Converging lenses and focal length
Key idea: A converging lens brings rays parallel to its axis to the principal focus. Focusing a distant object's sharp image on a screen gives an approximate focal length from the lens-to-screen distance.
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Guided lessons · coming soon
Guided lesson
Guided steps
- Trace the parallel rays through the lens and beyond its focus.
- Locate where the beam meets when you reverse the illumination.
- Follow the distant-object method for measuring focal length.
- Choose the object that gives a better focal-length estimate and explain why.
- Find the lens-to-screen distance from the two scale readings.
- Find each lens-to-screen distance, then calculate their mean.
- Find the focal length when the screen reading is smaller.
- Explain why a negative coordinate difference still gives a positive distance.
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ContinueHold a magnifying glass between a bright window and a sheet of paper, and at one distance a sharp, upside-down picture of the window appears on the paper. The lens does this by refracting every ray at its two curved faces. Where a parallel beam meets is a property of the lens, its focal length, and you can measure it with nothing more than a distant object, a screen and a metre rule.
What a converging lens does to a beam
A thin converging lens made of glass is thicker in the middle than at its edges. Light refracts at both faces. Near the edge, the two faces are tilted most, so rays there are bent the most; a ray through the centre crosses two nearly parallel faces and is hardly bent at all. The result is that an incoming parallel beam is brought together. In this model the lens is thin and the rays stay close to the principal axis.
Three words describe where things are:
- the principal axis is the line through the centre of the lens at 90° to it;
- the optical centre, C, is the centre of the lens, on the principal axis;
- the principal focus, F, is the point on the principal axis where rays parallel to the axis meet after passing through the lens.
So the action of a thin converging lens on a beam is this: a beam of light parallel to the principal axis converges to the principal focus on the other side of the lens. A ray along the principal axis passes straight through. After F, the rays carry on and spread out again.
Light can pass through a lens either way, so a lens has a principal focus on each side. For a thin lens, both are the same distance from C. In diagrams, the rays are drawn bending once, at the centre line of the lens.
Check your understanding 1 Predict the beam’s destination
Show answer
A beam parallel to the principal axis converges to the far-side principal focus. Reversing the incoming beam uses the principal focus on the new far side, at the same distance from the optical centre in this thin-lens model. It does not return to the same physical point on the old far side.
Focal length
The focal length, f, of a converging lens is the distance between its optical centre and its principal focus. Measure it in cm or m.
Measure f with a distant object
Rays from one point of a very distant object arrive almost parallel to one another. The lens brings them to one point near its focal plane. Different points of a window form different image points; the whole window does not focus to a single dot.
- Point the lens at a distant object, such as a building or a tree outside the window. Never use the Sun, or look at the Sun through a lens: concentrated sunlight can injure eyes and burn materials.
- Hold a white screen behind the lens and move it until the image of the object is as sharp as possible.
- Measure the distance from the centre of the lens to the screen with a metre rule. This distance is approximately f.
- Repeat two or three times and take the mean.
The object is not infinitely far away, so the sharp image forms slightly beyond the focal plane. The further away the object, the closer the reading is to f.
Check your understanding 2 Why a distant object?
A lens forms a sharp image of a nearby object 22 cm from its centre, and a sharp image of a distant building 16 cm from its centre. Which distance is the better estimate of the focal length? Would using an even more distant object make the estimate larger or bring it closer to the true focal length?
Show answer
The 16 cm distance is the better estimate. Rays from each point of the distant building arrive almost parallel, so its image forms near the focal plane. The nearby object’s diverging rays form an image further beyond F, giving the larger 22 cm distance. An even more distant object gives rays closer to parallel, so the image distance becomes slightly smaller and approaches the true focal length. The image of a finite-distance object forms slightly beyond the focus, so 16 cm is still an estimate.
Worked example 1
Focal length from two scale readings
Problem
On an optical bench, the centre of a lens is at the 25.0 cm mark. A sharp image of a distant window forms on a screen at the 41.0 cm mark. Find the approximate focal length.
Show full solution
Identify the distance you need
Method
Focal length is measured from the optical centre to the principal focus.
Reason
Rays from each point of the distant window arrive almost parallel, so the sharp image forms near the focal plane. The lens-to-screen distance estimates f.
Working
Answer
You need the separation of the lens centre and the screen.
Subtract the readings
Working
Answer
f ≈ 41.0 cm - 25.0 cm = 16.0 cm
Quick check
41.0 cm is a position on the scale, measured from the bench’s zero, not the focal length. A distance always comes from subtracting two positions.
Guided practice 2
Your turn: three trials
Problem
A student keeps the lens centre at the 10.0 cm mark and focuses a distant tree three times. The sharp image appears with the screen at 25.2 cm, 24.8 cm and 25.3 cm. Find each lens-to-screen distance and their mean.
Try this before viewing the solution
Show the steps Optional working
Hints
Hint 1
distances first
Subtract 10.0 cm from each screen position to get three distances.
Show solution step by step
Convert positions to distances
Method
Subtract the lens position from each screen position.Reason
Each trial estimates f from the lens-to-screen distance.
Working
Answer
15.2 cm, 14.8 cm and 15.3 cm
Average
Working
Answer
f ≈ (15.2 + 14.8 + 15.3)/3 = 15.1 cm
Quick check
The image looks sharp over a small range of screen positions, so repeat readings differ a little. Averaging them gives a more reliable value.
Check your understanding 3 Measure with the light travelling the other way
On an optical bench, a thin converging lens is centred at the 42.0 cm mark. Light from a distant building enters from the right, and its sharp image forms on a screen at the 26.0 cm mark. Estimate the focal length.
Show answer
The lens-to-screen separation is 42.0 − 26.0 = 16.0 cm. This distance estimates the focal length.
Check your understanding 4 Explain the negative subtraction
Using the same bench readings, a learner reports −16.0 cm because the screen reading is smaller than the lens reading. Explain what the sign in that subtraction describes and how focal length is measured here.
Show answer
The approximate focal length is the separation of lens centre and screen: 42.0 − 26.0 = 16.0 cm. Subtracting screen position minus lens position gives −16.0 cm as a directed coordinate difference on this bench, showing that the screen is on the lower-reading side. It does not make the lens-to-focus distance negative. Light entering from the right uses the focus on the left; the same thin converging lens still has the same positive focal-length distance.
The images a lens forms, and how to describe them, are covered in Describing lens images.
Common mistakes
- Saying any beam converges at F. F is where rays parallel to the principal axis meet.
- Giving a scale reading as the focal length. f is a distance, so subtract the lens-centre position from the screen position.
- Measuring from the edge of the lens holder instead of the centre of the lens.
- Using a nearby object to find f. Its light is not parallel, so the sharp image forms well beyond F.
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Syllabus and review details
- SEC G3 Combined Science Physics component 2027 · 2027
Physics section, PDF pages 8–24
Last reviewed: