Conduction current and Ohm's law

In a conducting material, an applied Electric field $\mathbf{E}$ drives free charges into motion. The microscopic relation is the point form of Ohm’s law:

$\mathbf{J}=\sigma \mathbf{E},$

where $\mathbf{J}$ is the current density (A/m²) and $\sigma$ is the conductivity of the material (S/m, where S = siemens = 1/Ω).

Drift velocity

Inside a conductor, charge carriers (electrons in metals, electrons and holes in semiconductors) experience the force $\mathbf{F}=q\mathbf{E}$, but collisions with the lattice prevent indefinite acceleration. The result is a steady drift velocity ${\mathbf{u}}_e$ proportional to $\mathbf{E}$:

${\mathbf{u}}_e=-{\mu }_e\mathbf{E}\text{(electrons)},$

where ${\mu }_e$ is the electron mobility (m²/(V·s)). The negative sign reflects electron negative charge: applied $+\mathbf{E}$ drives electrons in $-\hat{\mathbf{E}}$.

Combining with $\mathbf{J}={\rho }_v\mathbf{u}$ (with ${\rho }_v$ the volume charge density of the carriers):

$\mathbf{J}=-{\rho }_v{\mu }_e\mathbf{E}=\sigma \mathbf{E},$

with $\sigma =-{\rho }_v{\mu }_e$. For electrons ${\rho }_v<0$, so $\sigma >0$. In a semiconductor with both electrons and holes, $\sigma =-{\rho }_{v,e}{\mu }_e+{\rho }_{v,h}{\mu }_h$ summing both contributions.

Two ideal limits

Perfect conductor: $\sigma \to \infty$. For finite $\mathbf{J}$, $\mathbf{E}=\mathbf{J}/\sigma \to 0$ — no electric field inside. Metals are well-approximated this way at frequencies below their plasma frequency.

Perfect dielectric: $\sigma =0$. Then $\mathbf{J}=0$ regardless of $\mathbf{E}$ — no current flows. Good insulators approach this limit.

Macroscopic Ohm’s law

For a uniform conductor of cross-section $S$, length $l$, with field $E$ producing current density $J=\sigma E$, the macroscopic relation $V=IR$ follows with $R=l/(\sigma S)$ — the familiar $V=IR$ recovered from the point form. See Resistance for the general integral formula in arbitrary geometry, the RC product relation, and the high-frequency skin-effect modification.

A perfect conductor is equipotential

In electrostatic equilibrium inside a perfect conductor, $\mathbf{E}=0$. So for any two points $P_1,P_2$ inside,

$V_{12}=-{\int }_{P_1}^{P_2}\mathbf{E}\cdot d\mathbf{l}=0,$

meaning $V$ is the same everywhere inside the conductor. This makes conductors useful as equipotential references — the surface of a conductor is an equipotential surface, and the field just outside is perpendicular to it. This boundary condition is what determines the shape of $\mathbf{E}$ near conductors.

Joule’s law

The power dissipated per unit volume in a conducting medium is

$\frac{dP}{dv}=\mathbf{E}\cdot \mathbf{J}=\sigma E^2{\text{(W/m}}^3\text{)}.$

Total dissipated power in a volume:

$P={\int }_v\sigma E^{2}dv.$

This is Joule’s law — the rate of conversion of electrical energy to heat. For a simple resistor with uniform $\mathbf{E}$ and $\mathbf{J}$, this reduces to $P=I^{2}R=V^2/R$.

When does Ohm’s law fail?

The linear $\mathbf{J}=\sigma \mathbf{E}$ relation holds when:

• Field strength is moderate (below breakdown).
• Frequencies are low enough that the carriers can follow $\mathbf{E}$ (mobility doesn’t have a phase lag).
• The material is in steady state (not pulse-heated, not just starting up).

At high fields or high frequencies — common in modern electronics — corrections to Ohm’s law become important (hot-carrier effects, drift saturation, displacement currents in semiconductors, etc.).