1. Title and aim
Ohm’s law and resistance from a current–voltage graph. Determine the resistance of an ohmic conductor from the gradient of current against potential difference.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of resistance of 100.0000 Ω, differing by 0.000% 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
- Variable DC supply: Apply 1–10 V across the resistor with the circuit switched off while wiring.
- 100 Ω model resistor: Provide a constant resistance for the simulated comparison. A physical resistor must have an appropriate power rating.
- Ammeter and voltmeter: Place the ammeter in series and the voltmeter in parallel across the resistor.
- Switch and leads: Complete secure connections and limit unnecessary heating between readings.
- Fixed ideal resistance: 100 Ω at constant temperature.
- Voltage is across the resistor; current is in series, rounded to 0.01 mA.
- Ideal supply and meters; maximum model power at 10 V is 1 W.
4. Theory and graph choice
Build the link between a circuit diagram, meter readings and a straight-line graph. Set a voltage across a resistor, measure its current and use several observations to estimate resistance. The readings here are generated by a fixed-temperature resistor model and are simulated data.
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 A V⁻¹ = Ω⁻¹.
5. Method
- With the supply off, connect the resistor and ammeter in series and the voltmeter across the resistor. Check polarity and meter ranges.
- Begin at low voltage. For real apparatus check power rating and current limit before using the full range; keep temperature stable.
- Set the voltage and obtain the current. Record voltage in V and the displayed current in mA, including the display resolution.
- Take at least five voltage settings across the permitted range. Repeat if possible and compare increasing and decreasing voltage runs in a physical experiment.
- Convert mA to A and fit I = mV + b without forcing the intercept to zero. Inspect residuals and intercept.
- Calculate R = 1/m in Ω. State fit-only error separately from meter, loading and temperature uncertainties, and label these observations simulated.
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. | Applied voltage V (V) | Current I (mA) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 1.0 | 10.00 | 1.00000 | 0.01000 | 0.01000 | 0.00000 |
| 2 | 3.0 | 30.00 | 3.00000 | 0.03000 | 0.03000 | 0.00000 |
| 3 | 5.0 | 50.00 | 5.00000 | 0.05000 | 0.05000 | 0.00000 |
| 4 | 7.0 | 70.00 | 7.00000 | 0.07000 | 0.07000 | 0.00000 |
| 5 | 9.0 | 90.00 | 9.00000 | 0.09000 | 0.09000 | 0.00000 |
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 = 40.00000
Σ(x − x̄)(y − ȳ) = 0.40000
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 0.01000000
b = ȳ − mx̄ = 0.00000000
R = 1/m = 1/(0.01000000) = 100.000000 Ω
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 0.00000; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00000. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00000.
Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00000 Ω. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Current and voltage calibration, contact resistance, ammeter burden and voltmeter loading affect real measurements. Heating changes the resistance. The inverse-gradient standard error omits calibration and temperature uncertainty. A perfect straight line from an ideal simulator does not imply zero physical uncertainty.
10. Result and discussion
Model comparison value: 100.0000 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
- Switch off before changing physical connections; use correct meter sockets and ranges.
- Never connect an ammeter directly across the supply.
- Check resistor power rating; the model’s upper setting dissipates 1 W.
- Minimize contact resistance and heating; record simulated observations as simulated.
This model assumes a constant 100 Ω resistance, exact applied voltage, an ideal ammeter and voltmeter, and no heating or lead resistance. Current is rounded to 0.01 mA. The chosen voltage grid can give perfectly collinear readings; a zero fit-only standard error is not a claim of zero laboratory uncertainty. The generator uses I = V/100 and displays current in mA. Graph fitting converts current back to A and estimates R = 1/gradient. The supplied 100 Ω value is a model reference, not an independently measured result.
12. Conclusion
The worked example illustrates how to obtain resistance from measurements, graph variables and a fitted coefficient. The estimate is 100.0000 Ω. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.