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Put a candle and a screen on a bench and try to focus the flame with a convex lens. Sometimes you find two sharp positions, sometimes one, and sometimes none at all. The reason is a neat piece of algebra: a convex lens can only form a real image on a screen at least four focal lengths from the object. This post proves it, turns it into a method for measuring f, and works three problems.
Is this on the syllabus?
The thin lens formula and magnification are in the 2026-27 reprint of the NCERT Class 12 chapter Ray Optics and Optical Instruments, and Exercise 9.19 asks the classic question: a screen is 90 cm from an object, the image forms at two lens positions 20 cm apart; find f. The Syllabus for JEE (Main) 2026 lists the thin lens formula and magnification in Unit 16 and the focal length of a convex lens by the parallax method in its experimental skills. The NEET (UG) 2026 syllabus has the same topics. Check the syllabus for your exam year.
Why can't the screen be any distance away?
Work with magnitudes. For a real image, u is the object distance and v the image distance, both positive, and the lens formula becomes
1/u + 1/v = 1/f
The object and screen are D apart, so u + v = D. Combine the two:
(u + v)/(uv) = 1/f, so uv = fD
You now know the sum and the product of u and v. They must be the two roots of
x² − Dx + fD = 0
Real roots need the discriminant to be zero or positive: D² − 4fD ≥ 0. Since D is positive, that means
D ≥ 4f
At D = 4f there is one root, u = v = 2f, and the image is the same size as the object (m = −1). For D greater than 4f there are two roots, and the lens has two focusing positions.
You can also see it with calculus. D(u) = u + uf/(u − f). Setting dD/du = 0 gives u = 2f and D = 4f, the smallest possible separation.
Worked example 1: does the image form, and where?
A convex lens has f = 12 cm.
(a) D = 40 cm. 4f = 48 cm, which is more than 40 cm. The discriminant is 40² − 4 × 12 × 40 = −320. No real image forms anywhere between the object and the screen.
(b) D = 60 cm. x² − 60x + 720 = 0. The discriminant is 3600 − 2880 = 720, and √720 = 26.8. x = (60 ± 26.8)/2, so u = 16.6 cm or u = 43.4 cm.
Position 1: u = 16.6 cm, v = 43.4 cm. |m| = v/u = 2.62. Large image.
Position 2: u = 43.4 cm, v = 16.6 cm. |m| = 0.382. Small image.
Check: 1/16.6 + 1/43.4 = 0.0833 = 1/12. The two positions are d = 43.4 − 16.6 = 26.8 cm apart, and the two magnifications multiply to 1.
How does this give a way to measure f?
In the two positions, u and v swap. If the lens moves a distance d between them,
u = (D − d)/2 and v = (D + d)/2
Put these into uv = fD:
f = (D² − d²) / (4D)
This is the displacement method. Its advantage is that you measure only D and d, not distances to the lens's optical centre, so you avoid the index correction the u-v method needs.
Worked example 2: finding f from two positions
A screen is 100 cm from an object. Two sharp images form with the lens positions 40 cm apart.
f = (100² − 40²)/(4 × 100) = (10000 − 1600)/400 = 21 cm.
The positions: u = (100 − 40)/2 = 30 cm and u = (100 + 40)/2 = 70 cm.
Check: 1/30 + 1/70 = 10/210 = 1/21. It agrees.
Now try NCERT Exercise 9.19 the same way: with D = 90 cm and d = 20 cm, you should get f ≈ 21.4 cm.
Worked example 3: the object's height from two images
In the two positions, the magnifications are v/u and u/v, so their product is 1. If the images are h₁ and h₂ tall, the object height is h = √(h₁h₂).
Images of 9.0 mm and 4.0 mm give h = √36 = 6.0 mm. The object is the geometric mean of the two images.
Common mistakes and how to avoid them
Using signed u in u + v = D. With Cartesian signs, u is negative. Write D = |u| + v, or work in magnitudes as above.
Mixing up D and d. D is object to screen. d is the distance the lens moves between the two sharp positions.
Expecting two positions at D = 4f. There is only one, at u = v = 2f.
Applying this to a concave lens. A concave lens alone never forms a real image of a real object, so no screen distance works.
For the u-v method on the bench, see our guide to the convex lens focal length experiment. For the lens maker's formula and power, read lens combinations and the lens maker's formula. If the quadratic step feels like a trick, the Fermi Concept Library covers every concept in physics and mathematics, each with its own lesson.
A practice routine
Derive D ≥ 4f from the quadratic once, without notes.
Repeat example 1 for D = 48 cm and D = 80 cm with f = 12 cm.
Solve NCERT Exercise 9.19 and two more displacement-method problems.
Do one image-height problem using h = √(h₁h₂).
Try it now: with f = 15 cm, find the smallest object-screen distance and the two lens positions when the screen is 80 cm away.
