A plano- convex lens of refractive index 1.5 and radius of curvature 30 cm is silvered at the curved surface. Now , this lens has been used to form the image of an object . At what distance from this lens, an object be placed in order to have a real image of the size of the object?

Options

A. 20 cm
B. 30 cm
C. 60 cm
D. 80 cm

Answer

A, B, C

Explanation

Ray coming out of object placed at O on axis of a plano convex lens will first refract from a, than reflect from b and than will refract from c and final image I will be obtained. If this lens is silvered from - one surface as shown, then it will work as mirror, here equivalent focal length is `f_e`.

`therefore(1)/(f_e)=-(n/f_1""-m/f_m)` .....(i)
where `f_m` = focal length of mirror after silvered.
`implies` From lens maker.s formula,
`(1)/(f_1)=((n_e-n_1)/(n_1))((1)/(R_1)-(1)/(R_2))`
`(1)/(f_1)=((1.5-1.0)/(1.0))((1)/(infty)-(1)/(-30))`
`=1/2(+(1)/(30))`
`=(1)/(60)`
`therefore f_1=60` cm
From equation (1),`(1)/(f_e)=-((n)/(f_1)-(m)/(f_m))`
[No. of lens =2,no. of nirrror=1]
`(1)/(f_e)=-((2)/(60)-(1)/((-15)))`
`(1)/(f_e)=-((1)/(30)+(1)/(15))`
`(1)/(f_e)=-(3)/(30)=-(1)/(10)`
`therefore f_e=-10` cm
`therefore` Magnitude `f_e=10` cm
If image is to be obtained of same size of objec then it should be placed at centre of curvature
`therefore` Object distance `u = 2f_e = 2 xx 10 = 20` cm