Theory — Coulomb's Law
The Electrostatic Force
Two electric charges exert a force on each other along the line joining them. Like charges (both positive or both negative) repel; opposite charges attract. The size of the force grows with the product of the charges and shrinks rapidly as the charges move apart.
F = electrostatic force (N)
q₁, q₂ = charges (C), r = separation (m)
k = 8.99 × 10⁹ N·m²/C²
The Inverse Square Law
The most striking feature of Coulomb's Law is the r² in the denominator. Because the force depends on 1/r², the way it changes with distance is dramatic: if you double the separation, the force drops to one quarter; triple it and the force drops to one ninth. This same inverse-square pattern appears in gravity and in the brightness of light.
r → 3r gives F → F/9
r → r/2 gives F → 4F
Linearising the Data
A graph of force against distance is a curve, which is hard to read. But if you plot force against 1/r² instead, the inverse-square relationship becomes a straight line through the origin. The slope of that line equals k·|q₁·q₂| — a neat way to test the law and even recover the constant k.
slope = k · |q₁ · q₂|
Electric Potential of a Point Charge
A charge also sets up an electric potential in the space around it. The potential V at a distance r is the electric potential energy per unit charge, measured in volts. For a single point charge it falls off as 1/r, one power gentler than the force, which falls off as 1/r².
V = electric potential (V), Q = charge (C), r = distance (m), k = 8.99×10⁹ N·m²/C²
A graph of potential against distance is a falling curve, but a graph of potential against 1/r straightens into a line whose slope equals k·Q. This is the same straight-line trick used for the force, except the force uses 1/r² while the potential uses 1/r.
Percent Error
To judge how well the measured (simulated) force agrees with the value predicted by Coulomb's Law, compute the percent error.
Bigger charges
Force grows with the product q₁·q₂ — double either charge and the force doubles.
Greater distance
Force falls as 1/r². Doubling r cuts the force to a quarter.
Like vs opposite
Like charges repel, opposite charges attract; the magnitude follows the same formula.
| Quantity | Relationship | Notes |
|---|---|---|
| Electrostatic force | F = k|q₁q₂|/r² | k = 8.99×10⁹ N·m²/C² |
| Distance dependence | F ∝ 1/r² | inverse square law |
| Linear plot | F vs 1/r² | slope = k|q₁q₂| |
| Electric potential | V = kQ/r, V ∝ 1/r | plot V vs 1/r, slope = kQ |
| Percent error | |(T−S)/T|×100 | compares sim to theory |
Apparatus
The equipment a real Coulomb's-law experiment uses. In the simulation these are modelled for you, but the readings correspond to what each instrument would measure.
Instructions — Running the Virtual Experiment
The Force tab lets you set the charges and move q₂ to read the force at each distance; the Inverse Square tab records your readings and plots force against 1/r²; the Electric Potential tab measures the potential of a single charge and plots it against 1/r. Record every reading, calculate the theoretical value, and compare. Include a labelled graph for each plot.
Simulation — Electrostatic Force and Electric Potential
Charges & distance
Record a distance (q₁=+4 μC, q₂=+8 μC)
| r (m) | 1/r² (m⁻²) | Force (N) |
|---|---|---|
| Click a distance to record it. | ||
Single charge & probe
Place probe at distance r (records the reading)
| r (m) | 1/r (m⁻¹) | Potential V (V) |
|---|---|---|
| Click a distance to record it. | ||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission, and include a clearly labelled graph of force against 1/r².
Electrostatic Force and the Inverse Square Law
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To measure the electrostatic force between two positive charges as a function of their separation, to verify the inverse square law by comparing measured forces with values calculated from Coulomb's Law, to compute the percent error at each distance, and to confirm that a graph of force against 1/r² is a straight line through the origin.
Theory
Coulomb's Law gives the force between two point charges as F = k|q₁q₂|/r², with k = 8.99×10⁹ N·m²/C². Because F depends on 1/r², doubling the separation reduces the force to one quarter. Plotting F against 1/r² gives a straight line through the origin whose slope is k|q₁q₂|.
% Error = |(T − S)/T| × 100
Calculations — Sample: r = 0.02 m
Theoretical force: F = (8.99×10⁹)(4×10⁻⁶)(8×10⁻⁶)/(0.02)² = (8.99×10⁹)(3.2×10⁻¹¹)/(4×10⁻⁴) = 719 N
Percent error (if simulated = 719 N): |(719 − 719)/719| × 100 = 0%
Results Table (q₁ = +4 μC, q₂ = +8 μC)
| Trial | r (m) | 1/r² (m⁻²) | Simulated F (N) | Theoretical F (N) | % Error |
|---|---|---|---|---|---|
| 1 | 0.02 | 2500 | 719 | 719 | 0.0 |
| 2 | 0.04 | 625 | 180 | 180 | 0.0 |
| 3 | 0.06 | 278 | 79.9 | 79.9 | 0.0 |
| 4 | 0.08 | 156 | 45.0 | 45.0 | 0.0 |
| 5 | 0.10 | 100 | 28.8 | 28.8 | 0.0 |
A plot of force against 1/r² was a straight line through the origin with slope ≈ 0.288 N·m² = k|q₁q₂|.
Discussion
The measured forces matched the values from Coulomb's Law, and the graph of force against 1/r² was a straight line through the origin, confirming that the force follows the inverse square law. The slope of that line, about 0.288 N·m², equalled k|q₁q₂| = (8.99×10⁹)(4×10⁻⁶)(8×10⁻⁶), providing an independent check of the law. Doubling the distance from 0.02 m to 0.04 m reduced the force from 719 N to 180 N — a factor of about one quarter — exactly as the inverse square law predicts.
The simulation is very precise, so the main limiting factors on accuracy are small ones: aligning the charge exactly on the ruler mark, and rounding in the simulation's force display. These can introduce a percent error of a fraction of a percent even when the physics is exact.
Conclusion
The electrostatic force obeyed Coulomb's Law and the inverse square law: force was proportional to 1/r², the F-vs-1/r² graph was linear through the origin with slope k|q₁q₂|, and doubling the distance quartered the force. Percent errors were negligible, limited only by ruler alignment and display rounding.
Practice Questions
Show all work and include units. Use k = 8.99×10⁹ N·m²/C².