Resistance vs. Temperature lab report

1. Title and aim

The Dependence of Resistance on Temperature: Temperature Coefficient of Resistance of Copper. To investigate the variation of electrical resistance of a metallic copper conductor as a function of temperature, determine its temperature coefficient of resistance (α) and reference resistance at 0 °C (R₀) via linear regression, and extrapolate the inferred absolute zero of resistance.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of temperature coefficient of resistance of 3.930 × 10⁻³ °C⁻¹, 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

• Precision Copper Resistance Thermometer Coil: Provides a non-inductively wound bifilar coil of high-purity electrolytic copper wire (calibrated nominal resistance R₀ = 100.00 Ω at 0.0 °C) sealed inside a thin-walled protective brass sheath for rapid thermal equilibrium.
• Thermostatically Controlled Liquid Heating Bath: Provides a thermally insulated fluid bath fitted with electric immersion heating elements, an ice-water cooling stage, and continuous mechanical stirring to maintain stable and uniform temperature from 0.0 °C to 100.0 °C.
• Calibrated Laboratory Mercury-in-Glass Thermometer: Measures bath temperature from −5.0 °C to +105.0 °C with 0.1 °C resolution, immersed alongside the copper coil sheath to track true equilibrium fluid temperature.
• Four-Terminal Digital Ohmmeter / Kelvin Bridge: Measures coil resistance with 0.01 Ω resolution using a Kelvin 4-wire sensing configuration (independent source current and sense voltage leads) to eliminate lead and contact resistances.
• Continuous Motorized Liquid Stirrer: Prevents thermal stratification, convective fluid layering, and localized temperature gradients between the heating elements, thermometer bulb, and resistance coil sheath.
• Thermal Insulating Enclosure & Immersion Collet: Supports the thermometer and coil assembly securely at the geometric center of the bath while minimizing evaporative heat losses and thermal drafts from the laboratory.
• Electrolytic copper wire purity: 99.95% annealed copper (IACS conductivity standard)
• Reference ice-point resistance R₀ = 100.00 Ω (at T = 0.0 °C)
• Literature temperature coefficient α_ref = 0.003930 °C⁻¹ (3.930 × 10⁻³ °C⁻¹)
• Temperature measurement range: T = 0.0 °C to 100.0 °C in 20.0 °C intervals
• Ohmmeter excitation current: I_test ≤ 1.0 mA (Joule self-heating error < 0.002 °C)
• Four-wire Kelvin connection geometry eliminating lead wire resistance offsets

4. Theory and graph choice

The electrical resistance of pure metallic conductors varies directly with temperature as a consequence of electron-phonon scattering. At temperatures above absolute zero, thermal vibrations of lattice ions increase in amplitude, presenting a larger scattering cross-section to conduction electrons and shortening their mean collision time. Over normal operating ranges (0 °C to 100 °C), this relationship is highly linear: R(T) = R₀(1 + α T). By measuring the resistance of a calibrated copper coil immersed in a thermostatically controlled bath across multiple temperatures and applying linear regression, students determine the baseline ice-point resistance R₀ from the intercept, the temperature coefficient of resistance α from the ratio of slope to intercept, and extrapolate the inferred absolute zero of electrical resistance.

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 °C⁻¹.

5. Method

1. Immerse the copper coil sheath and calibrated mercury thermometer into a vigorously stirred equilibrium mixture of crushed distilled ice and pure water at 0.0 °C. Connect the four terminals of the digital bridge (two current leads, two potential leads) to the coil binding posts. Wait 5 minutes for complete thermal equilibrium and record the ice-point baseline resistance R₀ = 100.00 Ω.
2. Transfer the coil and thermometer into the thermostatically controlled heating bath filled with distilled water. Activate the motorized circulation stirrer at moderate speed to ensure vigorous convective fluid mixing throughout the tank, eliminating vertical temperature stratification.
3. Energize the immersion heater to elevate the bath temperature to approximately 20.0 °C. Throttle the heater power and monitor the thermometer until the temperature stabilizes to within ±0.1 °C for at least 60 seconds, ensuring that the internal copper windings have reached isothermal steady state with the bath.
4. Record the stabilized bath temperature T (in °C) and the corresponding four-wire resistance reading R (in Ω). Re-energize the heater to raise the bath temperature sequentially to 40.0 °C, 60.0 °C, 80.0 °C, and 100.0 °C (boiling point), recording the equilibrium resistance at each station.
5. Tabulate the paired observations of temperature T (horizontal axis x) and coil resistance R (vertical axis y). Compute the ordinary least-squares linear regression coefficients to determine the slope m = ΔR/ΔT (in Ω °C⁻¹) and the vertical intercept b (in Ω).
6. Calculate the temperature coefficient of resistance as α = m / b. Compute the inferred absolute zero of electrical resistance by extrapolating the best-fit line to R = 0 (T_abs = −b / m = −1 / α), and compare the experimental values against standard copper reference tables.

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.

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.

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 = 1.2117e-27; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 2.0803e-16. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.2597e-14.

Propagating only the relevant fit coefficient gives a fit-only standard error of 2.0803e-15 × 10⁻³ °C⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Thermodynamic thermal lag between the protective brass sheath and the internal copper wire windings (minimized by maintaining steady temperature holding times ≥ 60 s before each reading), fluid bath thermal stratification and convective layering (eliminated by continuous motorized paddle circulation), digital bridge Joule self-heating (kept below 0.002 °C by restricting test excitation current to I_test ≤ 1.0 mA), partial-immersion mercury stem emergence cooling correction (ΔT_stem = 0.00016 · n · (T − T_ambient) ≈ 0.04 °C at 100 °C), lead wire and terminal contact resistance offsets (completely bypassed via four-terminal Kelvin 4-wire sensing), and secondary non-linear Callendar-Van Dusen curvature (< 0.05% below 100 °C) govern the uncertainty envelope. Ordinary least-squares regression of coil resistance R against temperature T averages out thermometer meniscus parallax and digital bridge quantization fluctuations, with the slope isolating the pure thermal rate of change dR/dT and the vertical intercept isolating the true ice-point baseline resistance R₀.

10. Result and discussion

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

• Maintain continuous mechanical bath stirring to prevent thermal stratification and ensure the thermometer and coil experience identical temperatures.
• Wait at least 60 seconds after reaching each thermal station before recording resistance to allow internal heat conduction through the coil sheath.
• Keep the bridge measuring current below 1.0 mA to prevent parasitic Joule self-heating of the fine copper winding.
• Use four-terminal Kelvin connections directly at the coil binding posts to eliminate lead resistance offsets.
• Avoid splashing water onto the binding posts or terminal heads to prevent leakage currents and electrolytic corrosion.

Calculations assume uniform thermal equilibrium between the bath and copper winding, negligible Joule self-heating (excitation current ≤ 1 mA), and neglect higher-order quadratic Callendar-Van Dusen curvature which contributes < 0.05% below 100 °C. Real copper specimens exhibit slight residual resistance at cryogenic temperatures (Matthiessen's rule: ρ = ρ_residual + ρ_phonon(T)), while industrial platinum RTDs incorporate small second-order coefficients (B ≈ −5.775 × 10⁻⁷ °C⁻²).

12. Conclusion

The worked example illustrates how to obtain temperature coefficient of resistance from measurements, graph variables and a fitted coefficient. The estimate is 3.930 × 10⁻³ °C⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.