The Electromagnetic Fields

The significance of E, D, B and H


Introduction

The purpose of this article is to clarify the meanings of the electromagnetic field vectors and to explain the Gaussian electromagnetic units. Please refer to the References for matters not treated in detail here. Familiarity with vector calculus is essential. In fact, vector calculus is normally introduced in the Physics curriculum through electromagnetic fields.

It is observed that electrically charged bodies exert strong mutual forces, as do electric currents. The idealizations of point charges and linear currents are very useful in explaining these phenomena. The charges and currents are considered to produce states of space called fields, which then act on other charges and currents to produce forces.

A point electric charge of magnitude q exerts on a charge of magnitude q' at a distance of r cm a force along the line joining the charges of magnitude F = qq'/r2 dynes. This relation allows the absolute measurement of the quantity of charge in electrostatic units (esu). The magnitude of the electric field at a distance r cm from a point charge of q esu is E = q/r2 dyne/esu. If a unit charge moves d cm in the direction of the field, the work done is Ed erg/esu or Ed statvolt, so that E is in statvolt/cm.

Two parallel linear currents I and I' a distance d cm apart attract with a force of 2II'/d dynes/cm, which defines the currents in electromagnetic units (emu). If we measure the currents I,I' in esu/sec, then the force is 2II'/c2d. That is, we can find experimentally that I in emu is I in esu divided by about c = 3 x 10SUP>10 cm/sec, which is the speed of light. In Gaussian units, current is expressed in esu/s. Where necessary, this current is divided by c.

Units based on the form of Coulomb's Law we have used are often prefixed stat-, or are called esu, while those based on the expression for magnitic force are prefixed ab- or are called emu. The usual SI MKS units are based on emu, with various factors of 10. A statcoulomb is c abcoulombs.The coulomb is 0.1 abcoulomb, while a volt is joule/coulomb. This allows us to express statvolts in terms of ordinary volts, since 1 statvolt = 1 erg/esu x 1 j/107 erg x c/10 esu = 300 j/coul = 300 volt. In the Gaussian system, the constant c is used to convert from esu to emu, so that currents in emu appear as I/c, where I is the current in esu.

The Field Equations

The electromagnetic field equations for the electric field E and magnetic field B. For the moment, we can neglect the time derivatives on the right-hand sides of equations (b) and (d) and consider static fields in empty space. Equations (a) and (b) express the sources of the fields in charges and currents.

If we integrate equation (a) over a sphere of radius r with a total charge q contained in a region very close to the origin, using the divergence theorem, we find 4πr2E = 4πq, where E is the magnitude of the outward radial field, so that E = q/r2, in statvolt/cm if q is in esu. This is Coulomb's Law.

If we integrate equation (b) over a circle of radius r with a linear current I flowing in a wire through the centre of the circle and normal to the plane of the circle, we find using Stokes' theorem that 2πrB = 4πI/c, or B = 2I/c. This is Ampere's Law. Equation (b) follows directly from the Biot and Savart integral for the magnetic field of a current.

The scalar potential φ is introduced as shown in the figure at the left, starting from Coulomb's Law. The figure shows the usual conventions, with source point x and field point x' referred to the origin O, and the relative distance r. The del operator can operate on either the x or the x' coordiates of the function 1/r, with the results differing only in sign. This can be very useful as a del' can be expressed as a del and removed outside the integral sign, as is done here. This only works for 1/r, of course. It is much easier to calculate the field components from one scalar function than to express them directly. If the electric field is the gradient of a scalar potential, then the curl of the electric field is identically zero, which satisfies field equation (d) in the static case.

The magnetic field is treated in an analogous way in the figure at the right, starting from the Biot-Savart integral. Here the magnetic field is derived from a vector potential A, which is more easily found than the field itself. The field B is measured in gauss, and its flux in maxwells. That is, 1 gauss = 1 maxwell/cm2. If the magnetic field is expressed as the curl of a vector potential, then field equation (c) is identically satisfied.

Faraday's Law, equation (d) is a useful way to measure magnetic fields. Applying Stokes' Theorem, the line integral of the electric field (the emf) around a closed loop is equal to the time rate of change of the magnetic flux through the loop. For a change of 1 maxwell/sec, the line integral of the electric field is 1/c statvolts. For a change of 108 maxwells/sec (a weber), the voltage is 1/300 statvolt or 1 volt. 108 maxwells is a field of 104 gauss over an area of 104 cm2, so 1 tesla, or 1 weber/m2, is 104 gauss. Note that in the Gaussian equations, c appears in Faraday's law to convert abvolts to statvolts.

A maxwell is also called a line, referring to the graphical representation of the field by lines of force, which are normal to equipotential surfaces and give the direction of the field at a point. Of course, such lines do not actually exist, and are only a useful analogy. The field is greater where the lines are closer together. Lines of B do not begin or end, since div B = 0, while lines of H may begin or end on equivalent magnetization charge, since div H = -4πdiv M. Field stresses are tension along the lines of force, compression normal to them, so that field plots suggest the forces that are transmitted by the field.

Where the magnetic field produces a force, or is an applied field causing magnetization, it may be expressed in oersted, which is numerically equal to the flux density in gauss. We shall see that this unit is generally used for the field H, while gauss is used for B, but this represents no fundamental difference between the fields.

The Field Equations in the Presence of Matter

Matter contains charges and currents, and can significantly modify the electromagnetic field. The charges are normally equal and opposite, and the currents are atomic in size. In the absence of mobile charges, the electric field produces a polarization P, which is dipole moment per unit volume, by stretching charges in opposite directions, and the magnetic field can induce atomic currents or align electronic magnetic moments, creating a magnetization M, which is magnetic dipole moment per unit volume. These effects cause charges and currents to arise that are also sources of the electromagnetic fields E and B. We can separate charges into free charges and polarization charges, and currents into free currents and magnetization currents. It is convenient to define fields whose sources are only the free charges and currents, not those due to polarization.

The polarization charge density is derived in the figure at the left. We begin with the potential of a point dipole p, which is the limit of the product of the charge q times the distance from the negative to the positive charge a, as this distance approaches zero: p = lim qa. The dipole potential is the difference of the potentials due to +q and -q, which in the limit is just pgrad(1/r), as shown. The potential of a polarization density P is just the superposition of the potentials due to Pd3x'. We now move the differentiation from 1/r to P by an integration by parts, using the vector identity shown. The integral of div(P/r) can be transformed to a surface integral over a surface, which vanishes as the surface goes to infinity. The result is the potential due to a charge density -div P which is equivalent to the polarization. If we define D = E + 4πP, then div D = 4π(ρ - ρp) = 4πρt, which is the field we desire. D is called the "electric displacement", from the old mechanical model of the electromagnetic field.

The analogous result for the magnetic field is shown at the right. We begin with the vector potential for a magnetic dipole. Note that it is the same as in the electric case, with the scalar product replaced by the vector product. If you have not done so, calculate B from this vector potential and observe that the usual dipole fields result. The magnetic moment m is the product of a current I times the area of the current, with the direction given by the usual right-hand rule. The vector potential due to a distribution of dipole moment, the magnetization, then follows. Again, we integrate by parts using a vector identity. The volume integral of curl (M/r) can again be transformed to a surface integral over a bounding surface that vanishes when the surface goes to infinity. By comparison with the vector potential of a current density, the current density c curl M creates a field the same as that of the magnetization. If we define H = B - 4πM, then curl H = (4π/c)(J - Jm) = (4π/c)Jt, which is the desired field. H is usually called the "magnetic intensity" and B the "magnetic flux density".

Note carefully that D and H are analogous, not E and H, as is traditionally supposed. The MKS system, and its SI successor, which are now generally used, contain significant fossilized traces of the mechanical models of the electromagnetic field, which have been shown to be erroneous, especially in view of relativity. In this picture, E and H were regarded as driving funtions, and D and B as their results. They were even given different units, connected by the constants ε0 and μ0, the "permittivity" and "permeability" of free space, which have no physical reality, being only a way to factor the value of the speed of light. Though the fields are identical in free space, they have different units and magnitudes in the MKS system. This has caused significant mischief in magnetism, and even incorrect conclusions. In many cases, however, E and H can be regarded as applied fields, whose effects are to be studied.

The field equations can now be written as shown at the left. This is the usual form in which Maxwell's Equations are presented. J and ρ are here the free current and charge densities, not the total ones. D appears in Ampere's Law, since it includes not only the displacement current due to E, but also the actual current due to a change in polarization. The name "displacement current" indeed comes from this use, though it includes two very different effects. Note that besides the geometrical factor 4π, the only physical constant that appears is the speed of light, c, which is quite appropriate. It should be remembered that Gaussian units are based on the cgs system of units, so c is about 3 x 1010 cm/sec.

E is measured in statvolt/cm, which is equal to 30 kV/m. D and E have the same units, but statvolt/cm = dyne/esu, and dyne = (esu/cm)2, so the dimensions of D is also esu/cm2, which is appropriate for both E and D, since div D = 4πρ.

The electric polarization is quite commonly proportional to the applied electric field, P = χE, where χ is the electric susceptibility. The analysis of this relation is actuallly rather involved, in relating it to the microscopic structure of the material. The microscopic fields acting on the molecules are not necessarily the macroscopic fields, and the molecules interact with each other. The susceptibility usually depends on frequency, and there may be energy losses. Nevertheless, in many cases a simple constant value is a useful approximation. Then D = (1 + 4&piχ)E = κE, where κ is called the dielectric constant. In many practical cases it is considerably different from unity.

The magnetic response of matter is quite different. In most materials, the response to an applied magnetic field is a weak magnetization in the direction of the applied field due to alignment of permanent molecular magnetic moments, called paramagnetism, or a weak magnetization opposite to the field direction due to induced molecular currents, called diamagnetism. If the applied field is H, then M = χH, where the magnetic susceptibility χ is positive for paramagnetism and negative for diamagnetism, though always a small number. Then, B = (1 + 4πχ)H = μH, where μ is the permeability.

In ferromagnetic materials, the magnetic moments of certain electrons are locked parallel in macroscopic domains, which are strongly magnetized. When an external field is applied, these domains become aligned by several mechanisms, including movement of domain walls. The result is a very strong magnetization. This process is never linear, exhibiting a characteristic dependence on an alternating applied field in the form of a hysteresis loop.

Two Illustrative Examples

The use of the fields E, D, H and B can be illustrated by two simple examples, the parallel-plate capacitor and Rowland's ring, both of which can easily be studued in the laboratory.

The parallel-plate capacitor is shown at the right. The thickness of the dielectric betweent the plates can be made much smaller than the lateral extension of the plates, so that the fringing of the fields around the boundary of the plates is a very small part of the total field volume, in which the fields are uniform. A difference of potential V is established between the plates. Since there is no charge between the plates, the field must be constant there, and so its magnitude must be E = V/d. If the dielectric is linear with dielectric constant κ, the D = κV/d. This field terminates normally on the plates, and applying div D = 4πρ, there must be a positive charge density κV/4πd on the left-hand plate and an equal and opposite negative charge density on the right-hand plate. The total amount of this free charge is Q = (κA/4πd)V. The capacitance C of the system is defined by Q = CV, so C = κA/4πd cm.

The Gaussian (and esu) unit of capacitance is the cm, which is statcoulomb/statvolt. The SI farad is coulomb/volt, so we have 1 cm = 1 statcoul/statvolt = (10/c)coul/300 volt = (1/9 x 1011) F = 1.11 pF, a very conveniently-sized unit. The potential of a sphere of radius a carrying a charge Q is V = Q/a, so the capacitance of the sphere is just equal to its radius a. This potential is, of course, relative to zero at infinity, and represents the work per esu required to bring a unit charge in from infinity.

Note that there is a polarization charge at the surface of the dielectric opposite in sign to the free charge on the plate in front of it. The amount of this charge is -(1 - 1/κ)Q, so the net charge at a plate is Q/κ, just right to give a field E = V/d.

Suppose we have a permanently polarized dielectric with a polarization P normal to its surface, placed between two conducting plates. This is called an electret. Since there is no free charge, D = 0 between the plates. Therefore, E = -4πP. The potential difference between the plates is V = Ed = -4πPd. Note that the electric field is opposite in direction to the polarization, so it will act to reduce the polarization.

Rowland's ring is a torus of mean radius a and cross-sectional area A wound with N turns if wire that produces the applied field, denoted by H. There is no div M, so no field due to magnetization charge, so H due to the current in the N turns is always the applied field. There is usually a second winding of N' turns for the purpose of measuring the magnetic flux in the core when the flux is reversed. We shall assume for simplicity that the material of the core has a permeability μ, though the ring is used mainly for ferromagnetic cores that have a nonlinear relation between H and B. The ring can, in fact, be used for measuring this nonlinear relation. The curvature of the ring has a relatively small effect on the fields.

The applied field H is found by using Ampere's Law. The line integral of H around the mean circumference of the ring equals 4πNI/c, where I/c is the current in emu, and 10 times the current in amperes. The line integral is the magnetomotive force, mmf, in oersted-cm or gilbert. For a linear magnetic material (which only exists to a very rough approximation), B = μH. In general, B must be found from the hysteresis curve that gives the relation between B and H. The flux in the core is Φ = BA. The flux-turns linking the winding N' is given. Thhe time derivative of this flux linkage will give the induced voltage. If N' is just the main winding N, we have the flux linkage responsible for the self-inductance of this winding.

The winding N' is used to measure changes in the flux in the core. The voltage induced in this winding when the flux changes is e = (1/c)N'dφ/dt statvolts when φ is in maxwells. The induced emf in volts is 300e, so if the resistance of this winding and the circuit connected to it is R, the induced current is i = (300N'/Rc)dφ/dt amperes. The total charge that flows is then q = (300N'/Rc)Δφ coulombs. An integrating device is then used to measure q; this may be a ballistic galvanometer that responds to an angular impulse produced by the current, or an integrating op-amp circuit. The time constant of the galvanometer or the integrator should be large compared to the time in which the change of flux occurrs.

The exciting current I should be able to be easily reversed as well as controllable so the current can be varied quickly. The sample is demagnetized by subjecting it to a reversing field that is gradually reduced to zero. Then the magnetization curve can be traced by increasing the current in steps up to the desired maximum flux for the desired hysteresis loop. Then the current is decreased in steps following the upper half of the hysteresis curve to the same negative flux, and then increasing the current in steps back to the starting value, tracing the other half of the curve. This was, at one time, an excellent laboratory exercise that could be carried out with good accuracy.

A Google search for "Rowland's Ring" found a good picture of the device, but no good description of how it is used in any source. In fact, several comments were very careless and showed that the authors had no good idea of its use. A transformer core makes an available substitute, and may even have two windings already wound. Transformer iron is, however, specially selected to have small hysteresis loss and may not have a very satisfying curve to study. It may not be easy to determine the number of turns on a pre-wound core.

The article on Magnetostatics shows how to display a hysteresis curve on an oscilloscope, using an op-amp integrator. A typical transformer core material is 4% silicon steel, which is magnetically soft while having a high resistivity that discourages eddy currents. This steel begins to saturate at about 10,000 gauss when the magnetizing field is about 3 oersted. If the magnetizing field is removed, the remanent flux density is about 8100 gauss. The flux density is reduced to zero by a reverse field of 0.4 oersted, the coercive force. This steel saturates at just below 16,000 gauss, for a magnetizing field of 100 oersted. The incremental permeability reaches a maximum of about 5500 at flux density of 5500 gauss.

The Gaussian unit of magnetic flux density is the gauss, which is 1 maxwell or line per cm2. The integrated form of Faraday's Law, curl E = (1/c)dB/dt is ∫E·dl = (1/c)dΦ/dt, where Φ is the flux libnkage, the integral of the flux density times the number of turns linking the flux. This quantity is the electromotive force (emf) in statvolt, since E is statvolt/cm. Therefore, a time rate of change of flux linkage of 1 line/sec produces an emf of 1/c statvolt. If the rate of change is 108 lines/sec, the induced emf is 1/300 statvolt, which we know is 1 volt. In the MKS system, the factor 1/c is absent in Faraday's Law and a time rate of change of flux of 1 weber/sec induces an emf of 1 volt. Therefore, 1 weber = 108 line. A tesla is weber/m2, or 108 line/104cm2 = 104 gauss. These relations are the best way to keep the relations between these units in mind. The factor c in the Gaussian equations converts emu to esu. In the emu system (whichis closely related to MKS units) 1 line/s induces an emf of 1 emu of voltage, or abvolt, which is 10-8 volt.

The inductance is the ratio of the flux linkage to the current that produces it. In this case, L is the inductance in emu, maxwells/abamp. Since 108 maxwells is a weber, and 10 amperes is an abamp, we see that the emu is 10-9 weber/ampere, or henry. That is, the emu is a nH, a rather practical value. On the other hand, we may define the inductance by the equation e = L dI/dt. In this case, we must divide both the flux linkage and the current by c, so this definition gives a Gaussian L, statvolt-sec/statampere equal to c2Lemu, or 9 x 1011 henry. Jackson gives a rather obscure discussion of Gaussian units of inductance on pp. 820-821.

In traditional American engineering practice, flux density was expressed in kilolines per square inch, which was 155 gauss, and magnetizing force in ampere-turns per inch, or 0.495 oe (0.4π/2.54 = 0.495). Permeability is always gauss/oersted, or 313 times the ratio of kilolines per sq. in. to ampere-turns per inch.

It is easy to see that 1 oe = 2.02 ampere-turns/in = 79.577 ampere-turns/m, where ampere-turns/m is the MKS unit of magnetizing force. The flux density in vacuum is found by multiplying this value by 4π x 10-7, called "the permeability of free space" but actually just a numerical facor converting units. The result is 10-4 tesla, which is indeed just 1 gauss. Although the term "magnetizing force" is used for H, and different units of oersted or a-t/m, this is just the same magnetic field as B in free space.

If we define the volume magnetic susceptibility as the ratio of the magnetization per unit volume to the magnetizing force in either cgs or SI units, then a unit cgs susceptibility means that a magnetization of 1 abamp-cm2/cm3 is produced by a field of 1 oe. Converting to SI units, a magnetization of 1000 A/m is produced by 79.577 A/m, and the ratio of these is 1000/79.577 = 4π, so the SI susceptibility is, in general, 4π times the cgs susceptibility, and both are dimensionless. This can be a source of confusion in tabulated values. This relation follows directly by comparing the definitions B = (1 + 4πχ)H and B = μ0(1 + χ)H in the two systems of units.

Consider a permanant magnet made from a rigidly magnetized cylinder of uniform magnetization M. Outside the cylinder, the magnetic field is the familiar dipole field produced by a moment MV, where V is the volume of the cylinder. At the ends of the magnet there is an equivalent magnetic charge -div M that is the source of a field H that is the same as the field B outside the magnet, but is opposed to the magnetization inside the magnet. This field is a demagnetizing field that can affect the magnetization. The effect of a magnet "keeper" is easily seen--most of the demagnetizing field passes through it as the equivalent magnetic charges are compensated and the demagnetizing field is reduced to a small value. This also shows why disturbing a magnetic circuit (as in a permanent-nmagnet motor) can have a detrimental effect on the magnetization. When a long magnet is bent into a "U" form the dipole moments of the two legs nearly cancel and the external field is concentrated between the ends of the legs. If the ends of the legs are brought together, the external field vanishes and we have something similar to Rowland's Ring.

An instructive demonstration, apparently due to Faraday, but not often seen these days, is shown at the right. A long bar magnet is mounted to rotate about its axis. A wire attached to the middle of the magnet and dipping in an annular channel with mercury passes down to the end of the magnet, where there is a mercury contact. When a current I passes through this wire, the force exerted by the magnetic field causes the magnet to rotate in the direction indicated about its long axis. This is a unipolar motor, with no commutator or other switching necessary. If the magnet were rotated by an external torque, a current would be induced, and a unipolar generator would result, with no relative motion between the horizontal wires and the magnetic field lines imagined as fixed in the magnet, so no "cutting" of field lines.

References and Notes

In the text of this page, we use grad f for del f, div a for del· a and curl a for del X a, since the browser lacks the del character.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (New York: John Wiley & Sons, 1975). This excellent and authoritative work is my standard reference in these matters.

M. Abraham and R. Becker, Classical Electricity and Magnetism, trans. of 8th German Edition (London: Blackie and Son, 1947 and New York: Dover Publ.). Includes the elements of vector calculus.


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Composed by J. B. Calvert
Created 20 November 2010
Last revised 29 November 2010