# Question about lens maker's formula

#### kelsiu

I am trying to follow the derivation of lens maker's formula from the textbook "University Physics", p.1133 (https://books.google.com.hk/books?id...page&q&f=false)

I can understand the first equation because it is just the object–image relationship for spherical refracting surface. But for the second equation, why the left hand side is nb/s2+nc/s'2 instead of nc/s2+nb/s'2? s2 is the first image's distance and it is on the nc side. In addition, on the right hand side why it is nc-nb on the numerator instead of nb-nc? If we follow strictly the formula for spherical refracting surface, the nb should be the lens side and nc is the air side.

A more fundamental question is, why this kind of superposition principle can be applied? I mean why the lens can be expressed as two lens added together? In many books they directly apply the object–image relationship for spherical refracting surface twice and added together. But this formula is only for single spherical surface (e.g. one side is air only and the other side is water only). If it is a lens it is air on both sides but lens in the middle. Why the solution for single spherical surface can be superposed like this?

#### studiot

But for the second equation, why the left hand side is nb/s2+nc/s'2 instead of nc/s2+nb/s'2?
You always have to be careful which sign convention you are using, but your question does not involve this.

You have 3 media involved. R & H label these a,b and c.

a is air
b is the lens
c is air again.

So refraction at the first surface is air to lens.

A real image is formed within the lens by the first refraction.

so the first equation is a passing to b with
S1 in air (na) and S'1 in lens (nb)

So the equation is

na/S1 + nb/S'1 = etc

This real image becomes the object to the second refraction.

This refraction is lens (nb to air (nc)

So the equation is

nb/S2 + nc/S'2 = etc

R&H then make the point that

S2 = -S'1

(what horrible, clumsy notation)

and use this to make a substitution and perform some algebra to get the lensmaker's equation.

Does this help?

This becomes a virtual image for the