Q.A plano-convex lens of focal length $20\ \text{cm}$ has its plane face silvered. Explain, using the idea of combining a lens with a mirror, why the resulting system behaves as a concave mirror, and find its equivalent focal length.

Light entering through the un-silvered convex face is refracted once by the lens, then reflected by the silvered plane face (which, being flat, is an ordinary plane mirror with $f_{\text{mirror}}=\infty$, i.e. zero power), and finally refracted by the lens a second time on its way back out — two refractions through the lens and one reflection, all combined. Because two refractive "passes" through the same lens occur for every one reflection, the equivalent power is $\dfrac1F=\dfrac2{f_{\text{lens}}}+\dfrac1{f_{\text{mirror}}}=\dfrac2{20}+0=\dfrac1{10}$, giving $F=10\ \text{cm}$. Physically, the system as a whole redirects light back the way it came (as any mirror does, unlike a lens which transmits light through), so it must be equivalent to some single mirror; the calculation shows this equivalent mirror is converging, i.e. it behaves exactly like a concave mirror of focal length $10\ \text{cm}$, even though no curved mirror was actually used anywhere in building it.

Recognise a plane-silvered surface as a mirror of zero power, and combine two lens-refraction powers with one mirror-reflection power using $\frac1F=\frac2{f_{\text{lens}}}+\frac1{f_{\text{mirror}}}$.

• Forgetting the factor of 2 on the lens term (light passes through the lens TWICE).
• Treating the silvered plane surface as having no effect at all, rather than as a (zero-power) plane mirror that reverses the light's direction.
• Trying to use the ordinary lens formula on the silvered system instead of the mirror formula (once $F$ is found, mirror-formula rules apply, since the system reflects light).