### Summary

## 9.1 Electrical Current

- The average electrical current ${I}_{\text{ave}}$ is the rate at which charge flows, given by ${I}_{\text{ave}}=\frac{\text{\Delta}Q}{\text{\Delta}t}$, where $\text{\Delta}Q$ is the amount of charge passing through an area in time $\text{\Delta}t$.
- The instantaneous electrical current, or simply the current
*I*, is the rate at which charge flows. Taking the limit as the change in time approaches zero, we have $I=\frac{dQ}{dt}$, where $\frac{dQ}{dt}$ is the time derivative of the charge. - The direction of conventional current is taken as the direction in which positive charge moves. In a simple direct-current (DC) circuit, this will be from the positive terminal of the battery to the negative terminal.
- The SI unit for current is the ampere, or simply the amp (A), where $1\phantom{\rule{0.2em}{0ex}}\text{A}=1\phantom{\rule{0.2em}{0ex}}\text{C/s}$.
- Current consists of the flow of free charges, such as electrons, protons, and ions.

## 9.2 Model of Conduction in Metals

- The current through a conductor depends mainly on the motion of free electrons.
- When an electrical field is applied to a conductor, the free electrons in a conductor do not move through a conductor at a constant speed and direction; instead, the motion is almost random due to collisions with atoms and other free electrons.
- Even though the electrons move in a nearly random fashion, when an electrical field is applied to the conductor, the overall velocity of the electrons can be defined in terms of a drift velocity.
- The current density is a vector quantity defined as the current through an infinitesimal area divided by the area.
- The current can be found from the current density, $I={\displaystyle \underset{\text{area}}{\iint}\overrightarrow{J}\xb7d\overrightarrow{A}}$.
- An incandescent light bulb is a filament of wire enclosed in a glass bulb that is partially evacuated. Current runs through the filament, where the electrical energy is converted to light and heat.

## 9.3 Resistivity and Resistance

- Resistance has units of ohms $(\text{\Omega})$, related to volts and amperes by $1\phantom{\rule{0.2em}{0ex}}\text{\Omega}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}1\phantom{\rule{0.2em}{0ex}}\text{V/A}$.
- The resistance
*R*of a cylinder of length*L*and cross-sectional area*A*is $R=\frac{\rho L}{A}$, where*$\rho $*is the resistivity of the material. - Values of $\rho $ in Table 9.1 show that materials fall into three groups—conductors, semiconductors, and insulators.
- Temperature affects resistivity; for relatively small temperature changes $\text{\Delta}T$, resistivity is $\rho ={\rho}_{0}\left(1+\alpha \text{\Delta}T\right)$, where ${\rho}_{0}$ is the original resistivity and $\alpha $ is the temperature coefficient of resistivity.
- The resistance
*R*of an object also varies with temperature: $R={R}_{0}\left(1+\alpha \text{\Delta}T\right)$, where ${R}_{0}$ is the original resistance, and*R*is the resistance after the temperature change.

## 9.4 Ohm's Law

- Ohm’s law is an empirical relationship for current, voltage, and resistance for some common types of circuit elements, including resistors. It does not apply to other devices, such as diodes.
- One statement of Ohm’s law gives the relationship among current
*I*, voltage*V*, and resistance*R*in a simple circuit as*$V=IR$.* - Another statement of Ohm’s law, on a microscopic level, is $J=\sigma E$.

## 9.5 Electrical Energy and Power

- Electric power is the rate at which electric energy is supplied to a circuit or consumed by a load.
- Power dissipated by a resistor depends on the square of the current through the resistor and is equal to $P={I}^{2}R=\frac{{V}^{2}}{R}$.
- The SI unit for electric power is the watt and the SI unit for electric energy is the joule. Another common unit for electric energy, used by power companies, is the kilowatt-hour (kW $\xb7$ h).
- The total energy used over a time interval can be found by $E={\displaystyle \int Pdt}$.

## 9.6 Superconductors

- Superconductivity is a phenomenon that occurs in some materials when cooled to very low critical temperatures, resulting in a resistance of exactly zero and the expulsion of all magnetic fields.
- Materials that are normally good conductors (such as copper, gold, and silver) do not experience superconductivity.
- Superconductivity was first observed in mercury by Heike Kamerlingh Onnes in 1911. In 1986, Dr. Ching Wu Chu of Houston University fabricated a brittle, ceramic compound with a critical temperature close to the temperature of liquid nitrogen.
- Superconductivity can be used in the manufacture of superconducting magnets for use in MRIs and high-speed, levitated trains.