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

Answer

C

Explanation

Using lens maker's formula,
`(1)/(f)=(mu-1)((1)/(R_(1))-(1)/(R_(2)))`
`(1)/(f_(1))=((1.5)/(1)-1)((1)/(oo)-(1)/(-20))`
`impliesf_(1)=40cm`
`=(1)/(f_(2))=((1.7)/(1)-1)((1)/(-20)-(1)/(+20))`
`impliesf_(2)=-(100)/(7)cm`
and `(1)/(f_(3))=((1.5)/(1)-1)((1)/(oo)-(1)/(-20 ))`
`implies f_(3)=40cm`
`(1)/(f_(eq.))=(1)/(f_(1))+(1)/(f_(2))=(1)/(f_(3))`
`implies(1)/(f_(eq.))=(1)/(40)+(1)/(-100//7)+(1)/(40)`
`:. f_(eq.)=-50cm`
Therefore, the focal length of the combination if `-50 cm`.
Aliter `:`
`P_(convex)=(mu-1)/(R)=(1.5-1)/(20)=(1)/(40)`
`P_(concave)=(2(1.7-1))/(R)=(2(0.7))/(20)=(7)/(100)`
`P=2P_(convex)+P_(concave)`
`=(2)/(40)-(7)/(100)=(10-14)/(200)=(-4)/(200)=(-1)/(50)`
`F=-59cm`