Question

A planoconvex lens is polished with silvered find refractuve index iit

Answer 1

We have to consider the individual refractive surfaces only and not the planoconvex lens as a whole, but it so happens, that the final answer can be simply expressed in terms of the focal length the planoconvex lens would have without the silvering.

For a general derivation, consider a lens with radius (R1,R2), with R2 silvered. Cartesian sign convention are used here.

For an object distance (from the first surface, but for a thin lens, the distance is same for both the surfaces) u, we get for the first surface:-

μv1=μ−1R1+1u

(where μ is the refractive index of lens and that of air is 1)

Now for surface two, a mirror, v1 is the object:-

1v2=2R2−1v1

Now the second image v2 acts as the object for first surface again, but now with inverted sign conventions.

−1v=1−μ−R1+μv2

using the value of v1 and v2 from preceding equations you come up with

1v+1u=2μR2−2(μ−1)R1

This is similar to a mirror with focal length given by the left hand side of the equation. It can be rearranged to

1f=−(2(μ−1)R1−2(μ−1)R2)+2R2=−2/flens+1/fmirror