Draw a ray diagram showing the formation of the image by a point object on the principal axis of a spherical convex surface separating two media of refractive indices `n_(1)` and `n_(2)`, when a point source is kept in rarer medium of refractive index `n_(1)`. Derive the relation between object and image distance in terms of refractive index of the medium and radium of curvature of the surface.
Hence obtain the expression for lens-maker's formula in the case of thin convex lens.

Explanation

The incident rays coming from the object .O. kept in the rarer medium of refractive index `n_(1)`, incident on the refracting surface NM produce the real image at I.
From the diagram,
`/_ i = /_ NOM + /_ NCM`
`= ( NM )/( OM ) + ( NM )/( MC )`
`/_r = /_ NCM - /_ NIM `
`= ( NM )/( MC ) - ( NM ) /( MI )`
From Snell.s law,
`:. ( n_(2))/( n_(1)) = ( sin i)/( sin r ) ~ (i) /( r )` ( for small angles, sin` theta ~ theta )`
`:. n_(2) r = n_(1) i `
or `n_(2) ((NM)/( MC) - ( NM )/( MI)) = n_(1) ((NM)/( OM) + ( NM)/( MC))`
or ` n_(2) ((1)/( + R ) - ( 1)/( +v)) = n_(1) ((1)/( - u ) + ( 1)/( R ))`
or `(n_(2) - n_(1))/( R ) = ( n_(2))/( v ) - ( n_(1))/( u )`
Lens maker.s formula `:`

The first refracting surface ABC forms the image `I_(1)` of the object O. The image `I_(1)` acts as a virtual object for the second refracting surface ADC which forms the real image I as shown in the diagram.
`:`. For refraction at ABC
`(n_(2))/( v_(1)) - ( n_(1))/(u) = ( n_(2) - n_(1))/( R_(1))` ....(i)
For refraction at ADC
`(n_(1))/( v) - ( n _(2))/(v_(1)) = ( n_(1) - n_(2))/( R_(2))` ....(ii)
Adding equations (i) and (ii) , we get
`(n_(1))/( v) - ( n_(1))/( u ) = ( n _(2) - n_(1)) ((1)/( R_(1)) - ( 1)/( R_(2)))`
`(1)/( v) - ( 1)/( u ) = ( ( n_(2))/(n_(1)) - 1) ((1)/( R_(1)) - ( 1)/( R_(2)))`
`(1)/( f) = ( mu_(21) - 1) ((1)/( R_(1))- ( 1)/( R_(2)))`