26.1 - Electric Current

Electrons are always bouncing around at random in a circuit. Sometimes, however, these electrons travel generally in a common direction as a sort of biased movement. Electrons themselves move very slowly, however a conductor is always full of electrons, which move in a way very similar to a liquid in a pipe.

An electric current $i$ in a conductor is defined by:

where $dq$ is the amount of positive charge that passes in time $dt$.

By convention, the direction of electric current is taken as the direction in which positive charge carriers would move even though (normally) only conduction electrons can move.

Current is measured in coulombs per second, or as they are better known, amperes ($A$):

16.2 - Current Density

Current $i$ (a scalar quantity) is related to the current density $\vec J$ (a vector quantity), by:

where $d\vec A$ is a vector perpendicular to a surface element of area $dA$ and the integral is taken over any surface cutting across the conductor. The current density $\vec J$ has the dame direction as the velocity of the moving charges if they are positive and the opposite direction if they are negative.

If the current is uniform across the surface and parallel to $d\vec A$, then $\vec J$ is also uniform and parallel to $d\vec A$, simplifying the formula for $J$:

When an electric field $\vec E$ is established in a conductor, the charge carriers (assumed positive) acquire a drift speed $v_d$ in the direction of $\vec E$.

The drift velocity is related to the current density by:

where $ne$ is the carrier charge density.

26.3 - Resistance and Resistivity

The resistance $R$ of a conductor is defined as:

where $V$ is the potential difference across the conductor and $i$ is the current.

The resistivity $\rho$ and conductivity $\sigma$ of a material are related by:

where $E$ is the magnitude of the applied electric field and $J$ is the magnitude of the current density.

The electric field and current density are related to the resistivity by:

The resistance $R$ of a conducting wire of length $L$ and uniform cross section is:

where $A$ is the cross sectional area.

The resistivity $\rho$ for most materials changes with temperature. For many materials, including metals, the relationship between $\rho$ and temperature $T$ is approximated by the equation:

$$\rho - \rho_0 = \rho_0 \alpha(T - T_0)$$

Here $T_0$ is a reference temperature, $\rho_0$ is the resistivity at $T_0$, and $\alpha$ is the temperature coefficient of resistivity for the material.

26.4 - Ohm’s Law

A given device (conductor, resistor, or any other electrical device) obeys Ohm’s law if its resistance $R (= V/i)$ is independent of the applied potential difference $V$.

A given material obeys Ohm’s law if its resistivity $r (= E/J)$ is independent of the magnitude and direction of the applied electric field .

The assumption that the conduction electrons in a metal are free to move like the molecules in a gas leads to an expression for the resistivity of a metal:

where $n$ is the number of free electrons per unit volume and $\tau$ is the mean time between the collisions of an electron with the atoms of the metal.

Metals obey Ohm’s law because the mean free time $\tau$ is approximately independent of the magnitude $E$ of any electric field applied to the metal.

26.5 - Power, Semiconductors, Superconductors

Power

The power $P$, or rate of energy transfer, in an electrical device across which a potential difference $V$ is maintained is:

If the device is a resistor, the power can also be written as:

In a resistor, electric potential energy is converted to internal thermal energy via collisions between charge carriers and atoms.

Semiconductors

Semiconductors are materials that have few conduction electrons but can become conductors when they are doped with other atoms that contribute charge carriers.

Superconductors

Superconductors are materials that lose all electrical resistance. Most such materials require very low temperatures, but some become superconducting at temperatures as high as room temperature.