Electric current and resistance

Electric current

Electric current is the rate at which charge flows through a surface:

I=rac{dQ}{dt}

The SI unit is the ampere:

$1 A=1 C/s$

Current is conventionally defined as the direction positive charge would move. In metal wires, the mobile charges are electrons, which move opposite conventional current direction.

Current is not the same as electron speed

When a circuit is switched on, the electrical signal propagates through the circuit at a significant fraction of the speed of light, depending on the medium. But individual electron drift speeds in a metal are usually very slow, often fractions of a millimeter per second.

This is similar to pushing one end of a long line of marbles. The effect is transmitted quickly, even though each marble moves only slightly.

Current density

Current density describes current per unit area. If charge flows uniformly through cross-sectional area $A$,

J=rac{I}{A}

More generally, current through a surface is

I=int ec{J}cdot dec{A}

Current density is a vector pointing in the direction of conventional charge flow.

Drift velocity

In a conductor with mobile charge density $n$, charge per carrier $q$, cross-sectional area $A$, and drift speed $v_d$, the current is

$I=nqAv_d$

For electrons, $q$ is negative if using signed charge. Often we use magnitudes and handle direction separately.

This equation shows that a large current can occur even with small drift velocity because there are enormous numbers of charge carriers.

Resistance

Resistance measures how strongly an object opposes current:

R=rac{V}{I}

The unit of resistance is the ohm:

$1 Omega=1 V/A$

A large resistance allows little current for a given voltage. A small resistance allows large current.

Microscopic origin of resistance

In metals, electrons accelerate in response to electric fields but scatter from lattice vibrations, impurities, and defects. This scattering limits the average drift speed and converts electrical energy into thermal energy.

Resistance is not caused by electrons using up current. Charge is conserved. Instead, energy is transferred from electric fields to the material.

Electric power

The electric power delivered to a circuit element is

$P=IV$

Using Ohm's law for a resistor, this can be written

$P=I^2R$

or

P=rac{V^2}{R}

In a resistor, this power is usually converted into thermal energy. This is Joule heating.

Direct current

In direct current, or DC, charges flow steadily in one direction in the circuit. Batteries are common DC sources. In steady DC circuits, currents and voltages are constant in time after transients have died away.

This module focuses on DC circuits before later extending to time-varying AC behavior.

The big idea

Electric current is charge flow rate, while current density describes local flow through area. Resistance relates voltage to current and arises from microscopic scattering. Circuits transfer energy through electric fields, and resistors convert electrical energy into thermal energy according to $P=IV$.