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

Explanation

OM = u = object distance
MI = v = image distance
MC = R = radius of curvature
Angle i = angle of incidence
Angle r = angle of refraction
ON = incident ray
NI = refracted ray

NC=normal & `n_1,n_2` are the refractive indices
From the figure for small angles
tan`angleNOM=(MN)/(OM)`, tan `angleNCM=(MN)/(MC), tan angleNCM=(MN)/(MI)`
In the triangle NOC, `hati` = exterior angle = sum of the interior opposite angles
`anglei=angleNOC + angleNCO=angleNOM+angleNCM=(MN)/(OM)+(MN)/(MC)`
Similarly , `angleNCM = angleCNI + angleNIC = angleCNI+angleNIM`
`angler=angleCNI=angleMCN-angleNIM=(MN)/(MC)-(MN)/(MI)`
From Snell.s law `n_1` sin i = `n_2` sin r
For small angles `n_1 i = n_2 r`
Substituting the values of i and r we get
`n_1((MN)/(OM)+(MN)/(MC))=n_2((MN)/(MC)-(MN)/(MI))`
`n_1/(OM)+n_2/(MI)=(n_2-n_1)/(MC)`
Applying sign convention, OM=-u , MI=v , MC=R
`n_1/v-n_2/u=(n_2-n_1)/R`
This is the required expression.