Charge, conservation and quantisation

Charge is not just an amount; it is an amount with an algebraic sign, and that is the whole reason ordinary matter is quiet. Rub glass with silk and wax with fur and the two rubbed objects attract; rub two pieces of glass the same way and they repel — the eighteenth-century evidence that there are exactly two kinds, labelled positive and negative by Franklin's convention. The convention is arbitrary and, as it turned out, inconvenient: the particle that actually moves in a metal, the electron, came out negative. What matters is that charges combine as signed numbers, so a net charge is a sum with cancellation allowed: q = q₁ + q₂ + q₃ + … . A speck of dust holding 10¹⁸ protons and 10¹⁸ electrons has q = 0 and behaves, from a distance, exactly as if it held nothing. Build the habit now of writing the sign into the symbol rather than carrying it in words; later, when forces are superposed, the sign does the work of a direction.

One coulomb is (1 C)/e = 6.24 × 10¹⁸ elementary charges, which makes it a wildly oversized unit for static electricity. A rubbed rod carries a few nanocoulombs, and 1 nC is 6.24 × 10⁹ electrons — a huge count, but nothing beside the electrons already present. Take 3.0 g of copper: 0.047 mol, 2.8 × 10²² atoms, 29 electrons each, so 8.3 × 10²³ electrons carrying 1.3 × 10⁵ C, matched by as much positive charge in the nuclei. A 5 nC imbalance on that copper is a fractional miscount of 5 × 10⁻⁹ ÷ (1.3 × 10⁵) ≈ 4 × 10⁻¹⁴. Two lessons follow. Net charge is hard to build up, which is why electrostatic charges live in the nanocoulomb range. And moving charge is easy: a 1 A current pushes a whole coulomb along a wire every second, because the wire stays neutral while the electrons drift through it.

Nothing in the idea of charge requires it to come in lumps; experiment does. In an oil-drop cell the charge on each droplet is measured one drop at a time, and the values do not spread smoothly. They cluster on a ladder, with the differences between drops coming out as whole multiples of one common step. The practical move is to work with those differences, because a common calibration offset cancels in a difference and any true quantum has to divide every gap. Find the largest number that divides all the gaps, then confirm it by dividing it into the raw charges and checking that every quotient lands on a whole number. Two cautions. With only a handful of drops the true quantum may be half the step you found, so you need enough gaps to be sure. And since 20 May 2019 the experiment no longer measures e at all: e is defined as exactly 1.602176634 × 10⁻¹⁹ C, so the oil drop now demonstrates that charge is quantised, with the size of the quantum fixed by definition.

Charge conservation says the algebraic sum of charge inside a closed boundary never changes. Used as a tool it is one equation: write the signed sum before, write the signed sum after, set them equal, and solve for whatever is unknown. Ionising a sodium atom gives 0 → (+e) + (−e). Beta decay of a free neutron gives 0 → (+e) + (−e) + 0, which is how you know the antineutrino must be neutral. Pair production turns one photon into an electron and a positron: 0 → (−e) + (+e). Rubbing a balloon on a wool jumper is the same identity at a different scale: 0 → (−12.0 nC) + (+12.0 nC). The word closed is doing real work here. Touch the balloon with a damp hand and charge leaks across the boundary you drew, so the sum inside is free to change — the conservation law has not failed, your boundary has.

Conservation is stronger than a global bookkeeping rule: charge does not vanish in one place and reappear in another, it flows. The charge of a region therefore changes only by what passes through its surface, and that flow is a current: Δq = ∫ I dt, which becomes Δq = I Δt when I is steady. This is what turns a conservation statement into a number. A charged sphere leaking 2.5 μA to the air for 4.0 s loses Δq = 2.5 × 10⁻⁶ A × 4.0 s = 1.0 × 10⁻⁵ C, or 6.2 × 10¹³ electrons. Read the same equation the other way and a transfer becomes a current: a spark carrying 3.0 nC across a gap in 12 ns averages 3.0 × 10⁻⁹ C ÷ (12 × 10⁻⁹ s) = 0.25 A. The sign convention comes with it: a positive current into a region raises its charge and a positive current out lowers it, while electrons drifting in lower it, because they carry negative charge.

Quarks break the staircase one level down: an up quark carries +2e/3 and a down quark −e/3, so a proton (uud) is 2/3 + 2/3 − 1/3 = +1 in units of e, and a neutron (udd) is 2/3 − 1/3 − 1/3 = 0. Confinement keeps them bound inside hadrons, and searches for free fractional charge in bulk matter have found none down to fewer than about one per 10²¹ nucleons, so every charge you can isolate and measure is still N e. Two further facts are worth carrying. Charge is invariant: unlike energy or mass it does not depend on how fast its carrier moves, which is why a hydrogen atom is neutral to better than 10⁻²¹ e even though its electron moves at about 1% of the speed of light. And conservation has survived every test aimed at it — an electron decaying into a photon and a neutrino would destroy charge, and its mean life is known to exceed 6.6 × 10²⁸ years.