The adjacent figure shows a thin plano-convex lens of refractive index `mu_1` and a thin plano-concave lens of refractive index `mu_2`, both having same radius of curvature R of their curved surfaces. The thin lens of refractive index ` mu_3` has radius of curvature R of both its surfaces. This lens is so placed in between the plano-convex and plano-concave lenses that the plane surfaces are parallel to each other.
The focal length of the combination is

Options

A. ` (R ) /(( mu _ 1 + mu _ 2 - mu _ 3 - 1 )) `
B. `(R ) /( ( mu _ 1 + mu _ 2 + mu _ 3 )) `
C. ` (R ) /( ( mu _ 1 - mu _ 2 ) ) `
D. ` (R ) /( (mu _ 1 - mu _ 2 - mu _ 3 - 3 ) ) `

Answer

C

Explanation

` ( 1 ) /( f _ 1 ) = ( mu_ 1 - 1 ) (( 1 )/ ( oo) + ( 1 ) /(R ) ) = ( mu _ 1 - 1 ) /( R ) `
` rArr ( 1 ) /( f _ 1 ) = ( mu _ 3 - 1 ) ( - ( 1 ) /(R ) + ( 1)/(R )) = 0 `
` ( 1 ) /( f _ 3 ) = ( mu _ 2 - 1 ) (- ( 1) /( R ) - ( 1) /(oo) ) =- ((mu _ 2 - 1 ) ) /(R ) `
` ( 1 ) /(f) = ( 1 ) /( f _1 ) + ( 1 ) / (f _ 2 ) + ( 1 ) /( f _ 3) = ( ( mu _ 1 - 1 ) ) /(R ) - ((mu _ 2 - 1 )) /(R ) - (1) /(R ) ( mu _ 1 - mu _ 2 ) `
` rArr f = ( R) /( ( mu _ 1 - mu _ 2 )) `