Why is the sign convention used in the derivation of the lens formula and yet used again when it is applied in numerical problems? Won't the whole idea of sign convention be eliminated if it is used twice?
[Physics] Sign convention of the lens formula and its application
geometric-optics
Related Solutions
The superposition is only approximately correct and the easiest way to understand both (1) why it works and (2) when it can be applied is to think in terms of waves and wavefronts, not rays. Since your link isn't working, let's write the equation down:
$$P_{lens}\approx \frac{n_{lens}-n_0}{n_o}\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$
With the fields thought of as waves, the lens surfaces become phase masks and the superposition of the two lens curvatures holds because the phase delays imparted by the phase masks one after the other are additive as long as the wavefront curvature (i.e. lateral phase distribution of the field) does not change much between the phase masks.
The power of the lens is the reciprocal of its focal length $f$, i.e. it is the reciprocal of the radius $f$ of curvature of the wavefronts that are output from the lens when a plane wave is input. A spherical wave of this radius converges to its diffraction limited focus after having propagated through this distance. So think of a plane wave input: the first surface represents a phase mask with phase delay as function of distance $r$ from the optical axis given by:
$$\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2 R_1}$$
the phase mask function for the second is:
$$-\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2 R_2}$$
and so the total phase delay is simply:
$$\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2} \left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$
Now ask yourself: what is the radius of curvature of the spherical wave that shows this (to terms of second order and lower) phase distribution. You'll find it is the reciprocal of the power given by the lensmaker's formula.
I give the full details of these kinds of calculations in this answer here. I show the calculation above as well as the transformer matrix method where one thinks of lens surfaces and other optical processing elements as operators in the group $SL(2,\,\mathbb{R})$.
I have a better explaination. You see we have used the formula for refraction on spherical surface $$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$$ in the derivation of lens makers formula. Now, the use of sign convention in any derivation is only to make the formula generalized. If we don't use sign convention, we can use the the derived formula in only the situation which we considered while deriving it. And since we have already used sign convention in the derivation of formula for refraction through spherical surfaces, we don't have to use it again in derivation of lens makers formula.
Best Answer
We derive the lens or mirror formula with the sign convention. Then in solving the problem we again use sign convention to de neutralize it.
Also we use sign convention to make the calculation simple on large scale.
To identify the nature of image.
To know where is the image or object is placed with respect to lens or mirror.