Coulomb Law Simulator — Electric Force Between Point Charges
Real-time Coulomb force F = k·q_1·q_2 / (ε_r·r²) between two point charges. Read off the electric field E, potential energy U, and attractive vs repulsive direction with field lines and a |F|-r log-log curve.
Parameters
Charge 1 q_1
μC
Charge 2 q_2
μC
Distance r
cm
Relative permittivity ε_r
—
Coulomb constant k = 1/(4πε_0) ≈ 8.988×10⁹ N·m²/C². Charge unit μC = 10⁻⁶ C, distance cm = 10⁻² m. Setting ε_r = 80 (water) shrinks the force by a factor of 80.
Results
—
Coulomb force |F|
—
Field E from q_1
—
Potential energy U
—
Force direction
Sign: —
Point charges & field lines
Two point charges separated by r. Red = positive, blue = negative. Field lines indicate repulsion for like signs and attraction for opposite signs. Arrows show the Coulomb force vector F.
|F| vs r (log-log)
x = distance r (cm), y = |F| (N), both on a log scale. F ∝ 1/r² gives a straight line of slope -2. The yellow dot marks the current (r, |F|).
Theory & Key Formulas
Coulomb's law gives the electrostatic force between two point charges as inversely proportional to the square of the distance.
Coulomb force ($k = 1/(4\pi\varepsilon_0) \approx 8.988\times10^{9}$ N·m²/C²):
Field of $q_1$ at distance $r$, and the two-charge potential energy:
$$E = \frac{k\,|q_1|}{\varepsilon_r\,r^{2}},\quad U = \frac{k\,q_1\,q_2}{\varepsilon_r\,r}$$
$F > 0$ (like signs) is repulsive, $F < 0$ (opposite signs) is attractive. $U$ uses $r \to \infty$ as the zero reference.
What is the Coulomb law simulator?
🙋
Coulomb's law — F = kq₁q₂/r², bigger charges push harder, falls off as 1/r². At the defaults (q₁ = +5 μC, q₂ = +3 μC, r = 10 cm) the simulator says 13.48 N. That feels surprisingly large — am I missing something?
🎓
No, that's the surprise of electrostatics. μC sounds tiny, but with k = 9×10⁹ N·m²/C², even 5 μC at 10 cm gives 13 N — about the weight of a 1.3 kg mass. That's roughly the pull of a small refrigerator magnet. The reason your hair stands up after combing or paper jumps to a charged plastic ruler is exactly this huge prefactor at work.
🙋
When I press the sweep button, |F| vs r is a straight line on log-log axes. What does slope -2 actually mean?
🎓
log F = log k|q₁q₂| − 2 log r. Each decade of r (10×) gives two decades of force (1/100). Move from 10 cm to 1 m and 13.48 N drops to 0.135 N. Inverse-square laws appear everywhere a quantity radiates from a point — Newtonian gravity and the Gauss law for light intensity follow exactly the same shape.
🙋
If I flip one charge negative the arrows pull together and U becomes negative. Is that the "binding energy" people talk about?
🎓
Exactly. Negative U means the system sits below the U(∞) = 0 reference: a bound state. To pull the charges apart you must supply |U| of energy, the binding energy. Hydrogen's electron is bound to its proton this way, and Na⁺/Cl⁻ in salt sits in a Coulomb well too. Set ε_r = 80 (water) and U drops by 1/80 — that is the physics behind "salt dissolves in water".
🙋
The field shows 4.49 MV/m — that's a huge number. What does it represent?
🎓
E = kq₁/r² is the force per unit positive charge if you placed a test charge 10 cm from q₁. 4.49 MV/m already exceeds the air-breakdown field (~3 MV/m). Of course a true point charge gives E → ∞ as r → 0 — that singularity is why high-voltage hardware always rounds electrode tips to spread the field and avoid corona discharge.
FAQ
Both are inverse-square laws, but three things set them apart. (1) Sign: charge has both signs and Coulomb forces can attract or repel; mass is always positive and gravity is always attractive. (2) Strength: between an electron and a proton the Coulomb force is roughly 10³⁹ times stronger than gravity. (3) Screening: dielectrics and conductors can screen electric forces, but gravity cannot be screened (in general relativity it isn't even a "force" — it's curvature of spacetime). Atoms, molecules and solids are held together by Coulomb forces; gravity dominates only at planetary, stellar and galactic scales.
Because k = 8.988×10⁹ N·m²/C² is enormous. Two 1 C charges 1 m apart would push each other with 9 GN — comparable to the weight of millions of jumbo jets. In practice 1 C is a colossal amount of static charge, so electrostatic phenomena live in µC (10⁻⁶ C) and nC (10⁻⁹ C). Conduction currents move plenty of coulombs per second, but they are charge-balanced (electrons move through a neutral lattice), so no macroscopic Coulomb force results.
In a NaCl crystal, Na⁺ and Cl⁻ are held together by a Coulomb attraction with binding energies of several eV — much larger than the thermal energy at room temperature (~0.025 eV), so the crystal stays solid in vacuum. In water, ε_r ≈ 80 reduces every Coulomb force by a factor of 80, dropping the binding energies into the same order of magnitude as kT. Polar water molecules then surround Na⁺ and Cl⁻ ("hydration"), making the dissolved state energetically favourable. Set ε_r = 80 in this tool to see the force and potential collapse to 1/80 of the vacuum values.
Common solvers include Ansys Maxwell (motors, electrical machines), COMSOL Multiphysics (multiphysics), JMAG (motor design), and CST Studio Suite (high-frequency, EMC). They solve Laplace/Poisson's equation ∇²φ = -ρ/ε on a finite-element mesh to obtain field, potential, force, capacitance and breakdown risk in 3D. Coulomb's law is the elementary point-charge superposition that underlies everything: boundary-element methods (BEM) with free-space Green's functions, MEMS electrostatic actuators, ESD analysis, high-voltage insulation design, and semiconductor process simulation.
Real-world applications
Electrostatic discharge (ESD) protection: Semiconductor chips can be destroyed by tens of volts of static, so production and assembly lines use ESD mats, wrist straps and ionisers to drain charge. Coulomb's law combined with the typical body charge (a few µC) and ground capacitance (100–1000 pF) gives a discharge energy E = Q²/(2C) that drives the design margin of TVS diodes and ESD suppressors. Try q = 1 µC, r = 1 mm (fingertip to IC pin) here and you will see a ~9 N local force and an extreme local field appear instantaneously.
MEMS electrostatic actuators: Micromirrors (DLP projectors, automotive LiDAR), accelerometers and RF MEMS switches use Coulomb attraction F = ε₀A·V²/(2d²) between parallel plates as the drive mechanism. Sub-micron gaps mean that a few volts produce µN–mN forces with essentially zero static current. Mapping q to the stored electrode charge and r to the gap turns this tool into a quick sensitivity check (V → force).
High-voltage insulation design: Transmission lines, transformers, surge arresters and gas-insulated switchgear avoid field concentration with rounded electrodes and selected dielectrics (air, SF₆, oil, epoxy). Air breakdown is at 3 MV/m, SF₆ at 9 MV/m, transformer oil around 15 MV/m. Once the Coulomb field exceeds a third of the breakdown value, corona discharge starts and ages the insulation. Industrial workflows use Ansys Maxwell or COMSOL to compute 3D field maps and reshape electrodes to minimise the local maximum.
Atomic and molecular stability: Hydrogen's electron is bound to its proton at the Bohr radius (5.29×10⁻¹¹ m) by a Coulomb attraction of about 8.2×10⁻⁸ N, with a 13.6 eV binding energy (the Rydberg). Chemical bonds, van der Waals forces, hydrogen bonds and ionic bonds all originate from Coulomb interactions. Biomolecular structures (proteins, DNA) are shaped by Coulomb interactions in water (ε_r ≈ 80) balanced against thermal motion — the basic physics that DFT and molecular-dynamics codes integrate at high speed.
Common misconceptions and caveats
The most common misconception is to assume "F → ∞ as r → 0 makes the law unphysical". Mathematically it is true, but it only signals that the point-charge idealisation breaks down. Real electrons (charge radius ~10⁻¹⁵ m) and protons live in a regime governed by quantum electrodynamics and the strong force. In this simulator, dropping r from 10 cm to 1 cm raises |F| by 100× — perfectly correct in the classical limit. CAE practice handles the singular field at electrode corners with very fine meshes and integral (virtual-displacement) force formulations.
A second pitfall is to treat ε_r as just a scalar reduction factor. Physically, the medium polarises microscopically: applied field aligns dipoles which in turn screen the field — a self-consistent process. ε_r is roughly constant at low frequency (water at 80) but drops sharply at GHz–THz frequencies because dipoles cannot follow (the principle of microwave heating). In ferroelectrics like BaTiO₃, ε_r depends on temperature and field history and the linear approximation fails. CAE codes therefore include Debye/Cole–Cole models for frequency-dependent permittivity.
Finally, do not imagine that field lines may cross or simply vanish. The electric field is a single-valued vector, so two field lines crossing would imply two field directions at one point — a contradiction. By Gauss's law, lines start on positive charges and end on negative charges (or at infinity); they cannot terminate in empty space. The figure here shows the textbook diverge/converge patterns of like and opposite charges, but real-world configurations (three or more charges, conducting boundaries) require numerical FEM or BEM solutions — Coulomb's bare formula is no longer enough.