Article — Coulomb's Law Calculator
Coulomb's Law Calculator
Coulomb's law gives the electrostatic force between two point charges: F = k × |q₁q₂| / r². With k = 8.988 × 10⁹ N·m²/C², a pair of 1 μC charges 10 cm apart push (or pull) each other with about 0.9 N — enough to feel through your fingertips.
What Coulomb's law states
Two charged particles exert an electrostatic force on each other along the line joining them. The force is proportional to the product of the charges and inversely proportional to the square of the distance. This is what Charles-Augustin de Coulomb showed experimentally in 1785 using a torsion balance — twin pith balls on a rotating beam, deflected by a charged sphere held nearby.
Three features of the law repay close attention. The strength depends only on charges and distance, not on whether the charges are moving (that needs magnetism). The force lies along the line connecting the charges. And the third charge in a system feels the vector sum of the forces from every other charge — Coulomb's law is the building block for every electrostatics problem.
Coulomb's torsion balance was sensitive enough to measure forces around 10⁻⁷ N. The same trick — long quartz fiber, mirror, optical readout — let Henry Cavendish measure the gravitational constant in 1798 and Hans Geiger detect alpha particles in 1909.
The Coulomb's law formula
The magnitude is F = k × |q₁q₂| / r². Both charges in coulombs, distance in meters, force in newtons. The Coulomb constant k = 1 / (4πε₀) = 8.9875 × 10⁹ N·m²/C², where ε₀ is the permittivity of free space.
For practical work, three derived quantities matter: the electric field E = k|Q|/r² (force per unit test charge), the potential V = kQ/r (energy per unit test charge, in volts), and the electrostatic potential energy U = kq₁q₂/r between two charges. All three follow from the same Coulomb constant.
k 8.988 × 10⁹ N·m²/C²ε₀ 8.854 × 10⁻¹² F/me (elementary) 1.602 × 10⁻¹⁹ C1 μC 10⁻⁶ C1 nC 10⁻⁹ C1 C 6.24 × 10¹⁸ electronsAttractive versus repulsive Coulomb force
The magnitude formula uses absolute values, but direction depends on the signs of the charges. Like signs (both positive, or both negative) repel. Opposite signs attract. A proton and an electron pull toward each other; two electrons push apart; two protons push apart. This sign rule is what makes ordinary matter stable: atoms bind because the proton (+) attracts the surrounding electrons (−), but the electrons (−) repel each other and settle into shells.
Coulomb force in dielectric media
Coulomb's law in vacuum is the strongest version. Embedding the charges in a dielectric (insulating) medium reduces the force by a factor of ε_r, the medium's relative permittivity. Air is essentially vacuum (ε_r = 1.00059). Glass sits around 6, paper at 3.7, water at 80. So two charges in water feel only 1/80 of the force they would in vacuum.
This is no minor effect. Salt dissolves in water because water's high ε_r weakens the Coulomb attraction holding Na⁺ and Cl⁻ together — strong enough in vacuum, weak enough in water to let thermal motion break them apart. Oils with low ε_r (around 2) preserve electrostatic forces almost completely, which is why insulating oils are used in transformers.
Coulomb force versus gravity
The Coulomb force dwarfs gravity at small scales. Take two electrons, 1 m apart. Coulomb repulsion is 2.3 × 10⁻²⁸ N. Gravity between them is 5.5 × 10⁻⁷¹ N. The ratio: 4.2 × 10⁴² — Coulomb wins by 42 orders of magnitude. Yet at planetary scale gravity dominates, because bulk matter is electrically neutral. Positive and negative charges cancel almost perfectly, leaving only mass to interact.
When working with atomic problems, the natural unit of force is e²/(4πε₀a₀²) ≈ 8.24 × 10⁻⁸ N, where a₀ is the Bohr radius. Express your answer as a multiple of this and the numbers stay manageable.
Coulomb's law applications
Three application areas show the law in industrial use. Electrostatic precipitators in power-plant smokestacks charge particulate matter and collect it on grounded plates — Coulomb attraction between charged dust and the plate is the whole mechanism. Laser printers and copiers transfer toner from a charged drum to paper using the same idea; xerography is electrostatics packaged for the office.
Particle accelerators are perhaps the cleanest application. The proton-proton repulsion in a beam (Coulomb's law again) sets a limit on beam intensity, while electric fields accelerate the protons to relativistic speeds. Every design decision in a Large Hadron Collider beam line ultimately reduces to managing Coulomb forces.
At molecular scale, Coulomb's law explains why DNA strands separate in gel electrophoresis (negative phosphate backbone pulls toward the positive anode), why proteins fold into specific shapes (charge interactions on the residues), and why ionic crystals like table salt have melting points in the hundreds of degrees Celsius (strong ion-ion attractions to break).
Common Coulomb's law mistakes
Four pitfalls dominate practical work.
r in Coulomb's law is the distance between the centers of the charges, not between their surfaces. For point charges these are the same. For larger objects, treat them as point charges only if they're far enough apart that the size doesn't matter; otherwise integrate.
Second mistake: dropping the sign convention. The magnitude formula uses |q₁q₂|, but if you forget which sign convention you started with, the attractive-versus-repulsive verdict goes wrong.
Third, ignoring the dielectric medium. A force calculation in vacuum and one in water differ by a factor of 80. Always state the medium. Fourth, using the formula at distances below the size of the charged object — for two charged balls touching, the law doesn't apply directly to their centers; the charge redistribution has to be solved first.