Question about lens maker's formula

[kelvin490]

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=nQZyAgAAQBAJ&pg=PA1133#v=onepage&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 n_b/s₂+n_c/s'₂ instead of n_c/s₂+n_b/s'₂? s₂ is the first image's distance and it is on the n_c side. In addition, on the right hand side why it is n_c-n_b on the numerator instead of n_b-n_c? If we follow strictly the formula for spherical refracting surface, the n_b should be the lens side and n_c 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?

 

[brainpushups]

I am unable to see pages in that reference. Can you post a screenshot or something?

 

[kelvin490]

I am unable to see pages in that reference. Can you post a screenshot or something?

I have got two pictures of it:

https://www.dropbox.com/s/mk4vxpdc7nk7t00/1.jpg?dl=0

https://www.dropbox.com/s/5jvq3siyntpwmvq/2.jpg?dl=0

Thank you

 

[brainpushups]

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.

I think that the author has simply just not taken the sign conventions into account when writing the second equation. This may seem a little silly because they only consider thin lenses, but it is perhaps safer to let the reader figure out the correct signs for any exercises rather than choose signs for them. For example, consider a very thick glass sphere for which the image produced by the first surface is to the left of the second vertex. The distance s2 would be positive in this case.

Otherwise, if the image for the first surface is produced to the right of the second vertex then the distance s2 will be negative based on the sign convention, as will the radius of curvature (for a convex lens) which will change the order of nb and nc as you mentioned.

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?

My suspicion is that this only works in the paraxial approximation. A more general expression for the relationship between the object and image axial distances will also involve the path length of light. For a spherical surface the rays do not converge to a point and that seems like it would add a significant complication, though I have never had a reason to attempt an analysis of this. As long as we're working in the paraxial approximation the geometry is rather simple and the similar triangles involved are what make the superposition of the two surfaces hold. Someone else can correct me if I'm wrong.