R_1}-\frac{1}{R_2})$$ $$\implies\frac{1}{f}=(\frac{n_{lens}-n_o}{n_o}) (\frac{R_2-R_1}{R_2\ R_1})$$

But for a convex-concave lens, if $R_2>R_1$ and $n_o>n_{lens}$, you can have a negative focal length.

• No there no necessity for a convex to have r2 greater than r1 Dec 29, 2016 at 16:01
• I didn't say that. You didn't mention the type of convex lens. If you consider it ti be biconvex in a denser medium then it can not have negative focal length irrespective of the relation between R1 and R2. I think you didn't undersrand what I said. Read it properly once again. Dec 29, 2016 at 16:13