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A lens with one face silvered looks like an awkward mix of a lens problem and a mirror problem. It is easier than it looks: the whole thing behaves as one mirror, and once you find that mirror's focal length, you just use the mirror formula. This post derives the rule, works three examples, and lists the traps.
Which syllabus covers this?
The JEE (Advanced) 2026 brochure lists "combinations of mirrors and thin lenses" under Optics. The JEE (Main) 2026 syllabus lists the mirror formula, the lens maker formula and combinations of thin lenses in contact. The CBSE Physics curriculum for 2026-27 includes a Class XII experiment that uses a convex lens with a plane mirror, to find the refractive index of a liquid. Check your exam year's syllabus.
Why does a silvered lens act as a mirror?
Follow the light. It enters the lens and refracts, reflects off the silvered face, and passes back through the lens before leaving. So it meets three elements in a row: lens, mirror, lens. They are all in contact.
The NCERT chapter Ray Optics and Optical Instruments shows that for thin lenses in contact, powers add: P = P₁ + P₂ + … (Eq. 9.32). Extending that idea to this path, and counting converging power as positive:
P = 2P_lens + P_mirror
where P_lens = 1/f of the lens and P_mirror = 2/R for a curved mirror surface (since f = R/2), or zero for a flat one. The result behaves as a concave mirror of focal length F = 1/P when P is positive.
Worked example 1: a plano-convex lens, silvered either side
A plano-convex lens has n = 1.5 and a curved face of radius R = 20 cm. By the lens maker's formula, 1/f = (n − 1)/R = 0.5/20, so f = 40 cm and P_lens = 1/40 per cm.
Flat face silvered. P_mirror = 0, so P = 2 × (1/40) = 1/20 per cm, and F = 20 cm. In general F = R / 2(n − 1).
Curved face silvered. The silvered curved face acts as a concave mirror with P_mirror = 2/20 = 1/10 per cm. P = 1/20 + 1/10 = 3/20 per cm, so F ≈ 6.67 cm. In general F = R/2n.
Silvering the curved face makes a much stronger mirror, because the curved surface now reflects as well as the lens refracting twice.
Worked example 2: where is the image?
An object stands 30 cm in front of each version. Use the mirror formula 1/v + 1/u = 1/f with NCERT's sign convention, where a concave mirror has negative f and u = −30 cm.
Flat face silvered (f = −20 cm): 1/v = −1/20 + 1/30 = −1/60, so v = −60 cm. The image is real, in front of the lens, inverted, and magnification m = −v/u = −2.
Curved face silvered (f ≈ −6.67 cm): 1/v = −3/20 + 1/30 = −7/60, so v ≈ −8.57 cm and m ≈ −0.29. Real, inverted and smaller.
Worked example 3: an equiconvex lens
An equiconvex lens with n = 1.5 and both radii 20 cm has 1/f = 0.5 × (2/20), so f = 20 cm. Silver one face: P = 2 × (1/20) + 2/20 = 1/5 per cm, so F = 5 cm.
A built-in check
For the flat-silvered lens, F = f/2, so the centre of curvature of the equivalent mirror sits at 2F = f. An object at the lens's own focal point therefore forms its image on top of itself. That makes sense: light from the focus leaves the lens parallel, hits the flat mirror at right angles, and retraces its path. A lens on a plane mirror finds its focal length this way.
Why do these problems feel hard?
Goldberg and McDermott (1987) interviewed students about real images formed by lenses and concave mirrors. Many who had just finished a geometrical optics course could not connect the formulas and ray diagrams to an actual lens, mirror and screen. Silvered lenses add a second element, so draw the ray path before you write any equation.
Which mistakes cost marks?
Counting the lens once. It is 2P_lens.
Treating a flat-silvered lens as a plane mirror. The lens still refracts twice.
Using the lens formula at the end. The combination is a mirror, so use 1/v + 1/u = 1/f.
Sign slips. Find F as a converging power first, then put f = −F into the mirror formula.
Using R for a flat face. A plane surface has no power.
A practice routine
Day 1: Derive F = R/2(n − 1) and F = R/2n without notes.
Day 2: Redo example 2 with the object at 10 cm and say whether each image is real or virtual.
Day 3: Solve a plano-concave lens silvered on its flat face, and decide whether it acts as a converging or diverging mirror.
Day 4: Revise the lens maker's formula and lens combinations, then do two mixed problems.
Day 5: Test yourself without notes. Dunlosky and colleagues (2013) rated practice testing and spaced practice as the most useful study techniques.
If you cannot picture the ray going in and back out, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Tonight, draw the ray path for a flat-silvered lens and mark where the equivalent mirror's focus and centre of curvature lie.
