Two identical thin planoconvex glass lenses (refractive index `1.5)` each having radius of curvature of `20cm` are placed with their convex surfaces in contact at the centre. The intervening space is filled with oil of refractive index `1.7`. The focal length of the combination is

Options

A. `-50cm`
B. `50cm`
C. `-20cm`
D. `-25cm`

Answer

A

Explanation

Given combination is equivalent to three lenses. In which two are plano-convex with refractive index `1.5` and one is concave lens of refractive index `1.7`.

Using lens maker formula,
`(1)/(f)=(mu-1)((1)/(R_(1))-(1)/(R_(2)))`
For plano-convex lens
`R_(1)=oo,R_(2)=-20cm`,
`:.(1)/(f_(1))=(1)/(f_(2))=(1.5-1)((1)/(oo)-(1)/(-20))=(0.5)/(20)=(1)/(40)`
So, `f_(1)=f_(2)=40cm`
For concave lens,
`mu=1.7,R_(1)=-20cm,R_(2)=20cm`
`:.(1)/(f_(3))=(1.7-1)((1)/(-20)-(1)/(20))`
`=0.7xx((-2)/(20))=-(7)/(100)`
So, `f_(3)=-(100)/(7)cm`
Equivalent focal length `(f_(eq))` of the system is given by
`(1)/(f_(eq))=(1)/(f_(1))+(1)/(f_(3))+(1)/(f_(2))=(1)/(40)+(1)/(-100//7)+(1)/(40)`
`=(1)/(20)-(7)/(100)=-(2)/(100)=-(1)/(50)`
`:.f_(eq)=-50cm`