Refraction by a lens(Lens makers' formula & Lens formula)
From the figure we can write
\(\frac{n_1}{OB} + \frac{n_2}{BI_1}= \frac{n_2-n_1}{BC_1}\) .................(i)
For refraction at the face ABC of the lens, image I' acts as virtual objects and produces red image I at a distance v from the lens.
\(\frac{-n_2}{DI_1}+\frac{n_1}{DI} = \frac{n_2-n_1}{DC_2}\) ...................(ii)
Adding (i) and (ii), we get
\(\frac{n_1}{OB}+ \frac{n_1}{DI}=[n_2-n_1][\frac{1}{BC_1}+\frac{1}{DC_2}]\) ...................(iii)
By defination, when OB=\(\infty\) , DI = f
Equation (iii) becomes
\(\frac{n_1}{f}= (n_2-n_1)[\frac{1}{BC_1}+\frac{1}{DC_2}]\) .....................(iv) [BC1 = R1, CD2= -R2]
\(\frac{n_1}{f}= (n_2-n_1)[\frac{1}{R_1}-\frac{1}{R_2}]\)
\(\frac{1}{f}= (\frac{n_2}{n_1}-1)[\frac{1}{R_1}-\frac{1}{R_2}]\)
\(\frac{1}{f}= (_1n_2-1)[\frac{1}{R_1}-\frac{1}{R_2}]\) \(\Rightarrow\) This is lens makers formula
