Electrostatics & Coulomb's Law lab report

1. Title and aim

Electrostatics: Coulomb's Law, Electric Induction, and Charge Quantification. To investigate electrostatic charge production by friction and induction, verify Faraday's ice-pail experiment and charge conservation, measure the electrostatic force between charged spheres as a function of distance using a Coulomb torsion balance, confirm the inverse-square law, and determine Coulomb's constant and the permittivity of free space using 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 Coulomb constant k_e of 8.9876e+09 N m² C⁻², 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

• Coulomb Torsion Balance: Precision electrostatic torsion balance housed in a draft-free cylindrical glass chamber with circular degree vernier scale, grounded conductive floor, and low-torsion phosphor bronze suspension fiber.
• Matched Conducting Spheres: Pair of precision-turned, lightweight gold-plated conducting spheres (radius a = 5.0 mm), one suspended on an insulating dielectric balance arm and the other mounted on a graduated sliding carrier.
• Regulated High-Voltage DC Source: Continuously variable low-current power supply (0 to 6.0 kV DC, 50 μA current-limited) for depositing calibrated electrostatic charges on the conducting spheres.
• Precision Proof Plane and Faraday Ice Pail: Polished metallic proof plane disc with amber insulating handle and an insulated nickel-plated Faraday ice pail connected to a digital electrometer for quantitative charge sampling.
• Gold-Leaf Electroscope: Enclosed glass-windowed electroscope with grounded metal shielding case, calibrated leaf divergence protractor, and high-insulation sulfur bushing.
• Desiccated Environmental Chamber: Hermetically sealed chamber enclosure with rechargeable silica gel desiccant column to maintain relative humidity below 25%, preventing atmospheric ion leakage.
• Ambient room temperature: T = 20.0 °C; Relative humidity: RH < 25% (Controlled desiccated chamber).
• Reference Coulomb electrostatic constant: k_e = 8.98755 × 10⁹ N m² C⁻² (Permittivity ε₀ = 8.85419 × 10⁻¹² F m⁻¹).
• Conducting spheres: Precision gold-plated brass (radius a = 5.00 mm, mass m_sphere = 0.445 g).
• Sphere electrostatic charge: q₁ = q₂ = 20.00 nC (Regulated contact charging at 3.00 kV DC).
• Torsion balance suspension: Phosphor bronze fiber (torsional constant κ = 1.450 × 10⁻⁸ N m rad⁻¹, lever arm b = 10.0 cm).
• Chamber boundary: Grounded copper wire Faraday mesh shielding (external stray field attenuation > 99.8%).

4. Theory and graph choice

Electrostatics constitutes the cornerstone of classical electromagnetism, describing the fundamental forces, fields, and potentials produced by stationary electric charges. First quantified systematically by Charles-Augustin de Coulomb in 1785 using his delicate torsion balance, the electrostatic force between two point charges q₁ and q₂ separated by distance r in vacuum obeys the fundamental inverse-square relationship: F = (1 / 4πε₀) (q₁ q₂ / r²) = k_e (q₁ q₂ / r²), where k_e = 1 / (4πε₀) ≈ 8.98755 × 10⁹ N m² C⁻² is Coulomb's constant and ε₀ ≈ 8.85419 × 10⁻¹² F m⁻¹ is the permittivity of free space. Like Newton's law of universal gravitation, Coulomb's law exhibits an inverse-square dependence on distance governed by the geometry of three-dimensional Euclidean space, as formalized by Gauss's flux theorem: ∮ E · dA = Q_enc / ε₀. However, unlike gravity, electrostatic forces are bipolar: charges of like sign repel each other, while charges of opposite sign attract. When two small, identical conducting spheres are charged to equal potentials and placed within a draft-free chamber, the repulsive electrostatic force F deflects a suspended balance beam. By rotating a calibrated torsion head to restore the beam to its initial null position, the applied restoring torque τ = κ θ directly balances the electrostatic torque F · b, yielding an absolute force measurement in the micronewton (μN) regime. Simultaneously, the electrostatic behavior of conductors is governed by Gauss's law: in electrostatic equilibrium, the electric field within the bulk of a conductor is identically zero (E = 0), and all excess charge resides exclusively on the outer surface, as demonstrated by Michael Faraday's historic 1843 ice-pail experiment. Furthermore, the surface charge density σ concentrates at regions of high surface curvature (σ ∝ 1/R), generating intense localized electric fields capable of ionizing ambient air into a luminous corona discharge. By recording the electrostatic force F across various separation distances r, ordinary least-squares regression of F against the geometric variable 1/r² yields a strictly linear plot with slope m = k_e q₁ q₂. From this slope and the known sphere charges, students determine Coulomb's constant k_e and the permittivity of free space ε₀ with remarkable precision, bridging classical electrostatic induction, Faraday shielding, and continuum field theory.

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 N m².

5. Method

1. Inspect the torsion balance glass chamber and verify that the internal silica gel desiccant is active. Clean the insulating glass surfaces and quartz suspension rod with anhydrous ethanol, and verify that the relative humidity inside the chamber reads below 25%.
2. Adjust the leveling screws of the balance until the suspended dielectric beam swings freely without touching the chamber walls. Rotate the top torsion head dial until the index hair on the suspended sphere A aligns exactly with the chamber zero mark when both spheres are uncharged.
3. Connect the charging probe to the 3.0 kV regulated DC supply. Touch the probe simultaneously to the suspended sphere A and sliding sphere B while they are in contact, depositing an equal charge of q1 = q2 = 20.0 nC on both spheres. Verify the charge magnitude with the Faraday ice pail and electrometer.
4. Advance the sliding carrier to position sphere B at a center-to-center separation distance of r = 4.0 cm from sphere A. As electrostatic repulsion deflects sphere A, rotate the torsion head dial until sphere A is restored precisely to its original zero mark. Record the restoring angle theta and calculate the electrostatic force F in micro-newtons.
5. Repeat the torsion null measurement for six discrete separation distances (r = 4.0, 5.0, 6.0, 7.0, 8.0, and 9.0 cm). For each station, record the separation distance r, the inverse square 1/r^2, and the measured repulsive force F.
6. Using the proof plane, sample the surface charge density from the sphere and transfer it into the Faraday ice pail to confirm zero interior charge and verify that total charge has not decayed significantly during the run.

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 = 0.0001%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.3315e-17; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 4.3352e-12. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.4903e-09.

Propagating only the relevant fit coefficient gives a fit-only standard error of 10837.96131 N m² C⁻². This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Torsion fiber restoring constant calibration uncertainty (drifting shear stiffness ±0.5%), vernier separation scale parallax and non-coaxial sphere alignment (±0.2 mm in r, creating magnified ±0.8% error in 1/r²), parasitic atmospheric charge leakage via airborne ions and humidity (drift rate ~0.1 nC/min, minimized by desiccated silica chamber), electrostatic image charge induction between the conducting spheres and the grounded metal chamber shield, finite sphere radius correction (deviations from ideal point charges: Maxwell image series correction factor 1 − 4(a/r)³ ≈ 0.3% at closest approach r = 4 cm for a = 5 mm radius), and air convection currents inside the chamber contribute experimental uncertainty. Ordinary least-squares regression of F against 1/r² averages out random angular scale reading errors, with the slope isolating the true Coulomb product k_e q²

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

• Ground all outer metallic shielding cases and observer wrist straps before operating the balance to prevent stray electrostatic induction from clothing or body capacitance.
• Never breathe directly toward the open balance chamber, as moisture in exhaled breath instantly compromises the high-insulation quartz surfaces.
• Ensure the separation distance r is always at least 4 times the sphere radius (r >= 4a) to keep finite-size multipole induction corrections below 0.7%.
• Approach the null reading position slowly to avoid setting the suspended torsion beam into persistent rotational swinging.

The analytical model treats the conducting spheres as mathematical point charges in an infinite, unbounded vacuum, neglecting electrostatic image charges induced in the chamber walls, minor finite-size multipole surface charge redistribution (Lord Kelvin image correction ~0.3% at r = 4 cm), and residual atmospheric ion leakage (~0.1 nC/min). In a physical laboratory, slight fiber mechanical hysteresis (zero drift after large twists), minute convective air currents inside the glass chamber, and environmental humidity fluctuations introduce small experimental scatter.

12. Conclusion

The worked example illustrates how to obtain Coulomb constant k_e from measurements, graph variables and a fitted coefficient. The estimate is 8.9876e+09 N m² C⁻². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.