1. Title and aim
Focal length of a convex lens by the u–v method. Determine the focal length of one thin convex lens using focused object–image pairs, and test the thin-lens relationship with a reciprocal-distance graph.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of focal length of 15.003 cm, differing by 0.021% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.
3. Apparatus and fixed settings
- Illuminated arrow object: Provides a recognisable shape so image orientation and sharpness can be judged.
- Thin convex lens and holder: Refracts light into a real image when the object is beyond the focal point. Keep the same lens throughout the series.
- Movable screen: Intercepts the image. Its location is adjusted manually until focus is achieved.
- Optical bench and distance scale: Keeps object, lens and screen aligned and provides object and image distances from the lens plane.
- Thin converging lens; all object distances exceed its focal length.
- u and v are positive real-object and real-image distance magnitudes.
- Distances are measured from the lens principal plane; screen scale spacing 0.1 cm.
4. Theory and graph choice
An image becoming larger does not mean that a lens has changed its focal length. In this experiment, keep one lens and move an illuminated object and a screen. Find several sharp images, measure both distances, and ask whether one focal length explains them all. The simulation uses a 15 cm ideal lens; the measurement task is to recover that value from observations.
The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units dimensionless.
5. Method
- Place the illuminated object, lens centre and screen centre on a common axis. Establish the lens plane as the distance origin. In a physical lab, use a low-power lamp and keep the screen perpendicular to the bench.
- Keep the same lens and set u = 25.0 cm. Predict the approximate image position before moving the screen. All suggested object positions exceed the 15 cm model focal length.
- Move the screen through the blurred image and bracket the sharpest position from both sides. The virtual bench accepts a reading only when the screen is within 0.05 cm of the ideal image plane; that tolerance is a software rule, not a claim about real visual precision.
- Record u and the focused screen distance v to 0.1 cm. Note the inverted image and its relative size. Preserve the measured numbers before calculating f.
- Use u = 30.0, 35.0, 45.0 and 60.0 cm. Refocus for every new object position; do not leave the previous screen distance unchanged. Keep lens selection and distance units consistent.
- For each pair compute f = uv/(u+v), then plot 1/v against 1/u with labelled reciprocal units. Report the fitted slope, intercept, reciprocal-intercept focal length and limitations. Distinguish rounding differences from actual evidence of experimental noise.
6. Observations and calculation table
All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.
| No. | Object distance u (cm) | Focused image distance v (cm) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 25 | 37.5 | 4.00000 | 2.66667 | 2.66780 | -0.00114 |
| 2 | 32 | 28.2 | 3.12500 | 3.54610 | 3.54224 | 0.00386 |
| 3 | 38 | 24.8 | 2.63158 | 4.03226 | 4.03535 | -0.00309 |
| 4 | 45 | 22.5 | 2.22222 | 4.44444 | 4.44444 | 4.4244e-06 |
| 5 | 60 | 20.0 | 1.66667 | 5.00000 | 4.99964 | 3.6066e-04 |
Download the sample observations (CSV) ↓
7. Worked calculation for one observation
8. Graph, gradient and final calculation
Dots show the sample observations. The line is the ordinary least-squares fit.
Σ(x − x̄)² = 3.16712
Σ(x − x̄)(y − ȳ) = -3.16509
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = -0.99935878
b = ȳ − mx̄ = 6.66523730
f = 1 / 6.66523730 = 0.150032 m = 15.0032 cm
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0214%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 2.5839e-05; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00165. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00469.
Propagating only the relevant fit coefficient gives a fit-only standard error of 0.01055 cm. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Both inverse-distance axes have measurement uncertainty. The equal-weight vertical regression shown here is a teaching calculation, not a full errors-in-variables analysis. Scale zero, lens thickness, alignment and subjective focus affect real distances. f = 1/b gives an intercept-only standard error uf = ub/b² when other effects are ignored.
10. Result and discussion
Model comparison value: 15.000 cm
The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.
11. Precautions and limitations
- In a physical lab, use a lamp or LED object. Never view the Sun through the lens or use sunlight as the focusing source.
- Measure from the lens principal-plane approximation and document holder offsets; read bench scales without parallax.
- Keep the same lens for every row, align component heights, and judge focus using the same image feature.
- Record observations before rounding calculations. A screen-focus acceptance indicator is a simulation aid unavailable on an ordinary optical bench.
The simulation uses a monochromatic paraxial thin-lens model with a fixed 15 cm focal length. The ray diagram illustrates geometrical construction; it does not calculate diffraction, lens thickness, spherical or chromatic aberration, detector response, scattered light or actual image contrast. Focus is tested against a calculated plane with a 0.05 cm software tolerance and observations are displayed to 0.1 cm. These choices teach the measurement workflow but cannot establish real apparatus accuracy. For a real lens, identify its principal-plane approximation, assess focus range and scale uncertainty, and check alignment before interpreting agreement with the model. The lens has a model focal length of 15.0 cm and is treated as thin and paraxial. The scene is a schematic ray diagram. A screen setting within 0.05 cm of the ideal image plane is accepted; this tolerance represents half a 0.1 cm scale step, not optical depth of focus. Spherical/chromatic aberrations, lens thickness and alignment errors are omitted.
12. Conclusion
The worked example illustrates how to obtain focal length from measurements, graph variables and a fitted coefficient. The estimate is 15.003 cm. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.