Lens and Mirror Formula — The Sign Convention That Stops Errors
Almost nobody loses marks in ray optics because they forgot the formula. They lose marks because they put a plus where a minus belonged. The formulas are two lines long; the sign convention is the actual subject. Get the convention right and every numerical in the chapter becomes routine — and the sign of your answer starts telling you whether the image is real or virtual, without you having to remember any case list.
The two formulas
Notice the two differences that catch people out: the mirror formula has a plus and the lens formula a minus; and the mirror magnification carries a negative sign while the lens magnification does not. Those are not typing errors — they follow from the geometry.
The Cartesian sign convention, in four rules
- All distances are measured from the pole of the mirror or the optical centre of the lens.
- Light is always drawn travelling left to right. Distances measured in that direction are positive; distances measured against it are negative.
- Heights above the principal axis are positive, heights below are negative.
- Because a real object sits to the left of the mirror or lens, u is always negative in ordinary problems. Write it as negative before you touch the formula.
| Quantity | Symbol | Sign | Unit |
|---|---|---|---|
| Object distance | u | Negative for a real object | cm or m |
| Image distance | v | Mirror: negative = real (in front). Lens: positive = real (behind) | cm or m |
| Focal length, concave mirror | f | Negative | cm or m |
| Focal length, convex mirror | f | Positive | cm or m |
| Focal length, convex (converging) lens | f | Positive | cm or m |
| Focal length, concave (diverging) lens | f | Negative | cm or m |
| Magnification | m | Negative = inverted (real); positive = erect (virtual) | no unit |
| Power of a lens | P | Same sign as f | dioptre (D) = m−1 |
The single most useful line in that table is the last-but-one. Once you have m, you do not need to recall any table of image cases: a negative m means the image is inverted and therefore real; a positive m means erect and therefore virtual. The mathematics reports the answer for you.
Worked example 1 — concave mirror
Question: An object is placed 25 cm in front of a concave mirror of focal length 15 cm. Find the image position, nature and size.
Assign signs first: u = −25 cm, f = −15 cm (concave mirror).
Mirror formula: 1/v + 1/u = 1/f → 1/v = 1/f − 1/u
1/v = 1/(−15) − 1/(−25) = −1/15 + 1/25
Common denominator 75: −5/75 + 3/75 = −2/75
So v = −75/2 = −37.5 cm
Read the sign: v is negative, so the image is 37.5 cm in front of the mirror — a real image you could catch on a screen.
Magnification: m = −v/u = −(−37.5)/(−25) = 37.5/(−25) = −1.5
Negative → inverted. Magnitude 1.5 → one-and-a-half times taller than the object.
Sanity check: the object at 25 cm lies between f (15 cm) and 2f (30 cm), and the textbook result for that case is a real, inverted, magnified image beyond 2f. Our image is at 37.5 cm, which is indeed beyond 30 cm ✔
Worked example 2 — convex mirror
Question: A convex mirror has focal length 10 cm. An object stands 20 cm from it. Where is the image?
Signs: u = −20 cm, f = +10 cm (convex mirror).
1/v = 1/f − 1/u = 1/10 − 1/(−20) = 1/10 + 1/20 = 2/20 + 1/20 = 3/20
v = 20/3 ≈ +6.67 cm
Read the sign: v is positive, so the image is behind the mirror — virtual, and cannot be caught on a screen.
Magnification: m = −v/u = −(6.67)/(−20) = +0.33
Positive → erect. Less than 1 → diminished.
Virtual, erect and diminished for every object position is exactly why convex mirrors are used as vehicle rear-view mirrors and at blind corners: a small, upright image of a very wide field.
Worked example 3 — convex lens, with power
Question: An object is 30 cm from a convex lens of focal length 20 cm. Find the image distance, magnification and the power of the lens.
Signs: u = −30 cm, f = +20 cm (convex lens).
Lens formula: 1/v − 1/u = 1/f → 1/v = 1/f + 1/u
1/v = 1/20 + 1/(−30) = 1/20 − 1/30
Common denominator 60: 3/60 − 2/60 = 1/60 → v = +60 cm
Read the sign: for a lens a positive v means the image forms on the far side, where the light actually goes — a real image, 60 cm from the lens.
Magnification: m = v/u = 60/(−30) = −2 → inverted and twice the object's height.
Power: convert the focal length to metres first: f = 20 cm = 0.20 m
P = 1/f = 1/0.20 = +5 D
Sanity check: the object lies between f and 2f (20 cm and 40 cm), so the image should be real, inverted, magnified and beyond 2f. 60 cm > 40 cm ✔
Worked example 4 — concave lens
Question: An object is 40 cm from a concave lens of focal length 25 cm. Describe the image.
Signs: u = −40 cm, f = −25 cm (concave lens).
1/v = 1/f + 1/u = (−1/25) + (−1/40) = −(1/25 + 1/40)
Common denominator 200: 8/200 + 5/200 = 13/200
1/v = −13/200 → v = −200/13 ≈ −15.4 cm
Read the sign: negative v for a lens means the image is on the same side as the object — virtual.
Magnification: m = v/u = (−15.4)/(−40) = +0.38 → erect and diminished.
Power: f = −25 cm = −0.25 m, so P = 1/(−0.25) = −4 D — the negative power printed on a short-sighted person's spectacle prescription.
Common mistakes
- Substituting u as a positive number. This is the number-one error in board answer sheets. Write u = −25 cm on the page before you open the formula, not while you are rearranging it.
- Using the mirror formula for a lens. Mirror: 1/v + 1/u = 1/f. Lens: 1/v − 1/u = 1/f. Mixing them gives an image on the wrong side entirely.
- Using m = −v/u for a lens. That negative sign belongs to mirrors only; for lenses m = v/u. Applying it to a lens flips "inverted" and "erect" in your conclusion even when v is correct.
- Taking the focal length of a concave mirror as positive. A concave mirror converges, but in the Cartesian convention its focus lies on the incoming-light side, so f is negative. Convex lens positive, concave mirror negative — the two "converging" cases have opposite signs, and that is not a contradiction, just a consequence of which way the light travels afterwards.
- Computing power with f in centimetres. P = 1/f demands metres. Using 20 instead of 0.20 gives 0.05 D instead of 5 D — a factor of 100.
- Forgetting to invert at the end. The formula gives you 1/v. Students routinely write down −2/75 and report it as v. Always take the reciprocal.
Where this appears in exams
| Exam | Typical question |
|---|---|
| CBSE/ICSE Class 10 | Light — reflection and refraction: image position and nature for concave/convex mirrors and lenses, ray diagrams |
| CBSE/ICSE Class 12 | Ray optics: lens maker's formula, combination of lenses (1/F = 1/f₁ + 1/f₂), power in dioptres |
| JEE & NEET | Lens–mirror combinations, silvered lenses, defects of vision and corrective power |
| Practical / lab work | Finding focal length by the u–v method and by plotting 1/v against 1/u |
Check your signs, not just your arithmetic. The Optics calculator applies the Cartesian convention to the mirror and lens formulas and returns v, the magnification and the nature of the image — so if your answer says "real" and the tool says "virtual", you know instantly that a sign, not a calculation, went wrong.
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