A glass hemispher of refractive index `4//3` and of radius `4 cm` is placed on a plane mirror. `A` point object is placed on axis of this sphere at a distance `'d'` from `O` as shown in the figure. If the final image is formed at infinity, then find the value of `d` in `mm` .

Answer

30

Explanation

The image of object `O` by refraction at plane surface is formed at `I` such that
`AI=(4)/(3)d`

I acts object for curved surved. The curved surface makes image of `I` at `I`
`(1)/(v) -(4/3)/(-(R+(4)/(3)d))=(1-4/3)/(-R) or (1)/(v)=(1)/(3R)-(4)/(3R+4d)`
I' acts as object for mirror. Mirror makes its image at `I`" distance `v` above `B` .
I" acts as virtual object for the curved surface which makes its image at infinity
`(4/3)/(infty)-[(1)/(3R)-(4)/(3R+4d)]=(1/3)/(R)`
Solving we get `d=(3)/(4)R(3)/(4)xx4=3 cm`