Electrodynamics

Electric Current

What actually is electric current? We start from the most basic definition — charges moving through a cross-section — and build up to Ohm's law, power dissipation, and Kirchhoff's circuit laws.

Definition — instantaneous current

I \;=\; \frac{dQ}{dt} \qquad \bigl[\,\text{A} = \text{C/s}\,\bigr]

What is electric current?

Electric current $I$ measures how much electric charge $Q$ passes through a cross-section of a conductor per unit time $t$. The SI unit is the Ampere (A), equal to one Coulomb per second. Conventional current flows in the direction a positive charge would move — opposite to the actual electron drift.

$I$ Electric current in Ampere (A)

$Q$ Electric charge in Coulomb (C)

$t$ Time in seconds (s)

For a constant current the expression simplifies to $I = \Delta Q / \Delta t$. In a metal, current is carried by conduction electrons with an average drift velocity $v_d$ that is surprisingly small — typically fractions of a millimetre per second — even though the electric field propagates at close to the speed of light.

Ohm's Law

U \;=\; R \cdot I

Ohm's Law

For an ohmic resistor (one whose resistance $R$ does not depend on $I$), the voltage $U$ across it is proportional to the current through it. $R$ is the electrical resistance, measured in Ohm (Ω).

$U$ Voltage (potential difference) in Volt (V)

$R$ Electrical resistance in Ohm (Ω = V/A)

$G$ Conductance $G = 1/R$ in Siemens (S)

The resistance of a uniform wire depends on its material and geometry:

Resistance of a uniform conductor

R \;=\; \rho \,\frac{\ell}{A}

$\rho$ Resistivity of the material (Ω·m) — e.g. copper: $\rho \approx 1.7 \times 10^{-8}\,\Omega\text{m}$

$\ell$ Length of the conductor (m)

$A$ Cross-sectional area (m²)

Electrical power dissipation

P \;=\; U \cdot I \;=\; I^2 R \;=\; \frac{U^2}{R}

Power & Joule heating

The power $P$ delivered to (or dissipated by) a circuit element equals voltage times current. For a pure resistor, all power is converted to heat — this is called Joule heating. The three equivalent forms above follow directly from substituting Ohm's law.

Kirchhoff's Current Law (KCL) — node rule

\sum_{k} I_k \;=\; 0 \quad \text{(at any node)}

Kirchhoff's Voltage Law (KVL) — mesh rule

\sum_{k} U_k \;=\; 0 \quad \text{(around any closed loop)}

Kirchhoff's Laws

KCL is a statement of charge conservation: the algebraic sum of all currents entering a node equals zero (inflow = outflow). KVL follows from energy conservation: the sum of all voltage drops around any closed loop is zero. Together, these two laws let you solve any linear resistive network.