Three Proofs — Question 7

Question 7

Newtonian image distances turn the lens equation into a product

For a thin lens in one medium, define signed focal-offset distances $x=p-f$ and $x^\prime=q-f$. Begin with $1/p+1/q=1/f$. Exclude the singular case p=f when using finite conjugates.

1. Derive the Newtonian relation between x and x prime.

2. Express transverse magnification in terms of each focal-offset distance and infer how the image offset changes if the object offset doubles.

Question 7 - Solution

Newtonian image distances turn the lens equation into a product

(a) Method and calculation

$$pq=f(p+q)\Rightarrow pq-fp-fq+f^2=f^2.$$$$\boxed{(p-f)(q-f)=xx^\prime=f^2}.$$

(b) Interpretation and further calculation

$$\boxed{m=-q/p=-f/x=-x^\prime/f}.$$$$x\to2x\Rightarrow\boxed{x^\prime\to x^\prime/2,\quad m\to m/2}.$$

Physical check

The offset distances are measured from the focal points, while p and q are measured from the lens; confusing these origins changes the equation.