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Derive th lens maker's formula....

Derive th lens maker's formula.

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Refraction at first surface ABC:
Suppose that the surface ABC is absent. Now I. can be treated as the as the real image. Here the object is in `n _(1) ` medium and the image is in `n _(2)` medium.
`therefore ( n _(2))/( v .) - ( n _(1))/( v ) = (n _(2) - n _(1))/( R _(1))`
Refraction at second surface ADC:
For refaction at second surface ADC, I. may be treated as virtual object. Its real image is formed at I. Here the object is in `n _(2)` medium and the image is in `n _(1)` medium.
`therefore ( n _(1))/( v ) - ( n _(2))/( v .) = (n _(1) - n _(2))/( R _(2))`
`or (n _(1))/( v ) - ( n _(2))/( v.) = (n _(1) - n _(2))/( - R _(2))`
Adding equation (1) and (2), we have
`(n _(2))/( v . ) - (n _(1))/( v ) + ( n _(1))/( v ) - ( n _(2))/(v.) = (n_(2) - n _(1))/( R _(1)) + (n _(1) - n _(2))/( - R _(2))`
`(n _(1))/( v ) - ( n _(1))/( v ) = ( n _(2) - n _(1)) ((1)/( R _(1))- (1)/( R _(2)))`
`n _(1) ((1)/(v )- ( 1 )/(u)) = (n_(3) - n _(1)) ((1)/( R _(1)) - (1)/( R _(2)))`
`(1)/( v) - (1)/( v ) = ((n _(2) - n _(1))/( n _(1)) ) ( (1 )/( R _(1)) - (1)/( R _(2 )))`
`(1)/(v) - (1)/(v) = ((n _(2))/( n _(1))-1) ((1)/( R _(1)) - (1)/( R _(2)))`
`1/v -1/v = ( n _(2i) -1) ((1)/( R _(1)) - (1)/( R _(2)))`
Supose the object is at infinity, then the parallel rays from the object will converge at the principal focus F.
i.e., `u = oo implies u =f`
`therefore 1/f - (1)/( oo) = ( n _(21)-1 ) ((1)/( R _(1)) - (1)/( R _2))`
` 1/f = ( n _(21) -1 ) ((1)/( R _(1 ))- (1)/( R _(2)))`
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