Electric and Magnetic Fields

Type of physical science: Classical physics

Field of study: Electromagnetism

Electric and magnetic fields are mathematical constructs that are useful in describing the force of interaction between charges and currents, respectively, at a distance and how electromagnetic energy is stored.

Overview

Charge is a fundamental property of matter. It is observed to exist only on particles of nonzero rest mass. Charge can be found in two varieties that are identified as positive and negative. Charges of equal sign repel one another. Charges of opposite signs attract one another.

The electrical force of attraction or repulsion was first quantitatively described by Charles-Augustin de Coulomb in 1785. He deduced correctly that the magnitude of the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between the centers of the two charges. Coulomb also determined the constant of proportionality, which involves the fundamental constant of nature called the permittivity of free space, and he determined that the force acts along the line connecting the centers of the charges.

The force law Coulomb determined now bears his name.

The electric force, like gravitation, is an example of action-at-a-distance. Two charges will experience instantaneously a Coulomb force without requiring physical contact. The greater the distance, the smaller (by the inverse square) the resulting force, and, by Newton's second law of motion [F = ma], the smaller the acceleration.

Magnetic attraction/repulsion between certain naturally occurring minerals was known from antiquity. Where magnetism is present, a moving charge experiences a sideways deflecting force perpendicular to the direction of the charge's velocity. A magnetic force does not work on a charged particle because the force is always perpendicular to the displacement of the particle. By the work-energy theorem, one can conclude that magnetic forces do not change the kinetic energy of charges. Electric forces accelerate charges, changing their kinetic energy because work is done on the charge. That is one fundamental difference in the nature of electric and magnetic forces.

Charge in motion represents a current. Charge is the cause of the Coulomb force, and current is the cause of magnetism. In the interaction of charges and currents, the charges and currents are the cause, and the forces on other charges and currents are the effect. To aid in the description of one charge interacting with another or one current with another, it is useful to introduce the concept of field vectors E and B, the electric field and the magnetic field, respectively. These vectors play the role of intermediaries in the interaction.

A field is any physical quantity that can be specified simultaneously for all points in a given region of space. A field is a mathematical construct of assigning a value to a variable that depends upon position. (In physics, that variable is usually a physical parameter of a system or an interaction.) Fields may be either scalar or vector in nature. A scalar field assigns only a number or magnitude (in a particular system of units) to a point in space, whereas a vector field assigns a magnitude and a directionality to a point in space. Examples of scalar fields in physics are temperature inside a solid or pressure within a fluid. Gravitation, fluid velocity, and both electric and magnetic fields are examples of vector fields.

Consider two charges, one located at point P and the other located at point Q. They are separated by a distance r. To describe the Coulomb force between these charges using the electric field concept, assume that the space including points P and Q is totally empty. If the first charge is brought in from infinity, it can be located at point P. The first charge sets up an electric field in the space surrounding it. Lines of the electric field point radially outward from the charge if it is positive and radially inward if it is negative. If the second charge is now brought in from infinity to point Q, it experiences a force determined by the magnitude of the two charges and the separation distance r; the charge will move inward along a line of force if the two charges are of opposite signs or move outward along a line of force if the two charges are of the same sign.

The above discussion pertained to point charges, which are of negligible extent in space. Any extended object possessing charge will also set up an electric field and exert forces on other charged objects. To determine the electric field, one must know how charge is distributed within the object and then determine the effect of each individual charge, a process known as integration over the charge distribution.

Operationally, the electric field can be defined at any region of space by means of a positive test charge q and by measuring experimentally the magnitude and direction of the force F exerted on the known test charge. The electric field E is the ratio of the force to the test charge (F/q) in the limit as q approaches zero. Because q itself is a charged particle, it will set up an electric field of its own. Thus, q should be as small as possible. Charge, however, is quantized, and its magnitude can never be less than that of the electron charge, e, without being identically zero. Thus, F = qE.

Where a high degree of symmetry is present, Gauss's law of electrostatics permits easy calculation of electric fields using a knowledge of the total enclosed charge and integral calculus.

Gauss's law states that the permittivity of the space multiplied by the total electric field flux contained within the space equals the total charge. (Flux is defined as the field strength multiplied by an element of surface area.) For example, it can be shown that for points outside a spherically symmetric distribution of charge, the electric field is equivalent to the field of a point charge whose magnitude is the same as the total charge contained within the spherical distribution.

Because of the basic difference in the nature of E and B, it is more difficult to define the magnetic field. For example, if a charged particle moving at speed v experiences a sideways deflecting force F in the presence of a magnetic field B, the relationship between these physical quantities is F = qv x B.

Whereas the units of the electric field are newtons per coulomb, the units of the magnetic field are webers per square meter, or teslas (if expressed in the International System of Units).

The space around a magnet or a current-carrying conductor is defined as the site of a magnetic field B. The lines of B are arranged in such a way that the total flux of B within a closed surface in space vanishes. This results from the fact that, unlike charge where positive charges can exist independent of negative charges, north poles are paired with south poles. No magnetic monopoles--isolated north or south poles--have ever been verified experimentally. If a charged particle enters a region of space that has both an electric field E and a magnetic field B, the total force acting on the particle, called the Lorentz force, is given by F = qE + v x B.

Applications

Theoretical descriptions of electromagnetism and practical discussions of electrical components such as capacitors and dielectrics use the concepts of electric and magnetic fields.

The space surrounding charges and currents contains electromagnetic energy. It is logical to connect the field concept with this energy and attribute to both the electric and magnetic fields the ability to store electromagnetic energy in the space between charged objects and currents.

A capacitor is a device consisting of a pair of conducting plates separated by a small distance, with the space between the plates filled either with a dielectric slab or a vacuum. The conducting plates have equal but opposite charges on them. This separation of charge sets up an electric field that is uniform and independent of distance from the plates. That electric field strength is directly proportional to the magnitude of the charge and inversely proportional to the cross-sectional area of the plates and is independent of the plate separation distance. Outside the capacitor (as can be shown by an application of Gauss's law of electrostatics), the electric field vanishes; thus, electromagnetic energy is stored only in the space between the capacitor plates.

The electric energy density can be shown to be proportional to the square of the magnitude of the electric field vector E, further justifying the association of energy storage with the space in which an electric field exists. From the electric field, energy density can be calculated, and from the energy density, the capacitance of a system of conductors can be obtained.

The magnetic analog of a capacitor is an inductor. A current-carrying loop sets up a magnetic field, the strength of which is proportional to the current flowing through the loop. If the current flow is static, the magnetic field will also be static. If the current flow is time-dependent, the magnetic field will also be time-dependent. Like the capacitor setting up an electronic field and storing electric energy, the inductor sets up a magnetic field and stores magnetic energy. The magnetic energy density is proportional to the square of the magnitude of the magnetic field vector B, verifying a symmetry in the nature of E and B. From the magnetic energy density, the inductance of a current-carrying wire can be calculated. (In the capacitor, the electric energy is stored in the electric field and not the conducting plates. In the inductor, the magnetic energy is stored in the magnetic field and not in the currents or current-carrying wires.)

Energy stored in electric and magnetic fields can be extracted to do useful electrical work in circuits to power other circuits, to charge batteries, to turn motors and generators, and the like. Energy stored in electric fields can be transformed into other forms of energy, even magnetic field energy. This effect is seen most easily when describing the behavior of an LC circuit, which is a capacitor (initially charged) connected to an inductor with very low resistance wire leads. Initially, the capacitor is charged with an amount of positive charge Q on one plate and an amount of negative charge -Q on the other plate. The electric field caused by this charge separation is at its maximum strength. Since there is no initial current flow, the magnetic field in the inductor is zero. Yet, charge is able to migrate from the capacitor plates through the wires connected to the inductor. This current will increase and set up a magnetic field in the space surrounding the inductor and will grow in strength until all the charge is drained from the capacitor plates. At this moment, the electric field E is zero and the magnetic field B is at its maximum strength. The energy stored in the magnetic field at this time is identical to the electrical field energy initially stored between the capacitor plates.

The current subsides as charge begins to be stored back on the capacitor plates. The plate that originally had positive charge now will store negative charge, and the plate that originally had negative charge now will store positive charge. This process will continue until the current dwindles down to zero and the maximum charge is restored to the capacitor plates, but in the reverse order. The electric field is back at maximum strength, but points in the direction opposite of its initial direction. All the system's energy is again stored in the electric field. The charge again bleeds off the capacitor, building up current flow through the inductor until no charge remains on the capacitor plates and the current is maximized once more. The magnetic field is again at its maximum strength, but the field lines are reversed from the direction they had when the field strength was maximized previously. All the circuit energy is stored in the magnetic field B.

The current dwindles again, and charge builds up on the capacitor plates, putting positive charge on the plate that was originally positive and negative charge on the plate that was originally negative. This process continues until the current vanishes and the capacitor is restored to its initial condition and all the energy is once again stored in the electric field. The LC circuit continues to oscillate back and forth in this manner, with electric field energy transformed conservatively into magnetic field energy and back. Oscillating electric and magnetic fields form the basis of the nature of electromagnetic radiation. Visible light is only one small portion of the electromagnetic spectrum. All electromagnetic radiations--from high-energy γ rays to low-energy radio waves--share certain characteristics. They all travel at the speed of light c, nearly 300 million meters per second in a vacuum. In addition, they consist of mutually perpendicular oscillating electric and magnetic fields propagating through space in a direction perpendicular to both the magnetic and electric fields. The frequency of oscillation of both fields is the same, and the energy of the radiation is proportional to that frequency. Those field strengths are related to each other and can be described by James Clerk Maxwell's equations of magnetism that relate electric fields to magnetic fields and their spatial and temporal variations.

Context

Action-at-a-distance, a direct interaction between particles without direct contact, was a concept that was difficult for many in the scientific community to accept. This tendency slowed progress in research into the true nature of gravitation and electric and magnetic forces. The introduction of the field concept makes use of the space between two interacting particles. The field plays an intermediary role in the application of a force of attraction or repulsion between two interacting particles. It is the force that remains the true physical result of the interaction between two particles, and the two particles remain the cause of the interaction.

In the case of the electric field, the space surrounding a given charged particle is thought to be altered in such a manner that if a second charge is placed in that space, it experiences a force. By Newton's law of action and reaction (third law of motion), the force of the first charge on the second is equal but opposite to the force of the second charge on the first; therefore, the field concept must be symmetric. The electric force of interaction between these two particles can also be described as the second particle alters the space around it in such a way as to exert a force on the first particle at its position in the space around the second particle.

The concept of charge, in the context of static electricity, was known to the ancients.

Static attraction or repulsion between various materials was observed, but the nature of charge and its relation to current and naturally occurring lightning remained a mystery until the eighteenth century, when a large group of impressive researchers began investigating electricity.

Naturally occurring magnetism was also known, but not understood. Indeed, the term "magnetism" originates from the mineral magnetite, found in abundance in the Magnesia district of Asia Minor. Naturally occurring magnetism was utilized in navigation with the invention of primitive compasses that would indicate the direction of north.

Electric and magnetic forces were studied as separate physical phenomena until the early nineteenth century. The benchmark experiment that led ultimately to the unification of electricity and magnetism--under the term electromagnetism--was the observation by Hans Christian Ørsted that a current-carrying wire produces effects similar to naturally occurring magnetism. He found that if a coil of wire is wrapped around a compass and current is passed through the coil, the compass needle will be disturbed from the direction of north and with sufficient current will point in a direction perpendicular to the plane of the coil.

Maxwell summarized the work of other experimentalists into a complete theoretical framework that is referred to as Maxwell's equations of electromagnetism. It is a set of four laws that holds a position in physics equivalent in importance to Newton's laws of motion. Maxwell's equations explain how time-varying electric fields can create magnetic fields and time-varying magnetic fields can create electric fields. Without the field concept, the unification of electromagnetism would have been more difficult.

Principal terms

ACTION-AT-A-DISTANCE: electromagnetic forces of interactions that occur without physical contact between charges or currents

CHARGE: a fundamental property of material particles that manifests itself by attraction or repulsion to other material particles possessing charge

COULOMB FORCE: the force of interaction involving charged particles separated by distance

CURRENT: charges in motion representing a time rate of change of charge

ELECTRIC FIELD: a vector field describing the Coulomb force per unit of positive charge

LORENTZ FORCE: the combined effect of an electric and magnetic field acting on a moving charge

MAGNETIC FIELD: a vector field describing the force generated by currents of either macroscopic or microscopic origin

SCALAR: any mathematical or physical quantity that assigns a particular magnitude to a point in space

VECTOR: any mathematical or physical quantity that has both a magnitude and direction

Bibliography

Frankyl, Daniel R. ELECTROMAGNETIC THEORY. Englewood Cliffs, N.J.: Prentice-Hall, 1986. Although targeted for the undergraduate physics major, this text contains qualitative sections on the significance of electric and magnetic fields in understanding the electromagnetic force of interaction between charged particles in motion.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 3d rev. ed. New York: John Wiley & Sons, 1988. This version of the classic undergraduate physics text contains descriptive discussions of current topics in physics research as well as thorough coverage of basic physical concepts. Includes excellent descriptions of electromagnetic fields and their interactions with matter. Well illustrated, with numerous sample problems.

Lorrain, Paul, and Dale Corson. ELECTROMAGNETIC FIELDS AND WAVES. San Francisco: W. H. Freeman, 1990. A classic text geared for undergraduate instruction in electromagnetism. Many sections are accessible to the layperson with moderate mathematical skills. Descriptions of field concepts are excellent.

Nayfeh, Munir H., and Morton K. Brussel. ELECTRICITY AND MAGNETISM. New York: John Wiley & Sons, 1985. Undergraduate level instruction in mathematical and experimental aspects of electromagnetism. Accessible to those with modest skills in calculus of vectors.

Ohanian, Hans C. PHYSICS. New York: W. W. Norton, 1985. Although calculus-based, the text is not mathematically rigorous. Accessible to those with modest math skills. Excellent descriptions and illustrations of difficult physical concepts. Provides practical examples.

Reitz, John R., and Frederick J. Milford. FOUNDATIONS OF ELECTROMAGNETIC THEORY. Reading, Mass.: Addison-Wesley, 1987. A classic text for undergraduate instruction of electromagnetism. Excellent field concept descriptions. Accessible to the nonscience major with moderate mathematical skills.

Tomboulian, D. H. ELECTRIC AND MAGNETIC FIELDS. New York: Harcourt, Brace & World, 1965. Although somewhat dated, this text is ideal for the amateur interested in furthering an understanding of electromagnetism without requiring higher-level mathematics. Well illustrated, with diagrams and sample problems.

Wangness, Ronald K. ELECTROMAGNETIC FIELDS. New York: John Wiley & Sons, 1986. Highly mathematical, but descriptions of basic concepts are illuminating. For the seriously interested reader with good calculus skills.

Wilson, Jerry D., and John Kinard. COLLEGE PHYSICS. Boston: Allyn & Bacon, 1990. Basic text; excellent for those not familiar with calculus. More qualitative than rigorous. Well illustrated. Contains a thorough description of basic concepts without resorting to advanced mathematics. Excellent for high school physics instruction.

Electric and magnetic fields

Charges and Currents

The Fundamental Constants of Nature

Generating and Detecting Electromagnetic Waves

Forces on Charges and Currents

The Measurement of Magnetic Fields