Derive referaction formula (for object in air and image in the denser medium ) for refraction of light at a spherical surface

Explanation

From the figure,
`tan angleNOM=(MN)/(OM)`
`tan angle NCM=(MN)/(MC)`
where, `angle NOM and angle NCM` are interior angles
similarly, `angleNCM=r+angleNIM`
i.e., `r=angleNCM-angleNIM`
For small angles `tan angleNCM=angleNCM` in rad
similarly `i=(MN)/(OM)+(MN)/(MC)" " ` .....(1)
and `r=(MN)/(MC)-(MN)/(MI)" "` .......(2)
From snell.s law,
`n_(1)sin i = n_(2) sin r`
for small angle sin i = i (rad) and sin r = r (rad)
i.e., `n_(1)i = n_(2)r " "`.......(3)
using (1) and (2) in (3) we write,
`n_(1)((MN)/(OM)+(MN)/(MC))=n_(2)((MN)/(MC)-(MN)/(MI))`
or `(n_(1))/(OM)+(n_(1))/(MC)=(n_(2))/(MC)-(n_(2))/(MI)`
i.e., `(n_(1))/(OM) +(n_(2))/(MI)=(1)/(MC)(n_(2)-n_(1))`
where, OM = -u, MI = +v, MC = +R by using sign conventions.
`(n_(1))/(-u)+(n_(2))/(v)=(1)/(R)(n_(2)-n_(1))`
`therefore (n_(2))/(v)-(n_(1))/(u)=(1)/(R) (n_(2)-n_(1))`