ELECTROMAGNETIC FIELDS

Introduction. In 1864 the English physicist, James Clerk Maxwell (1831-1879), presented to the Royal Society his famous paper "On a Dynamical Theory of the Electromagnetic Field," in which he set forth his four, now famous, equations that describe all electromagnetic radiation. Building on the work of Coulomb, Ampere, Faraday, and their contemporaries who set forth the intimate relationship between electricity and magnetism, that an electric charge and a magnetic pole can exert a force on each other, provided that they are in relative motion, Maxwell translated these relationships into a mathematical form. Using as a model the concept of the field that Michael Faraday (1791-1867) had introduced and the lines-of-force he used to describe them, Maxwell converted the descriptions of electric and magnetic phenomena into mathematical terms. Faraday's great discovery of electromagnetic induction, which he expressed clearly in terms of his lines-of-force model of the electric and magnetic fields, became the fundamental postulate of Maxwell's theory. The work of Faraday, who is regarded as one of greatest experimentalists, was complemented by the work of Maxwell, who ranks among the greatest theoretical physicists. Even though Faraday expressed his discoveries in non-mathematical form his concepts are eminently mathematical. As Maxwell writes in the preface to his paper,

"As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. For instance, Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centers of force attracting at a distance; Faraday saw a medium where they saw nothing but distance; Faraday sought the seat of the phenomena in real actions going on in the medium, [whereas] they were satisfied that they had found it in a power of action at a distance impressed on the electric fluid."

Maxwell began by developing a clearer picture of how the lines of force functioned in transmitting electric and magnetic forces through the aether. Using the lines of force model, he converted the descriptions of electric and magnetic phenomena into mathematical symbols. Maxwell wrote, "When I had translated what I considered to be Faraday's ideas into a mathematical form, I found that in general the results of the two methods coincided, so that the same phenomena were accounted for, …."

The set of field equations that Maxwell obtained by the translation into mathematical notation was expressed in the language of the calculus; not just in the calculus of functions of one independent variable, but the calculus of functions of more than one independent variable, called partial differentiation and multiple integration. Maxwell did not have the concise notation of the later calculus of vectors that is called vector analysis. The form of vector analysis found in present day American texts was developed by one of the outstanding mathematical physicist of the nineteenth century, the American, John Willard Gibbs (1839-1903). In lecturing to his students at Yale, Gibbs thought that a simpler mathematical language was needed for the theoretical aspects of such subjects as electromagnetics and thermodynamics. His familiarly with the work of the famous Irish theoretical physicist, William Rowan Hamilton (1805-1865), on quaternions and of the German mathematician Herman Grassmann (1809-1877) on generalized multiplication enabled him to pick out those aspects of their work that seemed to apply best to the needs of theoretical physics. Apparently Gibbs did not think that his development of vector analysis was original enough to be worthy of publication; his notes on the subject were circulated among only his students and those interested in the subject. It was twenty years after the development of his original notes, that he allowed them to be published in book form. In 1901, J. B. Wilson published the first book on the subject. Maxwell field equations today are usually expressed most concisely in the notation of the vector calculus. The following four equations in the notation of vector analysis are Maxwell's four equations of electromagnetic fields.
∇ · D = ρ, (1)
∇ · B = 0, (2)
∇ × E = -∂B/∂t, (3)
∇ × B = J + ∂D/∂t, (4)
where D = εE and B = μH. These four equations (1) through (4) are called Maxwell's electromagnetic equations, and they express concisely the experimental knowledge of electromagnetic fields as contained in the laws of Coulomb, Ampere, Faraday, and Maxwell.

Maxwell's First Equation. The first 190 years of the modern study of electricity and magnetism that began with the publication in 1600 by Sir William Gilbert of his book De Magnete was qualitative until Charles Augustin Coulomb in 1790 quantitatively verified the inverse square law of the forces between two electric changes. This general law, known as Coulomb's Law of Electrostatic, opened the way for the quantitative study of electrostatics. Coulomb also verified a similar law for the forces between magnetic poles and opened the way for the quantitative study of magnetostatics.

Coulomb's Law of the Electrostatic Forces between two charged bodies may be stated as follows.

Bodies with like charges repel each other and those with unlike charges attract each other.

Expressed mathematically in vector notation, Coulomb's Law states that the force F on a point charge q₂ due to the presence of another point charge q₁ called the source charge is
F = k_e[(q₁q₂)/ r²]r₁, (5)
where k_e is the constant of proportionality, r is the distance between q₁ and q₂, and r₁ is the unit vector directed from q₁ toward q₂. The unit vector is introduced to give the proper direction to the force; if the bodies have like charges (both positive or both negative), then the force is positive and its direction is away from the source charge and if bodies have unlike charges (one positive and the other negative), then the force is negative and its direction is toward the source charge. The unit vector is
r₁ = r/r, (6)
where r is the vector from q₁ to q₂, whose magnitude is r, the distance between the two charges. The magnitude of the unit vector r₁ is one and is dimensionless. In the rationalized MKS (RMKS), where the unit of charge is the coulomb (as defined by Ampere's Law), the distance r is measured in meters and the force F in newtons, the constant of proportionality k_e has a value nearly equal to 9 × 10⁹ newton-meter² per coulomb².

This constant is usually defined in terms of another constant called absolute permittivity ε (Greek letter epsilon) that is the product to two other constants ε₀ and ε_r, that is,
ε = ε₀ε_r, (7)
where ε₀ is called the permittivity of empty space or the permittivity of vacuum and is equal to
8.854 × 10^-12 coulomb² per newton-meter² and ε_r is called the relative permittivity or dielectric constant whose value is dimensionless and depends upon the material (vacuum, air, glass, etc.) between the two charges; its value is greater than unity for all materials except a vacuum where it is equal to one. Hence,
k_e = 1/4πε = 1/4πε₀ε_r. (8)

The Field Intensity of an Electric Field, called Electric Field Intensity E, is defined as the force per unit charge that would be exerted on a small positive test charge if placed at a point in an electric field, that is,
E = F_t/q_t, (9)
where q_t is the quantity of charge on the positive test charge measured in coulombs and F_t is the force exerted on the test charge by the electric field measured in newtons. Hence, electric field intensity is measured in newtons per coulombs. At a distance r from a charge Q, the force on the test charge q_t is given by equation (5), so that the electric intensity at q_t due to Q is
E = F_t/q_t = F_t(1/q_t) = {k_e[(Qq_t)/ r²]r₁}(1/q_t) = (1/4πε)(Q/r²)r₁, (10)
where the unit vector r₁ is defined by equation (6) and is directed away from Q. This equation (10) gives the field intensity E due to a point source charge Q only; it is not definition of field intensity E nor a general expression that applies to all electric phenomena.

Gauss' Law for Electrostatics gives us an alternate form of Coulomb's Law and is a theorem derived from it. Consider a surface S which surrounds a volume V in which is located a point charge Q. At any element area dS let θ be the angle between the direction of dS and that of the electric intensity E due to Q. Using equation (10), we get
∫∫_SεE · dS = ∫∫_S {ε[Q/(4πεr²)]r₁} · dS = (Q/4π)∫∫_S(cos θdS/r²) =
(Q/4π)∫∫_SdΩ = (Q/4π)(4π) = Q,
where the integration is taken over the whole surface, since the surface S completely encloses the charge Q, and the differential of the solid angle dΩ = (cos θdS/r²) is the solid angle subtended by dS at point Q. When integrated over the whole surface, the solid angle is completely about the point and 4π steradians is the complete solid angle about a point. Thus
∫∫_SεE · dS = Q.
If more than one point charge is present within S, then we find that
∫∫_SεE · dS = ∑_iQ_i, (11)
where the right side of this equation represents the total charge contained in S. If the volume V contains a continuous charge distribution, then ρ denotes the charge density, or net charge per unit volume. Then equation (11) then becomes
∫∫_SεE · dS = ∫∫∫_VρdV, (12)
which is Gauss' law for electrostatics.

With the aid of Gauss' law, expressions for the electric field in the neighborhood of charged spheres, cylinders, and between two parallel charged plates may be found. We shall do this first for a charged sphere.

Field about a Charged Sphere. Suppose that a charge Q is uniformly distributed over the surface or volume of a spherical body. Let us take for the surface S a sphere of radius r outside and concentric with charged sphere. From symmetry considerations, the vector E must be constant in magnitude over S and directed radially outward if Q is positive, so that E · dS = EdS. Then we get
∫∫_SεE · dS = εE∫∫_SdS = εE(4πr²) = ε[Q/(4πεr²)](4πr²) = Q,
since the right side of equation (12) is just Q and the surface of the sphere is 4πr².

Field between two Parallel Charged Plates. Now let us find the electric field between two oppositely charged parallel plates. Let assume that the parallel plates are very large compared with the distance between them, so that the end effects may be neglected. Let the charge per unit area on one plate be σ and on the other -σ. From symmetry considerations, we may assume that σ is the same at all points on a plate, so that the field between the plates is uniform. Let the surface S be a closed surface shaped like a pillbox and enclosing an area A of the positive plate. A field inside a conductor will cause the free charges in the conductor to move. Here we assume that the charges are static, so that there is no field inside the plates. For the surface of our pillbox within the plate, ∫∫_SεE · dS will be zero. Since the cylindrical side of the pillbox are at right angle to the field, the electric field there will be zero. Then the only contribution to the surface integral comes from the area A of pillbox which is in the region between the plates. Let E be the field in that region. Then we get
∫∫_SεE · dS = εEA.
The charge Q enclosed within the pillbox is σA. From Gauss' Law we then have
εEA = σA, or
E = σ/ε = Q/εA. (13)

The Divergence Theorem of vector analysis states that for any vector function A
∫∫_SA · dS = ∫∫∫_V∇ · AdV,
where V is any volume enclosed by the surface S. If this theorem is applied to the left side to Gauss' Law, equation (12), this equation becomes
∫∫∫_Vε∇ · EdV = ∫∫∫_VρdV.

Since the limits of integration are the same on both sides of this equation, the integrands may be equated and we get
ε∇ · E = ρ or ∇ · εE = ρ. (14)

The divergence of E at any point in the electric field is thus proportional to the charge density enclosing the charge. The field emanates from any positive charge and converges toward a negative charge. In a region where ρ is zero, any field present will merely cross a closed surface, entering on one side and leaving from another. This, of course, is just what we expect from Gauss' Law, from which equation (14) is derived.

Lines of force represents the direction of a field by being drawn in such a way that the direction of the lines at a point is the direction of the field at that point. The direction of a line of force at a point is the direction of the tangent to it at that point. Since at any point in a field, the field can have but one direction, then only one line of force can be drawn through each such point of the field. In other words, lines of force can never intersect, for this would imply that the field would have two directions at the point of intersection. Thus lines of force can be used to represent the direction of an electrostatic field. But lines of force also represent the magnitude of an electrostatic field. This is accomplished by spacing of the lines in such a way that the number of lines per unit area crossing a surface perpendicular to the direction of the field is equal to the product of the absolute permittivity ε and the magnitude of the electric field intensity E at that surface. If the Greek letter ψ (psi) is used to designate the number of lines of force and if A is the symbol for the area of a surface perpendicular to the direction of the field, then the above rule may be expressed by the following equation.
ψ/A = εE. (15)
By use of this rule, the number of lines of force drawn to represent the electric field are limited in such a way as to represent the magnitude of the field. By making the number of lines of force per unit area of a surface perpendicular to the direction of the lines proportional to the electric field intensity, a suitable limiting of the number of lines to be drawn to represent the field. Thus the lines of force may be used to represent the magnitude of the field as well as its direction. Practically, this rule means that in a region where the field intensity is large, the lines of force will be closely spaced, and in the region where the field intensity is small, the lines of will be widely separated.

A new physical quantity was introduced in the equation (15) given above. It was symbolized by the Greek letter ψ, and is called electric flux; it represents the number of lines of force and in the RMKS system of units it is measured in coulombs. The ratio on the left side of equation (15), is taken as another new physical quantity. This quantity is called electric flux density and is defined as the ratio of the number of lines of force per unit area of the surface perpendicular to the direction of the field. That is,
D = ψ/A, (16)
where in the RMKS system of units ψ represents the electric flux measured in coulombs, A represents the area of the surface perpendicular to the direction of the field measured in square meters, and D represents the electric flux density measured in coulombs per meter². Hence, equation (15) becomes
D = εE. (17)
Since the electric field intensity is a vector quantity, then the electric flux density is also a vector quantity, whose direction is the direction of the electric field and its magnitude is the product of the absolute permittivity ε and the electric field intensity E, given by equation (16). That is,
D = εE. (18)
But since D = εE, substituting into equation (14), we get
∇ · D = ρ, (1)
which is Maxwell's First Equation of Electromagnetic Fields.

Maxwell's Second Equation.

Maxwell found, in general, that the laws of Coulomb, Ampere, and Faraday as stated were not compatible with the principle of continuity which states that electric charge can neither be created nor destroyed.

Electrostatic Discharge. In 1731, Stephen Gray communicated to the Royal Society, through one its members, a striking discovery that he had made while experimenting with a electrified glass tube. He had found, as others had before him, that if the glass tube is rubbed in the dark, tiny sparks are seen to pass between the glass tube and a finger held close to it. He reasoned that if the rubbed glass tube communicates sparks to a nearby object, perhaps it is at the same time communicating the "electric virtue" to the object. To test this conjecture, he procured a glass tube about 3 feet long and over an inch in diameter. He put a cork in each end of it "to keep the dust out." But being careful experimenter, he decided to check whether the corks would interfere with the operation of the tube. Upon rubbing the tube he found that it would attract a nearby feather just as well when stopped at both ends as when left open. Holding the feather near one end of the tube, he saw it approach and touch the cork and then be repelled, just as it had at the rubbed glass surface. Here was a new phenomenon; the cork had not been rubbed, but nevertheless it attracted the feather and hence was electrified. Seemingly, Gray concluded, the "attractive virtue" had been communicated to the cork, presumably because it was in contact with the electrified tube. He abandoned his original conjecture, and turned all his efforts to exploit his discovery. Continuing his experiments he replaced the cork with an ivory ball and suspended it from the glass tube by a string. To be able to use longer strings he stood on furniture and balconies. He succeeded in transmitting the "electric virtue" a distance of 34 feet along the vertical string. Gray decided to use a horizontal string tied at one end to the glass tube and an ivory ball at the other; the string was suspended near the far end by a piece of vertical cord looped over a nail in a ceiling beam. But the experiment failed completely. Gray concluded that the "electric virtue," when it reached the end of string, went up the cord to the beam. A friend suggested supporting the string along which the virtue moved by a thin silk cord and the plan worked. Gray thought that the smallness (thinness) of the supporting silk cords had prevented the "electric virtue" from being diverted from the transmission line. In order to increase the transmission distance the experimenters moved outdoors and succeeded transmitting the "electric virtue" a distance of 293 feet. When still longer distances were attempted, the silk supporting cords broke under the weight of the transmitting string. Gray replaced the silk cords with brass wires and the experiment failed completely. Gray realized he had been in error about the reason of the earlier failure; it was not the thinness of the supports but the kind of material of which they were made. So when the experimenters tried the silk supports again, but using many more of them fastened to a series of poles driven into the ground, they succeeded in transmitting the "electric virtue" a distance of 650 feet and even longer. Gray concluded that the silk was a non-conductor of the electric virtue, whereas the brass was a conductor of it.

Electrostatic discharges had been observed before Gray did; but it was not understood. It had been observed that when one attempted to build up a very large charge on an object, the charge tended to "leak off," especially at sharp points. Sailors since ancient times observed during storms at the points of masts and end of spars a glow that was called "St. Elmo's fire." But this phenomena was not understood as electrostatic discharge until Benjamin Franklin (1706-1790) performed his famous kite experiment. He recognized that lightning and other atmospheric electricity had properties identical to those of electricity produced by rubbing. And this recognition was the basis for his famous invention of the lightning rod. Having noticed in his experiments that electricity tends to discharge at points, Franklin concluded that a pointed rod would "probably draw the electrical fire silently out of a cloud before it came nigh enough to strike, and thereby secure us from the most sudden and terrible mischief."

Two technical developments during the eighteenth century helped to understand electrostatic discharge. One was the electrostatic generator, a device which produces an electric charge by rubbing an insulating material like glass mechanically instead of by hand. This made it possible to generate static electricity continuously. Sixty years after the publication of Gilbert's work the burgomaster of the German city of Magdeburg, Otto von Guericke (1602-1686), constructed a powerful electrostatic generator. He poured molten sulfur into a glass globe, let is solidify, and then broke the globe. The ball of sulfur, mounted on an axle, could be rotated by means of a crank. The rotating ball was electrified by rubbing it with a cloth. In later versions of the machine, the ball was rotated faster by a system pulleys. Von Guericke's machine was widely copied and improved. The sulfur sphere was replaced by the glass globe fitted with attachments to rub effectively the globe and to conduct the charge from the globe to other objects. In the eighteenth century electrostatic generators, like von Guericke's, became popular drawing room entertainment. Audiences were astounded by the spectacular maelstrom of feathers, paper, and whatever swirling about the electrified sphere. The air was filled with sparks and the smell of ozone, and those approaching the spinning globe felt their skin tingle. Electrostatic generators were also used for serious scientific investigation. Von Guericke, himself, noted that objects were attracted to the electrified globe until they came in contact with it, and then they were immediately repelled.

The other development was the "Leyden jar," that resulted from the discovery that a charge can be held, or "condensed" on a conductor by the presence of another, oppositely charged conductor. Consider two parallel metal plates separated by dry air, glass, or another good insulator. When these plates are oppositely charged, a large amount of charge can be "condensed" on the plates, because the opposite charges attract each other and hold each other in place. A device such as this is called an electric condenser or capacitor. The Leyden jar, invented by the Dutch mathematician, Professor Pieter van Musschenbroek (1692-1761) of the University of Leyden in Holland, in 1746, consisted of a thin-walled glass jar covered inside and out with a conducting material, such as metal foil, the inside and outside foils not connected together. The inside conductor is connected by a small chain to a rod through the stopper in the mouth of the jar. When a charge was place on inside conductor by contact at the end of the rod with a body charged by rubbing, the outside conductor also becomes charged. Sufficient charge can be accumulated, especially with the use of a mechanical electrostatic generator, to produce a severe electric shock, or a large visible electric spark. Franklin introduced the parallel plate condenser, that was an improvement on the Leyden jar.

The visible electric spark that passed between the two charged plates of the condenser so that they are no longer charged is called an electrostatic discharge. A condenser may also be discharged without generating a spark, by simply connecting the two plates with a metallic wire. During the process of discharge, the charge moves, or "flows," from one plate to other through the wire. Any movement of electric charge constitutes what is called an electric current. But this electric current during the discharge of the condenser is very brief and occurs during in an almost infinitesimally short time. This hindered the study of electric currents until it became possible to generate a steady and continuous flow of electric current.

Dynamic Electricity. The principle of the electric battery was discovered by an Italian anatomy professor at the University of Bologna, Luigi Galvani (1737-1798), who noticed by chance in 1786 that in certain circumstances convulsions were observed in frog's legs placed near an electric spark. On further investigation, he found that the electric spark was unnecessary; he found that the violent convulsions occurred when the frog was suspended from a brass hook into the spinal marrow and a rod made of a different metal was brought in simultaneous contact with the hook and frog's leg muscle. Galvani thought that the spasms was a manifestation of what was later called "animal electricity" or "galvanism." Galvani himself considered it to be the same as ordinary static electricity; others believed that galvanism to be a fluid different from ordinary electricity. It was Galvani's countryman, Alessandro Volta (1745-1827), Professor of Natural Philosophy in the University of Pavia, who in 1792 showed that animal nerve tissue served to detect, not to initiate, the electric effect that Galvani had observed. Later in 1793, Volta put forth the view that a steady electric current could be produced by connecting two dissimilar metals, such as copper and zinc, separated by any moist body, not necessarily organic. In 1800, Volta found that if he made a pile of metal disks, alternatively copper and zinc, separated by layers of a moist substance such as a moistened cloth or blotting paper soaked with a water solution of acid, lye or salt (so that the order was copper, zinc, paper, copper, zinc, paper, etc.), the effect of the pile was greater than any galvanic device. Touching the two ends of the voltaic pile, as it later was called, with moistened fingers would produce a strong continuous sensation. Furthermore, sparks could be drawn if the two ends of the pile were touched by a metal wire. When the two ends of the pile are connected by a wire, electric charge started to flow along the wire, and this flow of charge did not cease as long as chemical reaction continued between the moist substance and the two dissimilar metals. Later Volta made a device, called a battery, that consisted of a series of cups, each filled with brine and containing two separated strips of dissimilar metals, zinc and copper. The zinc strip in one cup is connected by a wire to the copper strip in another cup. When the connection was made between the copper strip in the first cup and the zinc strip in the last cup of the series, the effects of an electric current sparks and shocks, for example were observed. Thirty or more cups, arranged in the way, were required to produce the same kind of electric shock as that given by a Leyden jar or electrostatic generator. The flow of charge is continuous in Volta's battery in contrast to the instantaneous and intermittent discharges of these electrostatic devices. Each cup of the device was later called a cell of the battery.

Electric Circuit. In a single cell of Voltaic battery, the copper strip or terminal is positive with respect to the zinc terminal and the zinc terminal is negative respect to the copper terminal, and the current in the external wire can be considered to consist of a flow of positive charge from positive to negative, from the copper to the zinc terminal. The cell has the ability by the chemical reaction between the metals and the solution to impel a continuous flow of charge in the wire without altering the quantity of charge on either terminal. For this to happen, the charge must be considered to flow through the salt solution, and the cell and the wire connecting the terminals constitutes a cyclic path for electric current. Any such cyclic conducting path is called an electric circuit. The electric charge flows as long as the circuit is closed, that is, as long as the circuit is completed.

When a continuous and steady current exists in a electric circuit, the amount of electric charge flowing pass is the same at all points of its circuit; that is, electric charge is not lost and does not accumulate anywhere in the circuit. In other words, electric charge is conserved. Quantitatively this can be expressed by defining a physical quantity called electric current. Current may be defined as the amount of electric charge flowing pass a given point in a conductor per unit of time. That is,
current = charge/elapsed time or
I = Δq/Δt, (1)
where Δq is the quantity of electric charge measured in coulombs, Δt is the elapsed time measured in seconds that the quantity of charge Δq flows pass a given point in the conductor, and I is the symbol for the time rate of flow of the electric charge pass a given point in the conductor and is called electric current. According to this defining formula (1), the unit of electric current is coulombs per second, which is named the ampere. Several kinds of instruments have been invented to measure electric current, called ammeters (ampere measurers).

Electromotive Force. The flow of the electric charge must be caused by something in the circuit. The charge will not flow around the circuit of itself. Now if the battery is not present in the circuit there can be no current. What is it that the battery supplies? To arrive at the answer to this problem let us use an analogy. Just as a body, when free to move, will fall from a higher point to a lower point under the force of gravity, so the electric charge will move between two points, from the positive to the negative terminal of the battery under the force of an electric field. A body falls between the two points in a gravitational field because of .a difference of potential, meaning that there is difference of potential energy between the two points in the gravitational field. Similarly, a different of potential, that is, a difference of potential energy, between the two terminals of the battery. This difference of potential between the two terminals of the battery is called electromotive force (EMF) of the battery. That is, an electric battery is a source of electric potential energy. Since potential energy is the ability to do work by a field force, electric potential energy is the ability to do work by the forces of the electric field. In general, the electric potential difference between two points is defined as the quantity of work done by electric forces in moving a unit of charge from one point to the other, that is,
potential difference = work/charge
or, in symbols
V_ba = W_b→a/q. (2)
If the work is measured in joules and the charge in coulombs, then the unit of potential difference is one joule per coulomb, which is named one volt. This difference of potential is often called simply voltage. A device has been invented to measure the potential difference between two points in an electric circuit and it is called a voltmeter.

Each cell in Volta's battery has a certain small voltage across its terminals and as more and more identical cells are placed in the series, the battery becomes capable of producing greater currents in an external circuit. The essentials of an electric circuit are a source of potential difference, a connection between the terminals of the source and a "load" in which energy is expended, for example, the filament of a flashlight bulb or the heating elements of toaster. When the load in the circuit is by-passed by a connection of some kind between the terminals of the load, the circuit is said to be "short circuited," and is usually accompanied by sparking and intense heating. A switch in the connection between the source and load allows the circuit to be completed or broken. As long as the switch is open, no current can flow in the circuit and no work can be done in the load; when the switch is closed the potential difference produces a current in the entire circuit. If the load in the circuit is, for example, the filament of a light bulb, it has properties such that some of the work done by the charge moving through it is converted to heat but most of it is converted into light. The time rate that the electric energy is expended and converted into heat and light in the load may be calculated by multiplying the voltage (work per unit charge) by the current (the time rate of flow of charge) and is called electric power. In general, power is the time rate at which work is done. That is,
Power = work done/elapsed time or
P = ΔW/Δt, (3)
In the MKS system of units power is measured in watts and is equal to joules per second, since the work done ΔW is measured in joules and the elapsed time Δt in seconds. If both the potential difference across the load and the current flowing in the load is known, then the electric power can be calculated by the following formula, which is sometimes called Joule's Law. Since
VI = ΔW/Δq)(Δq/Δt) = ΔW/Δt = P,
then P = VI. (4)
The total electric energy expended and the work done in the load may be calculated by multiplying the electric power of the load by the length of time that the switch is closed. That is,
ΔW = PΔt, (5)
where electrical energy is measured in joules or more usually watt-sec, since electric power is measured in watts and time in seconds. Since this is a very small unit of electrical energy for commercial and practical use, a larger unit called the kilowatt-hour is commonly used. 1 kilowatt-hour = 3,600,000 watt-sec.

Electrical Resistance. The relationship between voltage and current in different conductors was first studied in the 1820's by Georg Simon Ohm (1787-1854), a German schoolmaster, who published his work in 1826. The ratio of the potential difference across the ends of a conductor to the current is called the resistance of the conductor. Ohm experimented with wires, which he made by hand, of different sizes and lengths and of different metals. He found that the resistance R of the wire varied directly with the length L and inversely as the cross sectional area A of the wire. That is,
R = ρ(L/A), (6)
where R is measured in ohms Ω (the capital Greek letter omega), the length L in meters, and the cross-sectional area A in square meters. The value of the constant of proportionality that is called resistivity ρ (the Greek letter rho) dependents upon the kind of metal as well as the system of units to measure the length and cross-sectional area of the conductor. The resistivity is measured in ohm·meters, as can be seen by solving formula (6) for ρ.
ρ = RA/L = ohms·meter²/meters = ohms·meters. (7)

The following table shows the approximate resistivities of some metals and non-metals at 0°C.

Substance (metals)	Resistivity (Ω·m)	Substance (non-metals)	Resistivity (Ω·m)
silver	1.47 × 10-8	carbon	4 × 10-5
copper	1.59 × 10-8	germanium	2 × 100
gold	2.27 × 10-8	silicon	3 × 104
aluminum	2.60 × 10-8	boron	1 × 106
tungsten	5.00 × 10-8	wood (maple)	3 × 108
iron	11.0 × 10-8	celluloid	4 × 1012
platinum	11.0 × 10-8	glass	1011 × 1013
constantan (60 Cu, 40 Ni)	49 × 10-8	amber	5 × 1014
mercury	94 × 10-8	sulfur	1 × 1015
Nichrome (60 Ni, 24 Fe, 16 Cr)	100 × 10-8	mica (colorless)	2 × 1015
		fused quartz	5 × 1017

Ohm's Law. Ohm's Law states that for given conductor the current in the conductor is directly proportional to the voltage across the ends of the conductor and inversely proportional to its resistance. In symbols,
I = V/R, (8)
where the voltage V is measured in volts, the current I is measured in amperes, and the resistance R is measured in ohms. The unit of electrical resistance, the ohm, is defined as that resistance which will allow a current of one ampere to flow through a conductor when the potential difference across its ends is one volt.

Joule's Law. The English physicist, James Prescott Joule (1818-1889), in his determination of the mechanical equivalence of heat, investigated the relationship between the heating effect and the current flowing in the conductor. He found that the rate of production of heat energy in a metallic conductor is directly proportional to the square of the current. Now the rate of production of energy is power, P, and is measured in the MKS system of units in joules per second, that is called the watt. Thus Joule's Law states that the power expended in a conductor is directly proportion to the square of the current flowing in it.
P = I²R. (9)
The constant proportionality is the resistance R of the conduction and is measured in the MKS units in ohms when the current I in the conductor is measured in amperes.
Since by formula (4) P = VI and, solving formula (8) for voltage V, V = IR, then
P = VI = IV = I(IR) = I²R.
Therefore, formula (4) is equivalent to formula (9), where the voltage V is the potential difference across the load in the circuit.

Series Resistance. If the load in a circuit consists of a series of resistance elements, like for example a string of series connected Christmas tree lights, then the potential difference across the entire string is equal to the sum of potential differences across each resistance element. That is,
V = V₁ + V₂ + V₃ + ..., (10)
where V is the voltage across the whole series, and V₁, V₂, V₃ and so forth, are the voltages across each resistance elements. Since by Ohm's Law,
V = IR, then V₁ = I₁R₁, V₂ = I₂R₂, V₃ = I₃R₃, and so forth.
Substituting into the equation (10), we get
IR = I₁R₁ + I₂R₂ + I₃R₃ + ....
But since the current in a series circuit is the same in each of the series element, that is,
I = I₁ = I₂ = I₃ = ..., then IR = IR₁ + IR₂ + IR₃ + ....
Therefore, dividing by the common current, we get
R = R₁ + R₂ + R₃ + .... (11)
That is, the total resistance of series of resistance elements is equal to the sum of the resistances of each element.

Parallel Resistance. If the load in a circuit consists of resistance elements connected in parallel, then the sum of the current in the branches is equal to the total current entering the parallel elements. That is,
I = I₁ + I₂ + I₃ + ..., (12)
where I is the total current entering the parallel arrangement, and I₁, I₂, I₃, and so forth, are the currents flowing in each of the branches of the parallel arrangement. Since by Ohm's Law,
I = V/R, then I₁ = V₁/R₁, I₂ = V₂/R₂, I₃ = V₃/R₃, and so forth.
Substituting into equation (12), we get
V/R = V₁/R₁ + V₂/R₂ + V₃/R₃ + ....
But since the voltage across the parallel arrangement of resistance elements is equal to the voltage across each of the branch elements, that is, V = V₁ = V₂ = V₃ = ..., then
V/R = V/R₁ + V/R₂ + V/R₃ + ....
Therefore, dividing by the common voltage V, we get
1/R = 1/R₁ + 1/R₂ + 1/R₃ + .... (13)
That is, the reciprocal of the total resistance of a parallel arrangement of resistances is equal to the sum of the reciprocals of the resistance of each element that are in parallel.

The Dry Cell. An ordinary dry cell consists essentially of a carbon rod and a zinc can separated by a moist paste containing ammonium chloride and manganese dioxide, which can be considered an electrolyte solution. When the dry cell is on the shelf with nothing connected to it, experiment shows that there is slight excess of positive charge on the carbon rod and a slight excess negative charge on the zinc can. To move a charge of one coulomb from the positive to the negative terminal through the air requires 1.5 joules of work; hence, the potential difference between the terminals of the dry cell is 1.5 joules per coulomb or 1.5 volts.