Thin lens equation and microscope

Introduction

The thin lens equation ${\frac {1}{i}}+{\frac {1}{o}}={\frac {1}{f}}$ to compound microscope with two lenses. The lens that is closer to the object is called objective and the the one closer to the eye is called ocular. The distance between the lenses is $L$.

Theory

We have ${\frac {1}{i_{1}}}={\frac {1}{f_{1}}}-{\frac {1}{o_{1}}}$. The distance between the lenses is $L$, thus $i_{1}+o_{2}=L$ which gives $o_{2}=L-i_{1}$. Thus we have for the image distance of the second lens

${\begin{aligned}{\frac {1}{i_{2}}}&={\frac {1}{f_{2}}}-{\frac {1}{o_{2}}}\\&={\frac {1}{f_{2}}}-{\frac {1}{L-i_{1}}}\\&={\frac {1}{f_{2}}}-{\frac {1}{L-{\frac {f_{1}o_{1}}{o_{1}-f_{1}}}}}\\&={\frac {1}{f_{2}}}-{\frac {o_{1}-f_{1}}{L(o_{1}-f_{1})-f_{1}o_{1}}}\\\end{aligned}}$

Magnification (for the thin lens) is $m=-{\frac {i}{o}}=-{\frac {i_{2}}{o_{1}}}$.