Electric Fields & Equipotentials lab report

1. Title and aim

Electric Fields: Equipotential Mapping, Electric Field Lines, and Potential Gradient. To investigate the electrostatic potential distribution in a two-dimensional conducting medium for uniform parallel-plate and non-uniform dipole electrode configurations, map equipotential contours, verify the strict spatial orthogonality between electric field lines and equipotential surfaces, measure the electric potential gradient (E = −dV/dx), and determine the uniform electric field strength using ordinary least-squares linear regression.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Uniform electric field strength E of 100.00 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

• Teledeltos Conducting Paper Mapping Board: Precision insulated mounting board supporting a 30 cm × 40 cm sheet of uniform carbon-graphite conducting paper, underlaid with a transparent 1 mm metric Cartesian coordinate grid for high-precision probe tracking.
• Parallel Linear Brass Electrodes: Pair of matched, heavy machined brass bar electrodes (length L = 20.0 cm, width 1.0 cm, thickness 0.5 cm) clamped securely onto the conducting paper with colloidal silver conductive paint to establish uniform parallel equipotential boundaries.
• Regulated Variable DC Power Supply: Low-noise laboratory DC power supply (0 to 15.00 V, 1.0 A current limit) providing an ultra-stable 10.00 V potential difference across the boundary electrodes.
• High-Impedance Digital Voltmeter (DVM): Digital multimeter with 10.0 MΩ input resistance and 1 mV voltage resolution, designed to sample local surface potentials without drawing significant loading current from the carbon sheet.
• Point-Contact Copper Tracking Probe: Insulated handheld copper-beryllium stylus probe with fine point contact for pinpointing exact Cartesian coordinates and sensing local surface potential without puncturing or tearing the conducting substrate.
• Interchangeable Dipole and Concentric Circular Electrodes: Set of brass point electrodes and concentric rings (inner cylinder r₁ = 1.0 cm, outer ring r₂ = 8.0 cm) for investigating non-uniform, radial logarithmic, and dipole field topologies.
• Applied electrode potential difference: V₀ = 10.00 V DC; Reference uniform electric field: E = V₀ / d = 100.00 V m⁻¹.
• Electrode geometry: Dual parallel brass bar electrodes (length L = 20.00 cm, inter-electrode gap d = 10.00 cm = 0.100 m).
• Conducting sheet substrate: High-uniformity carbon-impregnated Teledeltos paper (surface resistivity R_s = 2000 Ω/□).
• Measurement instrumentation: 4½-digit true-RMS digital voltmeter (input impedance Z_in = 10.0 MΩ, resolution 1 mV).
• Point tracking probe: Spring-loaded copper-beryllium tracking probe with hardened contact tip (radius r_tip = 0.5 mm).
• Ambient laboratory environment: Temperature T = 20.0 °C, relative humidity RH = 45% (controlled non-condensing atmosphere).

4. Theory and graph choice

The concept of the electric field, first introduced by Michael Faraday, provides an indispensable spatial framework for describing electromagnetic interactions across space. At every point in an electrostatic field, the electric field vector E is defined as the force F exerted per unit positive test charge q₀: E = F / q₀. In static conditions, Coulomb's force is strictly conservative, meaning the line integral of electric force along any closed path vanishes identically: ∮ E · dℓ = 0. Consequently, the electric field can be expressed rigorously as the negative gradient of a scalar electrostatic potential function V(r): E = −∇V = −(∂V/∂x î + ∂V/∂y ĵ + ∂V/∂z k̂). The scalar potential difference between two arbitrary spatial coordinates A and B represents the external work required to transport a unit positive charge from A to B against the electric field: V_B − V_A = −∫_A^B E · dℓ. An equipotential surface is defined as the locus of all contiguous points in space having the exact same electrostatic potential V. If a test charge is displaced along an equipotential surface by a differential displacement dℓ, no electrostatic work is performed: dW = −q₀ (E · dℓ) = −q₀ dV = 0. Because dV = 0 and |E| ≠ 0, it follows from the scalar product E · dℓ = |E| |dℓ| cos θ = 0 that cos θ = 0, which dictates θ = 90°. Hence, electric field lines are everywhere strictly perpendicular to equipotential surfaces. In a direct experimental laboratory, mapping static electric fields in a pure dielectric medium is notoriously challenging due to parasitic atmospheric ion neutralization and surface charge leakage. However, by virtue of the exact mathematical correspondence between electrostatics and steady direct current flow in a uniform conducting sheet (Ohm's law in point form: J = σ E, combined with current continuity ∇ · J = 0 in charge-free regions yielding Laplace's equation ∇²V = 0), the potential distribution in a thin conducting paper (such as carbon-impregnated Teledeltos paper of uniform surface resistivity R_s) is mathematically identical to the electrostatic field in a dielectric. When two parallel linear brass electrodes of length L separated by distance d are energized with a constant DC potential difference V₀, the electric field in the central inter-electrode region is remarkably uniform: E = V₀ / d = const. Along the symmetry axis perpendicular to the plates, the potential varies linearly with displacement x measured from the grounded electrode (V = 0 at x = 0): V(x) = (V₀ / d) x = E x. In contrast, for curved or dipole electrode geometries, field lines flare outwards and equipotential lines curve into families of orthogonal circular arcs and hyperbolas, concentrating potential gradients near points of high surface curvature. By systematically measuring the potential V at calibrated spatial positions x using a high-impedance digital voltmeter and point-contact tracking probes, students obtain an empirical dataset that maps equipotentials, constructs orthogonal flux lines, and applies ordinary least-squares linear regression (V = m x + b) to determine the uniform electric field strength E = m with rigorous metrological precision.

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. Fasten a fresh sheet of Teledeltos conducting paper smoothly onto the mapping board. Place the two parallel brass bar electrodes exactly d = 10.00 cm apart along the calibrated grid lines. Apply a thin line of colloidal silver conductive paint beneath each electrode to ensure flawless, uniform ohmic contact with the graphite substrate, and clamp firmly.
2. Connect the positive terminal of the regulated DC power supply to the right electrode (+10.00 V) and the negative terminal to the left electrode (0.00 V / Ground). Connect the common (COM) ground lead of the digital voltmeter directly to the left grounded electrode, and connect the positive voltmeter lead to the handheld tracking probe.
3. Set the voltmeter to DC millivolts/volts. Gently touch the tracking probe to the conducting paper and locate points having potentials of exactly 2.00 V, 4.00 V, 6.00 V, and 8.00 V. For each voltage level, locate at least 7 distinct (x, y) coordinates spanning across the board, and record their positions on the coordinate grid sheet.
4. Connect the points of equal potential with smooth continuous curves to produce equipotential lines. Using a drafting square, construct a family of electric field lines that start at the +10 V electrode and terminate on the 0 V electrode, ensuring every field line intersects every equipotential line at an angle of strictly 90°.
5. Position the tracking probe along the central longitudinal axis of symmetry (y = 0) where edge fringing fields are negligible. Record the local potential V for six discrete displacements x from the grounded electrode (x = 1.5, 3.0, 4.5, 6.0, 7.5, and 9.0 cm). Verify that the voltmeter reading stabilizes before logging data.
6. Replace the parallel plates with two cylindrical point electrodes (+q and −q separated by 10 cm). Trace the resulting curved equipotential ovals and observe the convergence of electric field lines near the point poles, verifying that electric field density is greatest where equipotentials are closest together.

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 = 5.7685e-30; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.9138e-14. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.1180e-15.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.9138e-14 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.

Contact resistance variations between tracking probe tip and graphite paper (fluctuations ±0.02 V depending on normal stylus contact force), slight paper manufacturing resistivity anisotropy (longitudinal vs transverse grain sheet resistance differences ~1 to 2%), finite electrode length fringing field curvature (distortion of equipotential linearity for |y| > 5 cm, minimized by sampling strictly along the central symmetry axis y = 0), voltmeter input impedance loading (negligible ~0.02% with 10 MΩ digital meter vs >5% with low-impedance meters), and coordinate grid parallax alignment tolerance (±0.5 mm in probe placement x, producing ±0.05 V gradient uncertainty) contribute experimental uncertainty. Ordinary least-squares regression of potential V against displacement x averages out random contact probe fluctuations, with the slope isolating the true uniform potential gradient dV/dx.

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

• Do not press the point tracking probe forcefully into the Teledeltos paper; excessive mechanical pressure tears or permanently indents the graphite surface, creating localized resistivity anomalies.
• Ensure colloidal silver paint is completely dry before applying voltage, as wet conductive paint creates unstable non-ohmic contact impedance along the electrode interface.
• Always collect potential gradient data along the central longitudinal axis of symmetry (y = 0) to avoid two-dimensional edge fringing distortions near the electrode extremities.
• Verify that the digital voltmeter has an input impedance of at least 10 MΩ to prevent circuit loading from depressing local potential readings.

The theoretical model assumes perfectly planar parallel electrodes of infinite length with zero contact resistance, perfectly uniform and isotropic graphite conductivity across the Teledeltos paper, and an ideal infinite-impedance voltmeter. In a physical laboratory, finite electrode length introduces edge fringing fields (causing outward bowing of equipotentials beyond ±6 cm from the central axis), paper manufacturing anisotropic grain introduces minor directional resistivity variations (~1 to 2%), and localized contact resistance at the probe tip introduces small experimental reading scatter.

12. Conclusion

The worked example illustrates how to obtain Uniform electric field strength E from measurements, graph variables and a fitted coefficient. The estimate is 100.00 V m⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.