Derive the equation for refraction at single spherical surface.

Explanation

Let us consider two transparent media having refractive indices `n_(1) and n_(2)` are separated by a spherical surface.Let C be the centre of curvature of the spherical surface. Let a point object O be in the medium `n_(1)`. The line OC cuts the spherical surface at the pole P of the surface. As the rays considered are paraxial rays, the perpendicular dropped for the point of incidence to thte principal axis is very close to the pole or passes through the pole itself.
Light from O falls on the refracting surface at N. The normal drawn at the point of incidence passes through the centre of curvature C. As `n_(2) gt n_(1)`, light in the denser medium deviates towards the normal and meets the principal axis at I where the image is formed.
Snell,s law in product form fot the refraciton at the point N could be written as,
`n_(1) sin i = n_(2) sin r`
As the angles are small, sin of the angle could be approximated to the angle itself.
`n_(1) i = n_(2) r`
Let the angles,
`angleNOP=alpha,angleNCP=beta,angleNIP=gamma`
`tanalpha=(PN)/(PO),tanbeta=(PN)/(PC),tangamma=(PN)/(PO)`
As these angle are small, tan of the angle could be approximated to the angle itself.
`alpha=(PN)/(PO)beta=(PN)/(PC),gamma=(PN)/(PI)`
For the triangle, `DeltaONC`,
`i = alpha + beta`
For the triangle, `DeltaINC` ,
`beta = r + gamma (or) r = beta - gamma`
Substituting for i and r from equations (4) and (5) in the equation (2).
`n_(1)(alpha+beta)=n_(2)(beta-gamma)`
Rearranging.
`n_(1)alpha+n_(2)gamma=(n_(2)-n_(1))beta`
Substituting for `alpha, beta and gamma` from equation (3)
`n_(1)((PN)/(PO))+n_(2)((PN)/(PI))=(n_(2)-n_(1))((PN)/(PC))`
Further simplifiying by cancelling PN,
`(n_(1))/(-u)+(n_(2))/(v)=((n_(2)-n_(2)))/(R)`
After rearranging, finally we get,
`(n_(2))/(v)-(n_(1))/(u)=((n_(2)-n_(1)))/(R)`
Equation (7) gives the ralation among the object distance u, image distance v, refractive indices of the two media `(n_(1) and n_(2))` and the radius of curvature R of the spherical surface. It holds for any spherical surface.
If the first medium is air then, `n_(1) = I` and the second medium is taken just as `n_(2) = n`, then the equation is reduced to.
`(n)/(v)-(1)/(u)=((n-1))/(R)`