Electric Current & Charge Flow

Ask what the current is here and the question is incomplete until a surface is named. Current is defined by drawing a cross-section through the conductor, choosing one of the two directions through it as positive, and counting the net charge that crosses in a time interval; divide by the interval and you have the current, in coulombs per second, which is the ampere. The word net does real work. A conduction electron in copper moves at roughly 10⁶ m s⁻¹ from thermal agitation alone, so in an unpowered wire enormous numbers cross the plane each way every nanosecond, and the current is still exactly zero because the two counts match. Connect a battery and the whole population acquires a small bias one way; the current measures only that imbalance. In practice nobody draws the surface, because the current in this wire is shorthand for the current through any cross-section of it — and it is charge conservation, taken up below, that makes the word any safe.

Iₐᵥ = ΔQ/Δt averages over the whole window Δt, and it is the honest answer whenever the flow is steady. When the flow is not steady, shrink the window: I = dq/dt, the slope of accumulated charge against time. The distinction is not academic. A lightning return stroke may move 15 C in 200 μs, which averages 15 C ÷ (2.00 × 10⁻⁴ s) = 7.5 × 10⁴ A, while the peak instantaneous current is several times larger, because the pulse rises and falls rather than sitting flat. Averages hide peaks, and components are destroyed by peaks. Read a charge-time graph exactly as you read a position-time graph in mechanics: the slope of the chord between two points is the average current over that interval, and the slope of the tangent is the instantaneous current there.

Current is a signed scalar, not a vector, and its sign means nothing until a reference direction is drawn on the circuit. Conventional current points the way positive charge would flow. In a metal the movers are electrons of charge −e drifting the other way, which changes no circuit answer: transporting −1 C to the left across a plane is the same net transfer as transporting +1 C to the right. That equivalence also lets both signs contribute at once, and add. In a salt solution the Na⁺ ions drifting right and the Cl⁻ ions drifting left both carry conventional current to the right. In a beam, 1.0 × 10¹² protons per second moving right together with 1.0 × 10¹² electrons per second moving left give a current of 3.2 × 10⁻⁷ A to the right, not zero. Finally, the sign lets an answer disagree with your guess: −0.40 A on a branch means 0.40 A flows against the arrow you drew, and reversing the arrow as well would be a double negative.

Take two cross-sections A₁ and A₂ on the same unbranched wire and watch the charge in the stretch between them. Charge cannot be created or destroyed, so the rate at which that stored charge changes is the current in at A₁ minus the current out at A₂. In a steady state nothing accumulates — a wire that tried to would charge up and its own field would shut the imbalance down within nanoseconds — so the two currents are equal. That is what licenses the sentence the circuit carries 1.5 A without naming a plane, and it survives a change of thickness. Solder a 2.0 mm² lead to a 0.50 mm² filament and both carry 1.5 A; what changes is the current per unit area, from 0.75 A mm⁻² to 3.0 A mm⁻². The filament is not given more current, it is given the same current through a quarter of the room.

Where wires meet, that same statement becomes the junction rule. A junction is a node of no size that stores no charge, so the currents entering must total the currents leaving: ΣIᵢₙ = ΣIₒᵤₜ. Send 2.00 A in on one wire and take 0.75 A out on a second, and the third must carry 1.25 A out. Take 2.40 A out on the second instead, and the third carries 2.00 − 2.40 = −0.40 A out, meaning 0.40 A into the junction; the arithmetic hands you the direction. The tidier convention is to draw every branch arrow pointing into the node and write ΣI = 0, after which a negative answer simply flags an outflow. Nothing here mentions batteries, resistance or steadiness of the source: the rule holds in any network at any instant, provided no charge is piling up at the node itself.

Where steady fails, one number per branch is not enough, because the current entering a region stops equalling the current leaving it. Drive 2.0 mA into an isolated capacitor plate and none of it leaves: the plate's charge grows at exactly the rate the wire delivers, dQ/dt = I, so after 5.0 ms it holds 1.0 × 10⁻⁵ C. That is not a breach of charge conservation, it is charge conservation with the storage term kept. The same caveat covers the first nanoseconds after a switch closes, while the surface charges that steer the field inside the wires are still arranging themselves, and it covers an antenna rod, where the current must be zero at the free tip and is not zero at the base. Everywhere else in this section the circuits are steady, the storage term vanishes, and one number per branch tells the whole story.