1. Title and aim
Unknown resistance using a metre bridge. Determine an unknown resistance using balance lengths on a metre bridge and a straight-line 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 unknown resistance of 4.9875 Ω, differing by 0.250% 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
- Uniform metre bridge wire: Provide a 100.0 cm wire whose resistance is proportional to length.
- Unknown resistor in the left gap: Connect the resistor R to the same side as the zero end used to measure balance length.
- Resistance box in the right gap: Set the known resistance S in ohms.
- DC cell, key, galvanometer and jockey: Detect a null using brief light contact; a physical galvanometer may require protective resistance.
- Unknown R is in the left gap; l is measured from the left end.
- Fixed unknown R = 5 Ω; wire length 100 cm and uniform resistance per length.
- Jockey scale resolution: 0.1 cm. Ideal galvanometer at null.
4. Theory and graph choice
Explore a null measurement instead of reading current directly. Change a known resistance, locate the balance point on a uniform wire and infer an unknown resistance from a ratio of lengths. All generated balances are simulated and use a 100.0 cm ideal wire.
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 Ω⁻¹.
5. Method
- Connect unknown R in the left gap and known S in the right gap. Mark the wire zero at the unknown side and use L = 100.0 cm.
- Select S, starting near 5 Ω in this model to place the expected null near the midpoint.
- Close the cell key and touch the jockey lightly at trial positions until galvanometer deflection passes through zero. In this simulation request the model balance reading.
- Record S and l to the shown resolution. Collect at least five distinct S values, preferably keeping physical balances around the central portion.
- For each reading calculate L − l and y = (L − l)/l. Fit y against S with a free intercept.
- Calculate R = 1/m in Ω and compare individual values S l/(L − l). In real apparatus interchange the gap resistors and use the reversed formula, then discuss end effects and uncertainties.
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. | Known resistance S (Ω) | Null balance length l (cm) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 2.5 | 66.7 | 2.50000 | 0.49925 | 0.49851 | 7.3993e-04 |
| 2 | 4.0 | 55.6 | 4.00000 | 0.79856 | 0.79926 | -7.0041e-04 |
| 3 | 5.0 | 50.0 | 5.00000 | 1.00000 | 0.99976 | 2.3769e-04 |
| 4 | 7.5 | 40.0 | 7.50000 | 1.50000 | 1.50101 | -0.00101 |
| 5 | 10.0 | 33.3 | 10.00000 | 2.00300 | 2.00227 | 7.3697e-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 = 35.30000
Σ(x − x̄)(y − ȳ) = 7.07768
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 0.20050075
b = ȳ − mx̄ = -0.00274142
R = 1/m = 1/(0.20050075) = 4.987513 Ω
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.2497%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 2.6662e-06; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.5867e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00101.
Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00395 Ω. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
The balance-length scale is rounded to 0.1 cm. Contact and end resistances, wire nonuniformity and resistance-box tolerances are omitted. The transform (100−l)/l has nonlinear length uncertainty. A vertical equal-weight fit does not model shared scale errors. Interchange the resistances and repeat in a real lab.
10. Result and discussion
Model comparison value: 5.0000 Ω
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
- Define l from the unknown-resistance end and keep both lengths in the same unit.
- Touch the jockey gently and briefly; do not drag it along a live wire.
- Use small currents and open the key between physical readings to limit heating.
- Prefer balances near the middle, check resistance-box contacts and distinguish model readings from real observations.
The simulated wire is exactly uniform and 100.0 cm long. End resistances, jockey resistance, heating, thermoelectric offsets and galvanometer sensitivity are omitted. The unknown resistor is fixed at 5 Ω and known resistance settings are exact. Balance lengths are rounded to 0.1 cm; resulting fit scatter reflects quantization rather than a complete measurement uncertainty. The model places unknown R = 5 Ω on the left and known S on the right. It computes l = 100R/(R + S), rounded to 0.1 cm. The fitted graph uses x = S and y = (100 − l)/l, so resistance is 1/gradient. Range endpoints are available for sensitivity exploration; real measurements should favor central balances.
12. Conclusion
The worked example illustrates how to obtain unknown resistance from measurements, graph variables and a fitted coefficient. The estimate is 4.9875 Ω. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.