Two plano-convex lenses of radii of curvature R and R and refractive index 1.5 will have equivalent focal length equal to R, if they are placed

Options

A. at distance R
B. at distance R/2
C. at distance R/4
D. in contact with each other

Answer

D

Explanation

For the first plano convex lens `R_(1)=R and R_(2)=oo`
`therefore (1)/(f) =(n-1)((1)/(R_(1))-(1)/(R_(2))) " but " (1)/(R_(2)) =0`
`therefore (1)/(f_(1)) =(1.5-1)(1)/(R )=(1)/(2R) therefore f_(1)=2R`
Similarly for the second lens, `f_(2)=2R`
If they are kept in closed contact, their equivalent focal length is given by
`therefore (1)/(F)=(1)/(f_(1))+(1)/(f_(2))=(1)/(2R)+(1)/(2R)=(2)/(2R)=(1)/(R )`
Thus, `F=R`
Thus the lenses are kept in contact with each other. If they are kept with some distance between them, then `F ne R`.