Lens–Mirror Systems & the Silvered Lens

Most of ray optics teaches lenses and mirrors as separate worlds — a chapter for one, a chapter for the other. Then a JEE problem quietly puts them in the same beam: light passes through a lens, reflects off a mirror, and passes back through the lens again. Suddenly a student who can solve any single lens and any single mirror freezes, because the question is not about either alone but about what they do together. Handle it as one object and the fog lifts: a lens-and-mirror combination behaves as a single equivalent mirror, with one focal length you can compute in a line and then use exactly like an ordinary mirror. The set-piece of this topic is the silvered lens — a lens with one face turned into a mirror by a thin silver coating. Light refracts through the glass, reflects off the silvered back, and refracts out again: three optical events, one compact device. The trick that tames it is to count each surface's power and add them up, remembering that the lens is crossed twice and the mirror once, so the combination has power $P = 2P_l + P_m$ . From that single relation everything follows: a plano-convex lens silvered on its flat face turns into a concave-equivalent mirror, and silvering the curved face instead makes a much stronger one. This is the Indian-board favourite that rewards clean bookkeeping. We build the equivalent-power rule from first principles, prove that a plane-silvered plano-convex lens acts as a concave mirror of half the lens's focal length, and extend the plain lens-in-contact rule $P = P_1 + P_2$ (established earlier in this chapter) to lenses separated by a gap. By the end you will look at any lens–mirror stack, collapse it to one equivalent element, and locate its image with the mirror formula you already trust.

Key takeaways

A lens–mirror combination — light refracting through a lens, reflecting off a mirror, refracting back — behaves as a single equivalent mirror of power $P = 2P_l + P_m$ : the lens is crossed twice, the mirror once.
A silvered lens is the special case where the mirror is a silvered face of the lens itself: $\dfrac{1}{F} = \dfrac{2}{f_l} + \dfrac{1}{f_m}$ , with $f_m = R_{\text{silvered}}/2$ .
Plane face silvered: $f_m = \infty$ , so $F = f_l/2$ — the device acts as a concave (converging) mirror, not a plane one. A plano-convex $f_l = 20\ \mathrm{cm}$ becomes $F = 10\ \mathrm{cm}$ .
From $R = 20\ \mathrm{cm}$ , $\mu = 1.5$ : the plano-convex lens has $f_l = 40\ \mathrm{cm}$ ( $P_l = 2.5\ \mathrm{D}$ ); plane-silvered gives $P = 5\ \mathrm{D}$ ( $F = 20\ \mathrm{cm}$ ), curved-silvered gives $P = 15\ \mathrm{D}$ ( $F = 6.67\ \mathrm{cm}$ ).
Two separated lenses use $P = P_1 + P_2 - d\,P_1 P_2$ : two $+5\ \mathrm{D}$ lenses $10\ \mathrm{cm}$ apart make $7.5\ \mathrm{D}$ ( $F = 13.33\ \mathrm{cm}$ ), weaker than in contact.
Key idea: collapse any lens–mirror stack to one equivalent mirror by adding the surface powers — lens twice, mirror once, $P = 2P_l + P_m$ — then treat it as an ordinary mirror.

The core insight

A lens–mirror system, and the silvered lens in particular, behaves as one equivalent mirror whose power is the sum of the powers of the surfaces the light crosses — the lens counted twice and the mirror once, $P = 2P_l + P_m$ — so once you compute that single equivalent focal length you locate its image with the same mirror formula you already know.

Where this lives today

Combining lenses and mirrors into one imaging element is a mainstay of real optics. The catadioptric telescope — the Schmidt and Maksutov designs prized by amateur astronomers and used in survey cameras — puts a thin corrector lens in front of a mirror, folding a long focal length into a short, portable tube; it is a lens–mirror system solved by exactly the equivalent-power thinking of this page. Compact camera lenses and long telephoto "mirror lenses" do the same, using a reflecting element to shorten the barrel. The humble silvered lens itself appears in optical benches and in some rangefinders and reflectors. Even a retroreflector and the corner optics of a bar-code scanner rely on collapsing several refractions and reflections into one predictable equivalent element. Wherever a designer needs the focusing power of a lens with the compact fold-back of a mirror, the rule $P = 2P_l + P_m$ — count the lens twice — is the arithmetic that sizes it.

The physical picture

Follow one ray through a silvered lens and the whole topic becomes visible. The ray enters the front face of the lens and refracts, bending as it crosses from air into glass; it travels through the glass to the back face, which is silvered, and there it reflects; it travels back through the glass to the front face and refracts out into the air again. Three events — refract, reflect, refract — and the light ends up going backwards, on the same side it came from. That last fact is the tell: the device sends light back the way it came, so it behaves not like a lens (which passes light through) but like a mirror. The power of a coaxial optical system, when its elements sit essentially in contact, is just the sum of the individual powers — the same rule that lets lenses in contact add their powers, extended to a reflecting surface. Here the light crosses the lens twice (once going in, once coming out) and the mirror once, so the equivalent power is

P = 2P_l + P_m \qquad (1)

where: $P_l = 1/f_l$ is the power of the lens (in dioptres, $f_l$ in metres) and $P_m = 1/f_m$ the power of the mirror surface, with $f_m = R_{\text{silvered}}/2$ . The factor of two on the lens is the heart of the topic and the single most forgotten detail: the lens does its refracting job on the way in and on the way out, so it counts twice. The combination's equivalent focal length is then $F = 1/P$ , and because the device returns light, it is a mirror's focal length — a converging (concave-equivalent) mirror when $P > 0$ .

Figure

A plano-convex lens with its flat back face silvered: an incoming ray refracts at the curved front, reflects off the silvered flat back, and refracts again at the front on the way out, so the light returns and the device acts as an equivalent mirror of power two P lens plus P mirror

The silvered lens, written as a mirror

Spell equation (1) out in focal lengths and it becomes the form most textbooks quote. Since $P = 1/F$ , $P_l = 1/f_l$ and $P_m = 1/f_m$ , dividing through gives

\frac{1}{F} = \frac{2}{f_l} + \frac{1}{f_m}, \qquad f_m = \frac{R_{\text{silvered}}}{2} \qquad (2)

where: $F$ is the equivalent mirror's focal length, $f_l$ the lens's focal length and $f_m$ the focal length of the silvered surface treated as a mirror, itself half the radius of that surface. The lens focal length, if not given, comes from the lensmaker's formula $1/f_l = (\mu - 1)(1/R_1 - 1/R_2)$ , quoted from earlier in this chapter; for a plano-convex lens one radius is infinite, so only the curved face contributes. The most examined case is the plane face silvered. A flat mirror has an infinite radius, so $f_m = \infty$ and $1/f_m = 0$ ; equation (2) collapses to $\dfrac{1}{F} = \dfrac{2}{f_l}$ , that is

F = \frac{f_l}{2} \qquad (3)

where: $F$ is the equivalent focal length when the plane face is silvered. The result is worth pausing on: a plane-silvered lens is emphatically not a plane mirror. It is a converging (concave-equivalent) mirror of half the lens's focal length, because the two passes through the converging lens do the focusing while the flat back merely folds the light. Once you have $F$ , locate the image with the ordinary mirror formula $1/v + 1/u = 1/f$ using $f = -F$ for the concave-equivalent (New Cartesian, as in the live mirror articles).

Extending the combination rule to separated lenses

The plain rule for two thin lenses in contact — $P = P_1 + P_2$ , established earlier in this chapter — is the $d \to 0$ corner of a more general one. When the lenses are pulled apart by a distance $d$ , the system weakens by a cross-term:

P = P_1 + P_2 - d\,P_1 P_2 \qquad (4)

where: $d$ is the separation in metres and $P_1, P_2$ the individual powers in dioptres. The subtraction says that spacing two converging lenses reduces the combined power below their simple sum, which is why a camera's zoom or a telescope's eyepiece uses the gap $d$ as a design knob. It is the same family as the silvered lens — a coaxial stack collapsed to one equivalent element — and the second figure shows it.

Figure

Two converging lenses on a common axis separated by a distance d: their combined power is P one plus P two minus d times P one P two, so two five-dioptre lenses ten centimetres apart make seven and a half dioptres, weaker than the ten dioptres they would give in contact

Derivation AThe equivalent power of a silvered lens

The goal is to derive the silvered-lens rule $P = 2P_l + P_m$ — equivalently $1/F = 2/f_l + 1/f_m$ — by counting the power the light picks up at each surface it crosses, and to see exactly why the lens power is counted twice. This uses only the fact that coaxial elements in contact add their powers, a rule quoted from the lens-combinations discussion earlier in this chapter; there is no calculus.

Step 1. List the events in order as the ray traverses the device. It refracts through the whole lens once on the way in (power $P_l$ ), reflects off the silvered surface (power $P_m$ as a mirror), and refracts through the whole lens once on the way out (power $P_l$ again). Three contributions: $P_l$ , then $P_m$ , then $P_l$ .

Step 2. Add the powers. For coaxial elements effectively in contact, the equivalent power is the sum of the individual powers. So $P = P_l + P_m + P_l = 2P_l + P_m$ . The lens power appears twice because the lens physically does its refracting work on both passes; the mirror is met only once.

Step 3. Translate to focal lengths. With $P = 1/F$ , $P_l = 1/f_l$ and $P_m = 1/f_m$ , the same statement reads $\dfrac{1}{F} = \dfrac{2}{f_l} + \dfrac{1}{f_m}$ , and $f_m = R_{\text{silvered}}/2$ because the silvered face is a mirror of that radius. This is equation (2).

Step 4. Test on the curved-silvered anchor. Take a plano-convex lens of $R = 20\ \mathrm{cm}$ , $\mu = 1.5$ : the lensmaker's formula gives $1/f_l = (1.5 - 1)(1/20) = 0.025\ \mathrm{cm}^{-1}$ , so $f_l = 40\ \mathrm{cm}$ and $P_l = 1/0.40 = 2.5\ \mathrm{D}$ . Silver the curved face: $f_m = R/2 = 10\ \mathrm{cm}$ , $P_m = 1/0.10 = 10\ \mathrm{D}$ . Then $P = 2(2.5) + 10 = 15\ \mathrm{D}$ , so $F = 100/15 = 6.67\ \mathrm{cm}$ — a strong converging mirror. Silvering the flat face instead makes $P_m = 0$ and $P = 5\ \mathrm{D}$ , $F = 20\ \mathrm{cm}$ : the same lens, a much weaker mirror, just by choosing which face to coat. Verified: $f_l = 40$ cm, $P_l = 2.5$ D, curved-silvered $P = 15$ D ( $F = 6.67$ cm) and plane-silvered $P = 5$ D ( $F = 20$ cm) are recomputed in verify_hs2-lens-mirror-systems.py.

Go deeper · Derivation B · 1 minWhy a plane-silvered plano-convex lens is a concave mirror of half the focal length

Derivation BWhy a plane-silvered plano-convex lens is a concave mirror of half the focal length

The goal is to prove that silvering the flat face of a plano-convex lens turns it into a concave (converging) mirror of focal length $F = f_l/2$ , and to confirm with the anchor $f_l = 20\ \mathrm{cm} \to F = 10\ \mathrm{cm}$ that it images a distant object exactly as a concave mirror of that focal length would. The point is to dispel the natural but wrong guess that a plane-silvered lens is a plane mirror.

Step 1. Start from the silvered-lens rule $\dfrac{1}{F} = \dfrac{2}{f_l} + \dfrac{1}{f_m}$ . The silvered face is flat, so its radius is infinite and $f_m = R/2 = \infty$ , giving $1/f_m = 0$ .

Step 2. The rule collapses: $\dfrac{1}{F} = \dfrac{2}{f_l}$ , so $F = \dfrac{f_l}{2}$ . The mirror term has vanished, yet the equivalent focal length is finite — the focusing is done entirely by the two passes through the converging lens, not by the flat back.

Step 3. Read the sign and nature. With a converging lens, $f_l > 0$ and so $P = 2P_l > 0$ : the equivalent mirror is converging, i.e. a concave-equivalent mirror. It is therefore not a plane mirror at all — a plane mirror would have $F = \infty$ and simply return an erect, same-size virtual image, whereas this device converges light to a real focus.

Step 4. Confirm with the anchor. For $f_l = 20\ \mathrm{cm}$ , $F = f_l/2 = 10\ \mathrm{cm}$ . A distant object ( $u \to -\infty$ ) then images at the focus of this equivalent concave mirror, $10\ \mathrm{cm}$ in front, real and inverted — exactly what a concave mirror of focal length $10\ \mathrm{cm}$ does with the Sun. To place a nearer object, use the mirror formula with $f = -10\ \mathrm{cm}$ (concave, New Cartesian). The device behaves in every way like a concave mirror of half the lens's focal length. Verified: plane-silvered $f_l = 20 \to F = 10$ cm, the concave-equivalent classification, and the distant-object image at the focus are recomputed in verify_hs2-lens-mirror-systems.py.

Limiting cases

Case	Result	Meaning
plane face silvered	$F = f_l/2$	concave-equivalent mirror, not a plane mirror
curved face silvered ( $f_m = R/2$ )	$F = 1/(2P_l + P_m)$	a much stronger converging mirror
lens power $\to 0$ (very weak lens)	$F \to f_m$	recovers the bare mirror alone
mirror plane and lens plane	$F \to \infty$	a true plane mirror, no focusing
two lenses in contact, $d \to 0$	$P \to P_1 + P_2$	equation (4) recovers the simple sum
separated lenses, $d = 1/P_2$	$P \to P_1$	the second lens sits at the first's focus, adds nothing

Common trapThe lens power counts twice, and a plane-silvered lens is not a plane mirror

Two misconceptions sink most silvered-lens problems. The first is forgetting the factor of two: the light passes through the lens on the way in and on the way out, so the equivalent power is $P = 2P_l + P_m$ , not $P_l + P_m$ . Drop the extra $P_l$ and every focal length you compute is wrong, usually badly. The second is imagining that silvering the flat face of a lens gives a plane mirror. It does not: with $f_m = \infty$ the rule gives $F = f_l/2$ , a concave-equivalent converging mirror that forms real, inverted images of distant objects — a plane mirror could never do that. A third, quieter slip is treating the whole device as a lens and looking for a transmitted image on the far side; there is no far side, because the silvered face sends the light back, so the image forms on the incoming side like any mirror. Count the lens twice, remember the device is a mirror, and locate the image with the mirror formula using $f = -F$ for the concave-equivalent.

Worked examples

Board level — a plane-silvered plano-convex lens (CBSE and JEE-Main pattern)

A plano-convex lens is made of glass of refractive index $\mu = 1.5$ with its curved face of radius $20\ \mathrm{cm}$ . Its flat face is silvered. Find the focal length of the lens alone, and the equivalent focal length of the silvered system. What does it do to a very distant object?

Step 1. Lens focal length from the lensmaker's formula (plano-convex, flat face has infinite radius): $\dfrac{1}{f_l} = (\mu - 1)\dfrac{1}{R} = (0.5)\dfrac{1}{20} = \dfrac{1}{40}$ , so $f_l = \boxed{40\ \mathrm{cm}}$ and $P_l = 1/0.40 = 2.5\ \mathrm{D}$ .

Step 2. The flat face is silvered, so $f_m = \infty$ and $P_m = 0$ . The equivalent power is $P = 2P_l + P_m = 2(2.5) + 0 = \boxed{5\ \mathrm{D}}$ .

Step 3. The equivalent focal length is $F = \dfrac{1}{P} = \dfrac{1}{5}\ \mathrm{m} = \boxed{20\ \mathrm{cm}}$ — equivalently $F = f_l/2 = 40/2 = 20\ \mathrm{cm}$ , the two routes agreeing.

Step 4. Because $P > 0$ , the device is a concave-equivalent mirror of focal length $20\ \mathrm{cm}$ . A very distant object ( $u \to -\infty$ ) images at its focus, $20\ \mathrm{cm}$ in front, real and inverted — the same as a concave mirror of $20\ \mathrm{cm}$ focal length. It is not a plane mirror, despite the flat silvered face.

Competitive level — silver the curved face instead (JEE-Advanced pattern)

The same plano-convex lens ( $\mu = 1.5$ , $R = 20\ \mathrm{cm}$ , $f_l = 40\ \mathrm{cm}$ , $P_l = 2.5\ \mathrm{D}$ ) now has its curved face silvered. Find the new equivalent focal length, and compare it with the plane-silvered case.

Step 1. The silvered surface is now the curved face of radius $20\ \mathrm{cm}$ , acting as a mirror of $f_m = R/2 = 10\ \mathrm{cm}$ , so $P_m = 1/0.10 = 10\ \mathrm{D}$ .

Step 2. Equivalent power: $P = 2P_l + P_m = 2(2.5) + 10 = \boxed{15\ \mathrm{D}}$ .

Step 3. Equivalent focal length: $F = \dfrac{100}{15} = \boxed{6.67\ \mathrm{cm}}$ — a strong converging mirror.

Step 4. Compare: plane-silvered gave $P = 5\ \mathrm{D}$ , $F = 20\ \mathrm{cm}$ ; curved-silvered gives $P = 15\ \mathrm{D}$ , $F = 6.67\ \mathrm{cm}$ — three times the power from the same lens, just by choosing which face to coat, because the curved mirror ( $P_m = 10\ \mathrm{D}$ ) adds far more than the flat one ( $P_m = 0$ ). This is the discriminating step examiners test: which face is silvered decides the strength. Timing note: about ninety seconds. The whole problem is the one line $P = 2P_l + P_m$ ; the only decisions are the lens power from the lensmaker's formula and which mirror power to add ( $0$ for a flat face, $1/(R/2)$ for the curved one). Keep everything in dioptres and metres and the arithmetic is trivial.

Exam traps

Forgetting the factor of two. The lens is crossed twice, so its power counts twice: $P = 2P_l + P_m$ . Writing $P_l + P_m$ is the number-one error and throws off every focal length that follows.

Calling a plane-silvered lens a plane mirror. With the flat face silvered, $F = f_l/2$ — a converging concave-equivalent mirror that makes real images, not the erect virtual image of a plane mirror.

Using simple addition for separated lenses. Two lenses a distance $d$ apart use $P = P_1 + P_2 - dP_1P_2$ , not $P_1 + P_2$ . Drop the cross-term and you overstate the power.

Treating the device as a lens. A silvered lens returns light, so it is a mirror: the image forms on the incoming side, and you locate it with the mirror formula ( $f = -F$ for the concave-equivalent), never by looking for a transmitted image.

Mixing centimetres and metres in powers. Power is $1/f$ with $f$ in metres. $f_l = 40\ \mathrm{cm} = 0.40\ \mathrm{m}$ gives $P_l = 2.5\ \mathrm{D}$ ; keep the units consistent before summing.

Board scope. The silvered lens and lens–mirror equivalent-power system are set by CBSE, ICSE, JEE-Main, JEE-Advanced and NEET — this is an Indian-board and JEE specialty. AP Physics 2 covers single lenses and single mirrors but not the combined silvered-lens system, so it is not on the AP-2 CED; IB SL/HL and A-Level 9702 carry no geometric optics of this kind at all. If you are sitting one of those exams, treat this as enrichment; if you are sitting JEE or NEET, it is a bankable set-piece.

Origins · 1 minPierre de Fermat and how this was discovered

Pierre de Fermat

1607–1665 · Toulouse

Fermat was a lawyer and councillor at Toulouse who did mathematics in his spare time — and changed it forever, founding analytic geometry alongside Descartes, probability with Pascal, and number theory almost single-handedly. His gift to optics is the principle of least time (1662): light travels between two points along the path that takes the least time. From this one variational idea both laws of ray optics follow — the law of reflection for a mirror and Snell's law of refraction for a lens are each the least-time path for their situation. That is exactly why a lens–mirror system is so natural to think about as a unit: a ray threading a silvered lens (refract, reflect, refract) is following a single least-time route through two elements at once. Honest scope: Fermat supplied the deep principle from which the lens and mirror laws spring; the silvered-lens equivalent-power trick you drill here is a much later, practical application of the ray optics his principle underwrites.

History

1621

— Willebrord Snellius states the law of refraction, the lens half of any lens–mirror system.

1637

— René Descartes publishes the sine law of refraction in La Dioptrique.

1662

— Pierre de Fermat derives reflection and refraction alike from the principle of least time.

1841

— Carl Friedrich Gauss reduces any coaxial optical system to equivalent focal lengths and cardinal points.

1930

— Bernhard Schmidt builds the catadioptric telescope, combining a corrector lens with a mirror in one imaging system.

Going further

Texts. NCERT Physics Class 12, Ch. 9 ("Ray Optics and Optical Instruments"), refraction at spherical surfaces, lens combinations and the silvered lens · H. C. Verma, Concepts of Physics Vol. 1, "Geometrical Optics" — combined systems · D. C. Pandey / Cengage optics problem sets for the JEE silvered-lens set-pieces.

Provenance. Standard geometric optics. Recompute: plano-convex $f_l = 1/[(1.5-1)/20] = 40$ cm, $P_l = 2.5$ D; plane-silvered $P = 2(2.5) + 0 = 5$ D, $F = 20$ cm $= f_l/2$ ; curved-silvered $f_m = 20/2 = 10$ cm, $P_m = 10$ D, $P = 15$ D, $F = 100/15 = 6.67$ cm; the direct anchor $f_l = 20 \to F = f_l/2 = 10$ cm; separated lenses $P = 5 + 5 - 0.10\times25 = 7.5$ D, $F = 100/7.5 = 13.33$ cm; distant object images at the focus of the equivalent mirror.

Prerequisites. The Mirror Formula and Sign Convention and Magnification by Mirrors — the mirror bookkeeping this page collapses the system down to. Leads to. Mirror and Lens Sign-Convention Discipline, which drills the signs used here; and the university seams below.

Go deeper (UG). The general treatment of coaxial multi-element systems — the matrix (ray-transfer) method that handles any stack of lenses and mirrors, and the cardinal points of a thick system — is developed in Thin Lens and Lensmaker's Equation and Plane and Spherical Mirrors in the university optics track. It is enrichment only; every result here follows from adding powers and the mirror formula, and no exam on this page asks for the matrix method.

Why is the lens power counted twice in a silvered lens?

Because the light physically passes through the lens twice — once on the way in, before it reflects, and once on the way out, after it reflects. Each pass bends the light by the lens's full refracting power $P_l$ , so the two passes contribute $P_l + P_l = 2P_l$ , while the mirror is met only once and contributes $P_m$ . That is why the equivalent power is $P = 2P_l + P_m$ . Think of the silvered surface as a fold in the light path: it turns the single lens into an effective double lens with a mirror sandwiched between, and the double crossing is what the factor of two records. Forgetting it is the commonest mistake in the whole topic, and it always makes the computed focal length wrong.

Is a plane-silvered lens the same as a plane mirror?

No. It looks like one — a flat silvered surface — but it is a converging mirror. When you silver the flat face of a plano-convex lens, the mirror term vanishes ( $f_m = \infty$ ) but the two passes through the converging lens remain, giving an equivalent focal length $F = f_l/2$ , finite and positive. A plane mirror has an infinite focal length and only ever returns an erect, same-size virtual image; this device, by contrast, converges light to a real focus and can form a real, inverted, magnified or diminished image of a real object, exactly like a concave mirror of focal length $f_l/2$ . The flat silvered face fools the eye; the physics is set by the lens it is attached to.

How do I locate the image once I have the equivalent focal length?

Treat the whole silvered lens as a single mirror and use the mirror formula $1/v + 1/u = 1/f$ with $f = -F$ for the concave-equivalent (New Cartesian, matching the live mirror articles). Sign the object distance as usual for a real object, $u < 0$ , substitute, and read off $v$ : a negative $v$ is a real image in front of the device, a positive $v$ a virtual image behind. The magnification follows from $m = -v/u$ , the mirror rule, because the device is a mirror. Do not use the lens formula and do not look for a transmitted image on the far side — there is no far side, since the silver sends the light back. One equivalent focal length, then ordinary mirror bookkeeping.

Why does silvering the curved face give a stronger mirror than silvering the flat face?

Because the curved silvered face is itself a converging mirror, adding its own power, whereas a flat silvered face adds nothing. Silvering the flat face gives $f_m = \infty$ , $P_m = 0$ , so the combination's power is just $2P_l$ . Silvering the curved face of radius $R$ makes it a mirror of $f_m = R/2$ , contributing $P_m = 2/R$ on top of the lens's $2P_l$ . For the worked lens ( $P_l = 2.5\ \mathrm{D}$ , $R = 20\ \mathrm{cm}$ ), the flat case gives $P = 5\ \mathrm{D}$ but the curved case gives $P = 5 + 10 = 15\ \mathrm{D}$ — three times stronger. The same piece of glass becomes a very different mirror depending on which surface you coat, which is exactly the discriminating detail exam questions probe.

When two lenses are separated, why does the power come out less than the sum?

Because the second lens intercepts light that the first has already started to converge, so the gap between them lets the beam narrow before the second lens acts, reducing how much extra bending it can add. The formula $P = P_1 + P_2 - dP_1P_2$ captures this with the negative cross-term $-dP_1P_2$ : the larger the separation $d$ , the more the combined power falls below the simple sum $P_1 + P_2$ . For two $+5\ \mathrm{D}$ lenses $0.10\ \mathrm{m}$ apart, the sum would be $10\ \mathrm{D}$ but the cross-term subtracts $0.10 \times 25 = 2.5\ \mathrm{D}$ , leaving $7.5\ \mathrm{D}$ . When the lenses touch ( $d = 0$ ) the cross-term disappears and you recover $P_1 + P_2$ , the in-contact rule.

Does the silvered lens work the same for a diverging lens or a concave lens?

The same rule applies — $P = 2P_l + P_m$ — but you must carry the signs. A diverging lens has $P_l < 0$ , so it subtracts from the mirror's power, and the equivalent system can end up converging or diverging depending on which term wins. For example, a diverging lens silvered on a curved face might give $2P_l + P_m$ that is still positive (a converging equivalent mirror) if the mirror term dominates, or negative (a diverging equivalent mirror) if the lens does. The bookkeeping never changes: compute each surface's power with its correct sign, count the lens twice, add, and the sign of the total tells you whether the equivalent mirror converges or diverges. The plano-convex examples here keep everything positive to isolate the method.