Current in Conductors — Electromagnetism

Electromagnetism / Current in Conductors

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Circuits — the rules

Conventional current flows from + to − outside the cell — opposite to the electron drift. Drift is glacial (mm/s); the field, and so the current, is set up almost instantly.

Kirchhoff. Junctions: ΣI = 0 (charge conserved). Loops: ΣV = 0 (energy conserved) — count a drop −IR along the current, an emf + when you enter − → +.

A real cell has internal resistance r: the terminal voltage is V = ε − Ir while delivering, and ε + Ir while being charged.

Series adds resistance R = ΣRᵢ; parallel adds conductance 1/R = Σ1/Rᵢ — the mirror image of capacitors.

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§01

Drift velocity & Ohm's law — why electrons crawl

Between collisions the field nudges each electron, giving a tiny average drift velocity on top of furious random motion: v_d = eEτ/m. Count the charge crossing a section per second and you get I = nAev_d — which, unpacked, is exactly V = IR with R = ρL/A. Dial the current and the wire's shape and watch how astonishingly slowly the electrons actually move.

CONDUCTOR · COPPER WIRE

current I{{ d1Ilab }} A

wire length L{{ d1Llab }} m

cross-section A{{ d1Alab }} mm²

v_d = InAe = {{ d1Vd }}

resistance R = ρL/A

across it V = IR

density J = I/A

to cross the wire

key: n = free-electron density (≈8.5×10²⁸ m⁻³) · A = cross-section area · e = electron charge (1.6×10⁻¹⁹ C) · I = current

Slow drift, instant light. A single electron would take {{ d1Time }} to walk the wire, yet every electron along it starts moving together the instant the field appears — so the current is set up in nanoseconds. Copper packs n ≈ 8.5×10²⁸ free electrons per m³.