Question

Drive an expression for equivalent focal length of two thin lenses kept in contact.

Answer 1

Let the focal lengths of the two lenses be $f_1$ and $f_2$.
$u_1=O{C}_1$ and
$v_1=I_1{C}_1$
$u_2=-C_1{I}_1$ which is approximately equal to $C_2{I}_1$ and
$v_2=C_{2}I$ which is approximately equal to $C_{1}I$
$\therefore \frac{1}{u_1}+\frac{1}{v_1}=\frac{1}{f_1}$
and $\frac{1}{u_2}+\frac{1}{v_2}=\frac{1}{f_2}$
$\frac{1}{O{C}_1}+\frac{1}{I_1{C}_1}=\frac{1}{f1}$ and
$\frac{1}{-C_1{I}_1}+\frac{1}{C_{1}I}=\frac{1}{f_2}$
So $\frac{1}{O{C}_1}+\frac{1}{I_1{C}_1}=\frac{1}{f_1}+\frac{1}{f_2}=\frac{1}{F}$.
Combined focal length of two thin lenses in contact is given by:
$F=\frac{f_1{f}_2}{f_1+f_2}$