Electric Charge (Q = It)

Current is not a thing that flows. It is a rate — the amount of charge passing a chosen cross-section of the conductor each second — and one ampere means one coulomb per second. Once that is clear, $Q = It$ needs no proof, because it is the definition read backwards: if charge crosses at a steady rate, the total that crossed is the rate multiplied by how long it kept up. The only condition the equation imposes is the word steady. A current that varies has to be integrated, $Q = \int I\,dt$, and this page is the special case where the integral collapses to a rectangle.

A 2 A charger running for one hour moves $2 \times 3600 = 7200\ \text{C}$. Battery ratings are the same arithmetic wearing different units: a phone cell marked 3000 mAh holds 3 Ah, and $3 \times 3600 = 10\,800\ \text{C}$ of deliverable charge. Divide by the elementary charge, $1.602 \times 10^{-19}$ C, and that is about $6.7 \times 10^{22}$ electrons — a number that only sounds absurd until you remember a gram of copper contains ten times as many free ones already sitting in the metal, drifting at well under a millimetre per second.

Since the 2019 redefinition of the SI, this relation is closer to the foundation than it used to be. The ampere is now fixed by declaring the elementary charge to be exactly $1.602176634 \times 10^{-19}$ C, which makes the coulomb a count of charges and the ampere a count per second. Michael Faraday got there experimentally in the 1830s: his laws of electrolysis measure the charge needed to plate out a mole of a substance, and that constant — 96 485 C per mole — is nothing but $Q = It$ run on a plating tank. Electroplating, anodising and battery capacity testing all still bill in ampere-hours for exactly this reason.

Two errors are worth naming, and one convention deserves an apology. The first error is treating milliamp-hours as energy. They are charge; a 3000 mAh cell at 3.7 V holds $3 \times 3.7 = 11.1\ \text{Wh}$, and the same 3000 mAh at 1.2 V holds a third of that, so comparing two batteries by mAh alone tells you very little. The second is applying the equation to a current that is not constant — a motor's inrush, a switching supply's chopped input, or anything on AC, where over a full cycle the net charge transferred is zero even though the current is real all along. As for the convention: current is drawn flowing from plus to minus, while in a metal the electrons actually travel the other way. Benjamin Franklin guessed the sign in the 1750s, a century before anyone knew a charge carrier existed, and he guessed wrong. Nothing in the physics breaks — a deficit of negatives moving left is indistinguishable from positives moving right — but it is a historical accident, not a discovery, and it is worth knowing that it is one.