Potentiometer EMF Comparison lab report

1. Title and aim

The Potentiometer: Accurate Measurement and Comparison of Electromotive Forces. To calibrate a slide-wire potentiometer using a Weston standard cell, determine the uniform potential gradient along the slide wire, compare the electromotive forces of dry and primary cells at null balance without drawing current, and verify that cell EMF is strictly proportional to balancing length.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of potentiometer potential gradient of 2.0000 V m⁻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 Slide-Wire Potentiometer Bench: Provides a 1.000-meter uniform constantan or manganin alloy resistance wire (R_wire = 10.00 Ω, length L₀ = 100.0 cm) stretched above a parallel metric scale graduated in millimeters, mounted between massive copper terminal blocks of negligible contact resistance.
• Regulated Lead-Acid Driver Accumulator & Rheostat: Supplies a steady, non-polarizing direct driving potential (ℰ₀ = 2.000 V) in series with a precision multi-turn wirewound rheostat to establish and maintain a perfectly constant potential gradient k = 2.000 V/m (0.02000 V/cm) along the slide wire.
• Saturated Weston Cadmium Standard Cell: Serves as the primary electromotive force calibration reference standard (ℰ_s = 1.0186 V at 20 °C) with an exceptionally low temperature coefficient of EMF (−0.00004 V/°C), balancing at ℓ_s = 50.93 cm.
• High-Sensitivity Center-Zero Reflecting Galvanometer: Detects microampere-level imbalance currents (sensitivity ~10⁻⁷ A/div) between the wire contact potential and the cell EMF to identify the exact physical null balance position.
• Protective 10,000-Ohm Series Resistor & Bypass Key: Protects the standard cell and delicate galvanometer suspension against destructive discharge currents during preliminary slider hunting, short-circuited during final delicate balance refinement.
• Precision Knife-Edge Slider Jockey & DPDT Reversing Switch: Enables instantaneous two-way switching between the standard cell and unknown test cells, and provides smooth, repeatable electrical contact along the potentiometer wire without scraping or local mechanical deformation.
• Driver Accumulator EMF: ℰ₀ = 2.000 V (regulated constant DC source)
• Potentiometer Wire Length & Resistance: L₀ = 100.0 cm, R_AB = 10.00 Ω (constantan wire)
• Potential Gradient: k = 2.0000 V m⁻¹ (0.02000 V cm⁻¹ = 20.00 mV cm⁻¹)
• Weston Standard Cell Reference: ℰ_s = 1.0186 V (balance length ℓ_s = 50.93 cm)
• Protective Ballast Resistance: R_prot = 10,000 Ω (tapped during coarse search)

4. Theory and graph choice

The slide-wire potentiometer, invented by Johann Christian Poggendorff in 1841, is the fundamental null-balance instrument for the precision measurement and comparison of electromotive forces. Unlike deflection voltmeters that draw operating current and depress terminal voltage via internal resistance drops (V = ℰ − I r), the potentiometer opposes the unknown cell EMF against a known, continuous potential drop along a uniform resistance wire. When the sliding contact is adjusted until the sensitive center-zero galvanometer indicates zero current (I_G = 0), the wire potential drop exactly equals the open-circuit cell EMF without perturbing the cell's thermodynamic equilibrium. In this experiment, students calibrate the wire potential gradient (k = 2.000 V/m = 0.02000 V/cm) using a saturated Weston cadmium standard cell (ℰ_s = 1.0186 V at ℓ_s = 50.93 cm), measure balancing lengths across six distinct cell EMF levels spanning 0.400 V to 1.500 V, apply linear regression to determine k and verify Poggendorff’s law (ℰ_x / ℰ_s = ℓ_x / ℓ_s), and explore the zero-current infinite input impedance principle.

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 V m⁻¹.

5. Method

1. Connect the regulated 2.000 V driver accumulator in series with a master knife switch, an adjustable wirewound rheostat, and the 1-meter potentiometer slide wire AB. Ensure the positive terminal of the driver accumulator connects directly to binding post A (the zero of the metric scale). Close the switch and adjust the rheostat to set the potential drop across wire AB to slightly greater than the maximum cell EMF to be measured (setting the potential gradient to exactly k = 0.02000 V/cm).
2. Connect the positive terminal of the Weston cadmium standard cell (ℰ_s = 1.0186 V) to the common positive post A. Connect the negative terminal through the protective 10,000 Ω resistor and sensitive center-zero galvanometer to the movable knife-edge jockey slider F. Verify that the standard cell electromotive force directly opposes the potential drop along the slide wire from post A.
3. Lightly touch the jockey contact to the extreme left of the wire near post A (ℓ ≈ 1 cm) and tap the galvanometer key, observing the direction of needle deflection. Next, touch the jockey to the extreme right near post B (ℓ ≈ 99 cm) and tap the key. Verify that the two deflections occur in opposite directions. If opposite deflections are not obtained, reverse the cell connections or reduce the driver rheostat resistance.
4. Use the method of successive approximations (bisection search) to locate the approximate null point with the 10,000 Ω protective resistor in series. Once the needle deflection is less than 1 scale division, depress the bypass key to short-circuit the protective resistor, and gently adjust the jockey position until the galvanometer rests at perfect zero. Record the standard cell balance length ℓ_s = 50.93 cm and compute the potentiometer potential gradient k = ℰ_s / ℓ_s = 1.0186 / 50.93 = 0.02000 V/cm (2.000 V/m).
5. Switch the secondary branch to the unknown test cells using the selector switch, ensuring that the driver circuit rheostat remains completely untouched. For each of the six cell EMF levels (spanning 0.400 V to 1.500 V, including standard cell, Daniell cell, and dry cells), determine the precise balancing length ℓ_x at galvanometer null. Record the balancing length in centimeters and convert to meters (L_x = ℓ_x / 100.0).
6. Plot the balancing length L (in meters) on the horizontal axis and the cell electromotive force ℰ (in Volts) on the vertical axis. Compute ordinary least-squares linear regression to determine the potential gradient k (the slope m in V/m) and the vertical intercept b. Calculate the cell EMF ratios ℰ_x / ℰ_s and compare against the corresponding length ratios ℓ_x / ℓ_s to verify Poggendorff’s law of the potentiometer.

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 = 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 V m⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Slide-wire cross-sectional non-uniformity and local mechanical wear from knife-edge jockey contact (~0.1% diameter variation producing subtle local potential gradient fluctuations), primary driver accumulator voltage drift and internal heating during prolonged continuous current flow (minimized by using a heavy-capacity lead-acid accumulator and monitoring standard cell balance ℓ_s periodically), parasitic thermoelectric Seebeck EMFs at the bimetallic copper-constantan binding post junctions (up to ±15 μV if terminal posts are touched by warm fingers, cancelled by current reversal), galvanometer pivot friction and visual crosshair null detection resolution (±0.2 mm along the metric scale, corresponding to ±40 μV in potential), knife-edge slider contact resistance fluctuations, and connecting lead resistance between the zero index and the binding post govern the uncertainty envelope. Ordinary least-squares regression of cell electromotive force ℰ against balancing length L averages out random slider visual alignment errors and local wire imperfections, with the slope isolating the pure potential gradient k and the vertical intercept isolating any end-resistance or zero-index offset.

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

• Always connect the positive terminals of both the driver accumulator and the test cells to the common binding post A at the zero of the metric scale.
• Keep the protective 10,000 Ω resistor in series with the galvanometer during initial slider hunting to prevent polarizing the standard cell, short-circuiting it only for final balance refinement.
• Never slide or scrape the knife-edge jockey along the wire under heavy downward pressure; touch it down gently at discrete points to avoid creating flat spots that ruin wire uniformity.
• Check the driver battery voltage and verify the standard cell balance length ℓ_s before and after the measurement series to confirm that the driver current has not drifted.
• Avoid touching the copper-constantan terminal posts with bare fingers during readings to prevent parasitic thermoelectric Seebeck voltages from biasing the galvanometer null.

Calculations assume a perfectly uniform wire cross-section and resistivity, an unvarying driver accumulator current throughout all runs, negligible thermoelectric EMFs at metal junctions, and negligible contact resistance at the jockey slider. Real slide wires exhibit minor local diameter variations (~0.1%) caused by manufacturing tolerances or mechanical wear from slider friction, as well as parasitic thermal Seebeck EMFs at the copper-constantan binding posts if touched by warm fingers.

12. Conclusion

The worked example illustrates how to obtain potentiometer potential gradient from measurements, graph variables and a fitted coefficient. The estimate is 2.0000 V m⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.