# Why don't we use sign convention during the derivation of a lens maker formula?

Please have a look at the Lens makers formula.

In any derivation of Geometrical optics, we use the sign convention twice: once while deriving it and next while using it for general cases.

But in the derivation of lens makers formula, we don't consider negative and positive values of the radius of curvature while solving for both spherical surfaces.

This should lead to a wrong answer and in fact I solved one example which through individual analysis of both spherical surfaces gave a different answer than while using the lens makers formula directly.

I think am not getting this.

From what I have read, it's because no matter which surface light hits first the net refraction is same. This sign convention doesn't play a major role. But I am still confused.

## 2 Answers

I posted this question a while ago and now have realised what I was missing.

In the derivation of any other formula for e.g. the mirror formula or magnification for mirrors or lenses etc., we use a specific case involving, usually but not necessarily, a convex lens.

Thus we use the sign convention during the derivation as well as later while solving problems involving different sets of lenses or mirrors

However in lens makers formula, we use straight and direct formula where the first and second radius can take any value. Why? You may ask. Why is it that using a specific case isn't important here?

This is because we never use any specific condition involving say a convex surface on both sides. We just say an object situated in the negative direction, undergoes refraction through first surface, forms an image which again undergoes refraction through second surface. No specific condition that image formed is going to be in the positive co-ordinate, or gonna be enlarged smaller etc. So during the derivation no mention of any case. We directly apply convention for the problem we are solving.

• Forgot to mention but the answer is really basic and simple. – tiffy Oct 2 '18 at 12:03

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.

• Hi user220718, note that it's typical to place a single space after all punctuation marks, not just periods. We also have MathJax enabled on this site to make equations look nice, search "notation" in help center to learn more. – Kyle Kanos Jan 23 at 15:18