Estimate focal length from repeated image measurements

Turn a prediction into a measurement plan

A matching simulation is not evidence that a physical lens has the assumed focal length. This activity estimates a converging lens’s focal length from real-image arrangements. Use a securely held converging lens, a small diffuse LED object or well-lit printed target, white card screen and ruler. Never focus sunlight or look into directed bright sources. Keep the lens, object and screen stable and at the same axis height.

If apparatus is unavailable, use the explicitly synthetic practice dataset below. Report that choice in your notes. Do not describe the supplied values as observations you made.

Record a reproducible procedure

1. Choose an object distance beyond the lens’s estimated focal length. A lens near +20 cm is suitable for the browser’s 30, 40 and 60 cm object settings. For a different lens, choose appropriate distances and calculate predictions separately; the browser’s limited presets do not cover every apparatus.
2. Measure object distance from the chosen lens reference plane. State how you identified that plane and keep the same convention for screen distance. A thick lens or an uncertain reference plane introduces model error.
3. Move the screen on the far side until the image is as sharp as you can judge. Record the screen distance, image orientation, ruler resolution and the range of positions that look similarly sharp. Do not force the screen to the predicted position.
4. Move the screen away and refocus five times, keeping the object fixed. Record every result before averaging. These repeated judgments assess refocusing repeatability, not five independently manufactured lenses.
5. Repeat at two other object distances. Keep raw readings and any excluded trial with its reason. Compare focal-length estimates rather than selecting only the arrangement closest to the nominal lens label.

The shared notebook stores free-text notes and model settings. It does not automatically parse these physical readings or certify that the apparatus procedure was performed. Keep a separate raw-data table if your report is longer than the note field.

Synthetic practice data

These invented readings are deliberately centered on ideal predictions for f = +20 cm. They are an arithmetic exercise, not measurements of a real lens. All distances are in centimetres.

Object distance	Five screen-distance readings	Mean screen distance
30.0	59.0, 61.0, 60.0, 60.5, 59.5	60.0
40.0	39.0, 41.0, 40.0, 40.5, 39.5	40.0
60.0	29.0, 31.0, 30.0, 30.5, 29.5	30.0

Use f = dₒdᵢ/(dₒ + dᵢ) with each row’s mean image distance. The first row gives 30 × 60/(30 + 60) = 20 cm. The other two also give 20 cm by construction. Compare these means with captures from the browser; matching them checks the exercise’s arithmetic, not experimental validity.

Attach uncertainty assumptions to the estimate

For a bounded-error exercise, assume the true object distance lies within ±0.2 cm of the stated value and true image distance within ±1.0 cm of the mean. These are stipulated bounds, not confidence intervals derived from five readings. In a real experiment you must justify bounds from ruler alignment, resolution, focus ambiguity and other evidence.

For positive distances, f increases when either distance increases. The first row therefore has a lower estimate (29.8 × 59.0)/(29.8 + 59.0) = 19.79955 cm and an upper estimate (30.2 × 61.0)/(30.2 + 61.0) = 20.19956 cm. Report about 19.80–20.20 cm under the stated bounds, not six-decimal experimental precision. The calculation assumes the two bounds together enclose the true distances and the thin-lens model is adequate.

Refocusing more times can characterize repeatability, but it does not remove a shared lens-reference-plane error or ruler offset. The range of repeated readings is not automatically the uncertainty of their mean. The two other rows give different focal-length sensitivity to the same assumed distance bounds; an identical nominal estimate does not imply identical precision.

Test a new condition

After estimating f, predict a fresh object distance before measuring it. For the synthetic +20 cm estimate, choose dₒ = 50 cm and calculate the screen position. Use the browser only after writing your prediction. A physical trial at this new arrangement tests transfer of the fitted model; repeated inspection of the original three rows does not.

Check the measurement transfer

dᵢ = 20 × 50/(50 − 20) = 33.333333 cm. Magnification is −2/3. For a 0.50 cm object the predicted image height is −0.333333 cm. In real measurements compare a defensible prediction interval and observed focus range, not exact agreement with every printed decimal.

Finish with your data source, apparatus and reference plane, raw readings, focal estimates, assumptions behind the uncertainty bounds, one possible systematic error, and a new-condition result or clearly labeled plan. A human review of procedure and reasoning is still needed; notebook completion does not establish physical skill.

The governing equation is described in OpenStax: thin lenses. The procedure, synthetic readings and bounded-error exercise above are original teaching activities.