How this instrument works
Charles-Augustin de Coulomb hung a silvered straw from a silk thread inside a glass cylinder, charged a small pith ball, and measured how far the thread twisted. Those torsion-balance memoirs, read to the Académie des Sciences from 1785, turned electricity from a parlour curiosity into arithmetic: force between two point charges rises with the product of their magnitudes and falls with the square of their separation. Henry Cavendish had reached that conclusion in the early 1770s using nested spheres, then left it unpublished until Maxwell dug his notes out a century afterwards.
One coulomb is a brutal amount of charge to keep in one place. Rub a balloon on your hair and perhaps ten nanocoulombs transfer; a whole lightning stroke moves around fifteen coulombs, and even that arrives smeared over kilometres of ionised channel. Bench work therefore lives at microcoulombs and below, where forces land in millinewtons — enough to bend a cantilever in electrostatic force microscopy, drag toner onto a photoconductor drum inside a laser printer, or steer droplets leaving an electrospray needle.
Two conditions must hold. Charges have to be point-like, or spherically symmetric and far enough apart that neither induces a lopsided distribution on its neighbour, which is precisely why two adjacent charged spheres attract harder than their centre-to-centre distance predicts. They also have to be at rest: set them moving and magnetic forces appear, retardation starts to matter, and only a full Lorentz treatment stays honest. Immersing a pair in some medium divides everything by relative permittivity, near 80 for water, so ionic solids that hold together dry come apart when dissolved.
- Put your figure into First charge and switch its unit to mC, µC or nC rather than typing exponents — a slip of a thousand here costs a million in the answer.
- Fill Second charge the same way. Signs are optional, since this sheet works from magnitudes: a proton beside an electron returns what two protons return.
- Set Separation as the centre-to-centre distance in mm, cm or m. Zero is rejected, because an inverse square has no value there.
- Read Electrostatic force in N, mN or kN, then supply the direction yourself: apart for matching signs, together for opposite ones.
Worked example — one coulomb against one coulomb
Set First charge to 1 C, Second charge to 1 C, Separation to 1 m. Arithmetic gets as bare as it ever gets: F = k × 1 × 1 ⁄ 1² = 8,987,551,787.37 N. Nothing rounds away in there. That answer is simply the Coulomb constant restated, which makes this pairing the standard sanity check on any implementation of the law.
Nine billion newtons weighs about 916,000 tonnes — roughly nine Nimitz-class carriers dangling off each ball. Hence the arrangement stays a thought experiment: long before assembly finished, dry air would break down near three million volts per metre and every bit of charge would leave in one flash. Coulomb-sized quantities exist only in motion, as one ampere flowing for one second.
Questions
Why does the result ignore whether my charges are positive or negative?
Because this sheet reports magnitude alone. Absolute values go into the expression, so +2 µC beside −3 µC and +2 µC beside +3 µC both return an identical number. Direction is not lost, merely left with you: matching signs repel along the line joining them, opposite signs attract along that same line. Any vector calculation needs that line's orientation regardless, and one numeric output field cannot carry it.
What units does the Coulomb constant k carry?
Newton metres squared per coulomb squared, N·m²/C², which is what delivers newtons when charges arrive in coulombs and separation in metres. Its value, 8.9875517873682 × 10⁹, equals c² × 10⁻⁷ in SI numbers — an exact identity until 2019, while μ₀ was fixed by definition at 4π × 10⁻⁷ H/m. Redefining the SI turned μ₀ into a measured quantity tied to the fine-structure constant, so k now carries relative uncertainty around 10⁻¹⁰. Nothing computed on this page will ever notice.
How well do we know that the exponent is exactly two?
Extraordinarily well. Writing it as 2 + δ, Cavendish's concentric-sphere null method already bounded δ under 0.02 during the 1770s; modern repetitions of that same experiment push the bound below 10⁻¹⁶. Stakes run deeper than tidiness — any genuine departure from an exact square would imply that photons carry rest mass, so these limits double as the tightest laboratory constraints on it.
Why does the same pair of charges pull far more weakly in water?
Because water molecules rotate to oppose an applied field, and that polarisation screens whatever sits inside. Swapping ε₀ for ε₀εᵣ divides everything by relative permittivity — near 80 for water at room temperature, against 1.0006 for air. Sodium and chloride ions gripping each other tightly in a dry crystal thus feel barely one per cent of that attraction once dissolved, which is a large part of why salt goes into solution at all.
How does electrostatic force compare with gravity between the same particles?
It dwarfs gravity utterly. Take two protons: electrical repulsion beats gravitational attraction by a factor near 1.24 × 10³⁶, and since both weaken as 1/r², that ratio never shifts with distance. Gravity rules planets purely because bulk matter sits almost perfectly charge-neutral, positives cancelling negatives to many decimal places, whereas mass only ever accumulates.
What mistake trips people up most often here?
Unit slips on charge. A figure quoted in microcoulombs but typed as coulombs inflates output by a million, and doing that on both fields inflates it by 10¹² — spot the error by asking whether newtons of push between two dust-sized specks sounds plausible. Second most common: measuring Separation between sphere surfaces. Two balls of 1 cm radius with a 1 cm gap sit 3 cm apart centre to centre, making real force nine times smaller than the surface figure suggests.