A transparent thin film of uniform thickness and refractive index `n_(1)``=1.4` is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index `n_(2)``=1.5`, as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance `f_(1)` from the film, while rays of light traversing from glass to air get focused at distance `f_(2)` from the film, Then `

Options

A. `|f_(1)|=3R`
B. `|f_(1)|=2.8R`
C. `|f_(2)|=2R`
D. `|f_(2)|=1.4R`

Answer

A, C

Explanation

We can use the refraction formula for two refractions which are taking place in each case.
`(mu_(2))/(v)-(mu_(1))/(u)=(mu_(2)-mu_(1))/(R )`
Air to `n_(1)`
`(1.4)/(v_(1))-(1)/(oo)=(1.4-1)/(R )" …(1)"`
Image from this refraction will serve as object for the next refraction.
`n_(1)" to "n_(2)`
`(1.5)/(f_(1))-(1.4)/(v_(1))=(1.5-1.4)/(R )" ...(2)"`
Adding equations (1) and (2) we get the following :
`(1.5)/(f_(1))=(0.4)/(R )+(0.1)/(R )=(0.5)/(R )`
`f_(1)=3R`
Let us proceed similarly for `f_(2)`.
`(mu_(2))/(v)-(mu_(1))/(u)=(mu_(2)-mu_(1))/(R )`
`n_(2)" to "n_(1)`
`(1.4)/(v_(1))-(1.5)/(oo)=(1.4-1.5)/(-R )" ...(3)"`
Image from this refraction will become object for the next refraction.
`n_(1)" to air"`
`(1)/(f_(2))-(1.4)/(v_(1))=(1-1.4)/(-R)" ...(4)"`
Adding equations (3) and (4) we get the following :
`(1)/(f_(2))=(1.4-1.5)/(-R)+(1-1.4)/(-R)`
`rArr" "(1)/(f_(2))=(-0.1)/(-R)+(-0.4)/(-R)=(0.5)/(R)`
`rArr" "f_(2)=2R`
Hence answer (a) and (c) are correct.