Electricity, phenomenon associated with stationary or moving electric charges. Electric charge is a fundamental property of matter and is borne by elementary particles. In electricity the particle involved is the electron, which carries a charge designated, by convention, as negative. Thus, the various manifestations of electricity are the result of the accumulation or motion of numbers of electrons.

## Electrostatics

Electrostatics is the study of electromagnetic phenomena that occur when there are no moving charges—i.e., after a static equilibrium has been established. Charges reach their equilibrium positions rapidly because the electric force is extremely strong. The mathematical methods of electrostatics make it possible to calculate the distributions of the electric field and of the electric potential from a known configuration of charges, conductors, and insulators. Conversely, given a set of conductors with known potentials, it is possible to calculate electric fields in regions between the conductors and to determine the charge distribution on the surface of the conductors. The electric energy of a set of charges at rest can be viewed from the standpoint of the work required to assemble the charges; alternatively, the energy also can be considered to reside in the electric field produced by this assembly of charges. Finally, energy can be stored in a capacitor; the energy required to charge such a device is stored in it as electrostatic energy of the electric field.

## Coulomb’s law

Static electricity is a familiar electric phenomenon in which charged particles are transferred from one body to another. For example, if two objects are rubbed together, especially if the objects are insulators and the surrounding air is dry, the objects acquire equal and opposite charges and an attractive force develops between them. The object that loses electrons becomes positively charged, and the other becomes negatively charged. The force is simply the attraction between charges of opposite sign. The properties of this force were described above; they are incorporated in the mathematical relationship known as Coulomb’s law. The electric force on a charge Q1 under these conditions, due to a charge Q2 at a distance r, is given by Coulomb’s law,Equation.

## Calculating the value of an electric field

In the example, the charge Q1 is in the electric field produced by the charge Q2. This field has the valueEquation.in newtons per coulomb (N/C). (Electric field can also be expressed in volts per metre [V/m], which is the equivalent of newtons per coulomb.) The electric force on Q1 is given byEquation.in newtons. This equation can be used to define the electric field of a point charge. The electric field E produced by charge Q2 is a vector. The magnitude of the field varies inversely as the square of the distance from Q2; its direction is away from Q2 when Q2 is a positive charge and toward Q2 when Q2 is a negative charge. Using equations (2) and Electricity and Magnetism. Electricity. Electrostatics. Static electricity. [Calculating the value of an electric field – equation 4], the field produced by Q2 at the position of Q1 isEquation.in newtons per coulomb.

## Superposition principle

This calculation demonstrates an important property of the electromagnetic field known as the superposition principle. According to this principle, a field arising from a number of sources is determined by adding the individual fields from each source. The principle is illustrated by Figure 3, in which an electric field arising from several sources is determined by the superposition of the fields from each of the sources. In this case, the electric field at the location of Q1 is the sum of the fields due to Q2 and Q3. Studies of electric fields over an extremely wide range of magnitudes have established the validity of the superposition principle.

## Electric potential

The electric potential is just such a scalar function. Electric potential is related to the work done by an external force when it transports a charge slowly from one position to another in an environment containing other charges at rest. The difference between the potential at point A and the potential at point B is defined by the equation

## Deriving electric field from potential

The electric field has already been described in terms of the force on a charge. If the electric potential is known at every point in a region of space, the electric field can be derived from the potential. In vector calculus notation, the electric field is given by the negative of the gradient of the electric potential, E = −grad V. This expression specifies how the electric field is calculated at a given point. Since the field is a vector, it has both a direction and magnitude. The direction is that in which the potential decreases most rapidly, moving away from the point. The magnitude of the field is the change in potential across a small distance in the indicated direction divided by that distance.

## Capacitance

A useful device for storing electrical energy consists of two conductors in close proximity and insulated from each other. A simple example of such a storage device is the parallel-plate capacitor. If positive charges with total charge +Q are deposited on one of the conductors and an equal amount of negative charge −Q is deposited on the second conductor, the capacitor is said to have a charge Q. As shown in Figure 11, it consists of two flat conducting plates, each of area A, parallel to each other and separated by a distance d.

## Principle of the capacitor

To understand how a charged capacitor stores energy, consider the following charging process. With both plates of the capacitor initially uncharged, a small amount of negative charge is removed from the lower plate and placed on the upper plate. Thus, little work is required to make the lower plate slightly positive and the upper plate slightly negative. As the process is repeated, however, it becomes increasingly difficult to transport the same amount of negative charge, since the charge is being moved toward a plate that is already negatively charged and away from a plate that is positively charged. The negative charge on the upper plate repels the negative charge moving toward it, and the positive charge on the lower plate exerts an attractive force on the negative charge being moved away. Therefore, work has to be done to charge the capacitor.

## Dielectrics, polarization, and electric dipole moment

The amount of charge stored in a capacitor is the product of the voltage and the capacity. What limits the amount of charge that can be stored on a capacitor? The voltage can be increased, but electric breakdown will occur if the electric field inside the capacitor becomes too large. The capacity can be increased by expanding the electrode areas and by reducing the gap between the electrodes. In general, capacitors that can withstand high voltages have a relatively small capacity. If only low voltages are needed, however, compact capacitors with rather large capacities can be manufactured. One method for increasing capacity is to insert between the conductors an insulating material that reduces the voltage because of its effect on the electric field. Such materials are called dielectrics (substances with no free charges). When the molecules of a dielectric are placed in the electric field, their negatively charged electrons separate slightly from their positively charged cores. With this separation, referred to as polarization, the molecules acquire an electric dipole moment. A cluster of charges with an electric dipole moment is often called an electric dipole.

## Direct electric current

### Basic phenomena and principles

Many electric phenomena occur under what is termed steady-state conditions. This means that such electric quantities as current, voltage, and charge distributions are not affected by the passage of time. For instance, because the current through a filament inside a car headlight does not change with time, the brightness of the headlight remains constant. An example of a nonsteady-state situation is the flow of charge between two conductors that are connected by a thin conducting wire and that initially have an equal but opposite charge. As current flows from the positively charged conductor to the negatively charged one, the charges on both conductors decrease with time, as does the potential difference between the conductors. The current therefore also decreases with time and eventually ceases when the conductors are discharged.

### Conductors, insulators, and semiconductors

Materials are classified as conductors, insulators, or semiconductors according to their electric conductivity. The classifications can be understood in atomic terms. Electrons in an atom can have only certain well-defined energies, and, depending on their energies, the electrons are said to occupy particular energy levels. In a typical atom with many electrons, the lower energy levels are filled, each with the number of electrons allowed by a quantum mechanical rule known as the Pauli exclusion principle. Depending on the element, the highest energy level to have electrons may or may not be completely full. If two atoms of some element are brought close enough together so that they interact, the two-atom system has two closely spaced levels for each level of the single atom. If 10 atoms interact, the 10-atom system will have a cluster of 10 levels corresponding to each single level of an individual atom. In a solid, the number of atoms and hence the number of levels is extremely large; most of the higher energy levels overlap in a continuous fashion except for certain energies in which there are no levels at all. Energy regions with levels are called energy bands, and regions that have no levels are referred to as band gaps.

### Electromotive force

A 12-volt automobile battery can deliver current to a circuit such as that of a car radio for a considerable length of time, during which the potential difference between the terminals of the battery remains close to 12 volts. The battery must have a means of continuously replenishing the excess positive and negative charges that are located on the respective terminals and that are responsible for the 12-volt potential difference between the terminals. The charges must be transported from one terminal to the other in a direction opposite to the electric force on the charges between the terminals. Any device that accomplishes this transport of charge constitutes a source of electromotive force. A car battery, for example, uses chemical reactions to generate electromotive force. The Van de Graaff generator shown in Figure 13 is a mechanical device that produces an electromotive force. Invented by the American physicist Robert J. Van de Graaff in the 1930s, this type of particle accelerator has been widely used to study subatomic particles. Because it is conceptually simpler than a chemical source of electromotive force, the Van de Graaff generator will be discussed first.

### Direct-current circuits

The simplest direct-current (DC) circuit consists of a resistor connected across a source of electromotive force. The symbol for a resistor is shown in Figure 15; here the value of R, 60Ω, is given by the numerical value adjacent to the symbol. The symbol for a source of electromotive force, E, is shown with the associated value of the voltage. Convention gives the terminal with the long line a higher (i.e., more positive) potential than the terminal with the short line. Straight lines connecting various elements in a circuit are assumed to have negligible resistance, so that there is no change in potential across these connections. The circuit shows a 12-volt electromotive force connected to a 60Ω resistor. The letters a, b, c, and d on the diagram are reference points.

### Resistors in series and parallel

If two resistors are connected in Figure 16A so that all of the electric charge must traverse both resistors in succession, the equivalent resistance to the flow of current is the sum of the resistances.

### Kirchhoff’s laws of electric circuits

Two simple relationships can be used to determine the value of currents in circuits. They are useful even in rather complex situations such as circuits with multiple loops. The first relationship deals with currents at a junction of conductors. Figure 17 shows three such junctions, with the currents assumed to flow in the directions indicated.

## Alternating electric currents

### Basic phenomena and principles

Many applications of electricity and magnetism involve voltages that vary in time. Electric power transmitted over large distances from generating plants to users involves voltages that vary sinusoidally in time, at a frequency of 60 hertz (Hz) in the United States and Canada and 50 hertz in Europe. (One hertz equals one cycle per second.) This means that in the United States, for example, the current alternates its direction in the electric conducting wires so that each second it flows 60 times in one direction and 60 times in the opposite direction. Alternating currents (AC) are also used in radio and television transmissions. In an AM (amplitude-modulation) radio broadcast, electromagnetic waves with a frequency of around one million hertz are generated by currents of the same frequency flowing back and forth in the antenna of the station. The information transported by these waves is encoded in the rapid variation of the wave amplitude. When voices and music are broadcast, these variations correspond to the mechanical oscillations of the sound and have frequencies from 50 to 5,000 hertz. In an FM (frequency-modulation) system, which is used by both television and FM radio stations, audio information is contained in the rapid fluctuation of the frequency in a narrow range around the frequency of the carrier wave.

### Transient response

Consider a circuit consisting of a capacitor and a resistor that are connected as shown in Figure 19. What will be the voltage at point b if the voltage at a is increased suddenly from Va = 0 to Va = +50 volts? Closing the switch produces such a voltage because it connects the positive terminal of a 50-volt battery to point a while the negative terminal is at ground (point c). Figure 20 (left) graphs this voltage Va as a function of the time.

### Alternating-current circuits

Certain circuits include sources of alternating electromotive forces of the sinusoidal form V = V0 cos(ωt) or V = V0 sin(ωt). The sine and cosine functions have values that vary between +1 and −1; either of the equations for the voltage represents a potential that varies with respect to time and has values from +V0 to −V0. The voltage varies with time at a rate given by the numerical value of ω; ω, which is called the angular frequency, is expressed in radians per second. Figure 22 shows an example with V0 = 170 volts and ω = 377 radians per second, so that V = 170 cos(377t). The time interval required for the pattern to be repeated is called the period T, given by T = 2π/ω. In Figure 22, the pattern is repeated every 16.7 milliseconds, which is the period. The frequency of the voltage is symbolized by f and given by f = 1/T. In terms of ω, f = ω/2π, in hertz.

### Behaviour of an AC circuit

The way an AC circuit functions can be better understood by examining one that includes a source of sinusoidally varying electromotive force, a resistor, a capacitor, and an inductor, all connected in series. For this single-loop problem, only the second of Kirchhoff’s laws is needed since there is only one current. The circuit is shown in Figure 23 with the points a, b, c, and d at various positions in the circuit located between the various elements. The letters R, L, and C represent, respectively, the values of the resistance in ohms, the inductance in henrys, and the capacitance in farads. The source of the AC electromotive force is located between a and b. The wavy symbol is a reminder of the sinusoidal nature of the voltage that is responsible for making the current flow in the loop. For the potential between b and a,

## Electric properties of matter

### Piezoelectricity

Some solids, notably certain crystals, have permanent electric polarization. Other crystals become electrically polarized when subjected to stress. In electric polarization, the centre of positive charge within an atom, molecule, or crystal lattice element is separated slightly from the centre of negative charge. Piezoelectricity (literally “pressure electricity”) is observed if a stress is applied to a solid, for example, by bending, twisting, or squeezing it. If a thin slice of quartz is compressed between two electrodes, a potential difference occurs; conversely, if the quartz crystal is inserted into an electric field, the resulting stress changes its dimensions. Piezoelectricity is responsible for the great precision of clocks and watches equipped with quartz oscillators. It also is used in electric guitars and various other musical instruments to transform mechanical vibrations into corresponding electric signals, which are then amplified and converted to sound by acoustical speakers.

### Electro-optic phenomena

The index of refraction n of a transparent substance is related to its electric polarizability and is given by n2 = 1 + χe/ε0. As discussed earlier, χe is the electric susceptibility of a medium, and the equation P = χeE relates the polarization of the medium to the applied electric field. For most matter, χe is not a constant independent of the value of the electric field, but rather depends to a small degree on the value of the field. Thus, the index of refraction can be changed by applying an external electric field to a medium. In liquids, glasses, and crystals that have a centre of symmetry, the change is usually very small. Called the Kerr effect (for its discoverer, the Scottish physicist John Kerr), it is proportional to the square of the applied electric field. In noncentrosymmetric crystals, the change in the index of refraction n is generally much greater; it depends linearly on the applied electric field and is known as the Pockels effect (after the German physicist F. R. Pockels).

### Thermoelectricity

When two metals are placed in electric contact, electrons flow out of the one in which the electrons are less bound and into the other. The binding is measured by the location of the so-called Fermi level of electrons in the metal; the higher the level, the lower is the binding. The Fermi level represents the demarcation in energy within the conduction band of a metal between the energy levels occupied by electrons and those that are unoccupied. The energy of an electron at the Fermi level is −W relative to a free electron outside the metal. The flow of electrons between the two conductors in contact continues until the change in electrostatic potential brings the Fermi levels of the two metals (W1 and W2) to the same value. This electrostatic potential is called the contact potential ϕ12 and is given by eϕ12 = W1 − W2, where e is 1.6 × 10−19 coulomb.

### Thermionic emission

A metal contains mobile electrons in a partially filled band of energy levels—i.e., the conduction band. These electrons, though mobile within the metal, are rather tightly bound to it. The energy that is required to release a mobile electron from the metal varies from about 1.5 to 6 electron volts, depending on the metal. In thermionic emission, some of the electrons acquire enough energy from thermal collisions to escape from the metal. The number of electrons emitted and therefore the thermionic emission current depend critically on temperature.