Deduce the relation between n,u,v, Q, R for refraction at a spherical surface, where the symbols have their usual meaning.

Explanation

From the figure,
`tan angleNOM=(MN)/(OM)`
`tan angleNCM=(MN)/(MC)`
where, `angleNOM` and `angleNCM` 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_1i=n_2r`...(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_3-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)`