The aim of exact science is to reduce the problems of nature to thedetermination of quantities by operations with numbers. James Clerk Maxwell(1831-1879) On Faraday's Lines of Force (1856)
Arguably, the original equations of Maxwell (1864) were essentiallythe so-called macroscopic equations, which describe electromagnetism in a dense medium. The microscopic approach (which is now standard) is due to H.A. Lorentz (1853-1928). Lorentz showed how the introduction of densities ofpolarization and magnetization reduces the macroscopic equations ("in matter") to the more fundamentalmicroscopic ones ("in vacuum") statedbelow.
Except in thefirst article,we consider only one flavor of electromagnetic quantities and use onlythe MKSA units introduced byGiovanni Giorgi(1871-1950) in 1901, which are the basis of all modern SI electrical units: ampere (A), ohm (), coulomb (C),volt (V), tesla (T), farad (F), henry (H), weber (Wb)...
(2005-07-22)
A science which hesitates to forget its founders is lost. Alfred NorthWhitehead (1861-1947)
The first consistent system of mechanical units was themeter-gram-second system advocated by CarlFriedrich Gauss in 1832. It was used by Gauss and Weber (c.1850)in the first definitions of electromagnetic units in absolute terms.
However, the term Gaussian system now refers to a particularmix of electrical C.G.S. units (discussed below) once dominantin theoretical investigations.
James Clerk Maxwell himself was instrumental in bringing about the cgs system in 1874 (centimeter-gram-second). Two sets of electrical units were made part of the system. An enduring confusion results from the fact that the quantities measured by thesedifferent units have different definitions (in modern terms, for example,the magnetic quantity now denotedB could be eitherB or cB). Following Maxwell's own vocabulary, it's customary to speak of either electrostatic units (esu) or electromagnetic units (emu). However, one must appreciate the obscure fact that these two are notonly different system of units, they are different traditions in which symbols may have different meanings...
At first, no C.G.S. electromagnetic units had a specific name. On August 25th, 1900, theInternationalElectrical Congress (IEC) adopted 2 names:
Gauss for the CGS unit of magnetic field (B) : 1 G = 10 4 T.
Maxwell for the CGS unit of magnetic flux () : 1 Mx = 10 8 Wb.
The maxwell, still known as a line of force,is called abweber (abWb) using the later naming of CGS electrical units after theirMKSA counterparts. Likewise, the gauss (1 maxwell per square centimeter) is also called abtesla (abT). For electrostatic CGS units (esu) the prefix stat- is used instead...
In 1930, the Advisory Committee on Nomenclature of the IEC adopted the gilbert (Gb) as a CGS-emu unit equal to the magnetomotive force around the border of a surface through which flows a current of (1/4) abA. The relevant values in SI units are:
1 abA = 10 A 1 Gb = (10/4) A-t = 0.795774715459... A-t 1 A-t = 1 A
The last expression is to say that no distinction is made in SI units betweenan ampere-turn and an ampere. Although the gilbert seems obsolete, the oersted (equal to one gilbertper centimeter) is still very much alivein the trade as a unit ofmagnetization (density of magnetic dipole momentper unit volume) and/or magnetic field strength (the vectorial quantityusually denotedH). The oersted was introduced by the IEC in the plenary convention at Oslo, in 1930.
Electrodynamic units are now based on a properindependent electrical unit,as advocated by the Italian engineer Giovanni Giorgi (1871-1950) in 1901: The addition of the ampère to the MKS system has turned itinto a consistent 4-dimensional system (MKSA)which is the foundation for modern SI units.
Paradoxically, this mess comes from a great clarification of Maxwell's: The ratio of the emu value to the esu value of a given fieldis equal to the speed of light (c = 299792458 m/s). Scholars from bygone days should be creditedfor accomplishing so much in spite of such confusing systems.
(2005-07-15) The Lorentz force on a test particle defines the electromagnetic field(s).
The expression of the Lorentz force introduced here defines dynamically the fieldswhich are governed byMaxwell's equations,as presented further down. Neither of these two statements is a logical consequence of the other.
In electrostatics, the electric field E present at the location of a particle of charge q summarizes the influence of all other electric charges, by stating thatthe particle is submitted to an electrostatic force equal to q E. This defines E.
This concept may be extended to magnetostatics for a moving test particle. More generally, the electromagnetic fields need not beconstant in the following expression of the force acting on a particle of charge q moving at velocity v.
The Lorentz Force (1892)
F = q (E +vB )
The average force exerted per unit of volume may thus be expressed in terms ofthe density of charge and the density of currentj.
Density of Force
F / V = E + jB
Another way to define the magnetic fieldB (best called "magnetic induction") would involve the concept of a pointlike magnetic dipole.
Torque on a Magnetic Dipolem
mB
Potential Energy of a Dipolem
m . B
The force exerted on a dipoleis grad (m.B). It vanishes in a uniform field.
(2005-07-18) Coulomb's inverse square law translates into the local differential property of the field expressed by Gauss,namely: divE =o
The SI unit of electric charge is named after the French military engineer CharlesAugustin de Coulomb (1736-1806). Using a torsion balance, Coulomb discovered, in 1785, that theelectrostaticforce between two charged particles is proportional to each charge,and inversely proportional to the square of the distance between them. In modern terms, Coulomb's Law reads:
Electrostatic Forcebetween Two Charged Particles
||F || =
| qq' |
4o r 2
The coefficient of proportionality denoted 1/4o (to match the modern conventions about the rest of electromagnetism) is called Coulomb's constant and is roughly equal to 9 109 if SI units are used (forces in newtons,electric charges in coulombs and distances in meters). More precisely, the modern definitions of the units of electricity (ampere) and distance (meter) give Coulomb's constant an exact value in SI units whose digits are thesame as the square of the speed of light (itself exactly equal to 299792458 m/s because of the way the meter is defined nowadays):
1
= 8.9875517873681764 10 9 m / F 9 10 9 N. m 2 / C 2
4o
The direction of the electrostatic forceis on the line joining the two charges. The force is repulsive between charges of the same sign (both negative or both positive). It's attractive between charges of unlike signs.
In the language of fields introducedabove,all of the above is summarized by the followingexpression, which gives the electrostatic field E produced at position r by a motionless particle of charge q located at the origin:
Electrostatic Field of aPoint Charge at the Origin
E =
qr
4o r 3
Since r / r 3 is the opposite of the gradient of 1/r, we may rewrite this as :
E = grad
where =
q
4o r
The additivity of forces means that the contributions to the local field E of many remote charges are additive too. The electrostatic potential we just introduced may thus be computed additively aswell. This leads to the following formula, which reduces the computation ofa three-dimensional electrostatic field to the integrationof a scalar over any static distribution of charges:
The Electrostatic Field Eand Scalar Potential
E = grad
where (r) =
(s)
d3s
4o||r - s ||
The above static expression of E would have to becompleted with a dynamic quantity (namely A/t, as discussedbelow) in the nonstatic case governed by the full set ofMaxwell's equations. Also, the dynamical scalar potential involves amore delicate integration than the above one.
In 1813,Gauss bypassedboth dynamical caveats with a local differential expression, also valid in electrodynamics :
divE =
o
A similar differential relation had been obtained by Lagrange in 1764 for Newtonian gravity (which also obeys an inverse square law). This can be established with elementary methods...
One way to do so is to approximate any distribution of chargesby a sum of pointlike sources: For each point charge q, we can check that theaboveelectric field has a zero divergence away from the source. Then, we observe that our relation does hold on the average in any tiny sphere centered on the source, becausethe integral of the divergence is the flux of E through the surface of such a sphere, which is readily seen to beequal to q /o
Gauss's Theorem of Electrostatics (1813)
In electrostatics, we call Gauss's Theorem theintegral equivalentof the above differential relation, namely:
Q /o = V (o) dV = SE . dS
This states that the outward flux of the electric field E through a surface bounding any given volume is equal to the electric charge Q contained in that volume, divided by the permittivity o.
Thenext section features a typicalexample of the use of Gauss's Theorem.
Another nice consequence is that the field outside any distribution of charge with spherical symmetry has thesame expression as the field which would be produced ifthe entire charge was concentrated at the center.
(2005-07-20) [ electrostatics, or low frequency ] The static charges on conductors are proportional to their potentials.
Consider an horizontal foil carrying a superficial charge of C/m2. Let's limit ourselves to pointsthat are close enough to [the center of] the plate to make it look practically infinite. Symmetries imply that the field is vertical (the electrical flux through any vertical surface vanishes) and that its value depends onlyon the altitude z above the plate (also, if it's E at altitude z, then it's -E at altitude -z).
Let's apply Gauss's theorem to a vertical cylinderwhose horizontal bases are above andbelow the foil, each having area . This pillbox contains a charge and the flux out of it is 2 E . Therefore, we obtain for E a constant value,which does not depend onthe distance z to the plate: E = ½ o.
Capacitor consisting of two parallel plates :
For two parallel foils with opposite charges, the situation isthe superposition of two distributions of the type we just discussed: This means an electric field which vanishes outside of the plates,but has twice the above value between them.
Assuming a small enough distance between two plates of a large surface area , the above analysis is supposed to be good enough for most points between the plates (what happens close to the edges is thus negligible). The whole thing is called a capacitor and the following quantity is its electric capacity.
Capacity of Two Parallel Plates
C =
o
Because E = o = qo = z , the difference U between the potentials of the two plates is qo = q/C. In other words:
Charge on a Capacitor's Plate
q = C U
This is a general relation. In a static (ornearly static) situation, the potential is the same throughout the conductive material of each plate. The proportionality between the field and its sources implythat the charge q on one plate is proportional tothe difference of potential between the two plates. We define the capacity as the relevant coefficient of proportionality.
Permittivity of Dielectric Materials :
The above holds only if the space between the capacitor's plates is empty (air being a fairly good approximation for emptiness). In practice, a dielectric material may be used instead, which behavesnearly as the vacuum would if it had a differentpermittivity. This turns the above formula into the following one. In electrodynamics, the permittivity may depend a lot on frequency.
C =
A capacity is times a geometrical factor, homogeneous to a length.
The SI unit of capacity is called the farad (1 F = 1 C/V) in honor ofMichael Faraday. It's such a large unit that only its submultiples (F, nF, pF) are used.
(2008-03-24)
Consider the electric field created by static chargeslocated near the origin. The electric potential (r) seen by an observer located atposition r is:
is the angle between s and r. The Legendre polynomials(A008316)are:
P0(x)
=
1
Pn(x) = (2-1/n) x Pn-1(x) (1-1/n) Pn-2(x)
P1(x)
=
x
2
P2(x)
=
-1
+
3x2
2
P3(x)
=
-3 x
+
5x3
8
P4(x)
=
3
30x2
+
35x4
8
P5(x)
=
15 x
70x3
+
63x5
16
P6(x)
=
-5
+
105x2
315x4
+
231x6
16
P7(x)
=
-35 x
+
315x3
693x5
+
429x7
Let's define the electric multipole moment (of order n) as the following function of the unit vectoru (where cos = u.s/ s ).
Qn(u) =
(s) s n Pn(u.s/ s) d3s
This yields the so-called multipole expansion of the electrostatic potential:
V(r) = V(ru) =
-G
Qn(u)
4o
r n+1
. So, it may not be valid inside a sphere whose radiusequals the distance from the origin to the most distant source (i.e., r > s is "safe").
The first term (n=0) corresponds to the field created by a pointcharge (equal to the sum of all the charges in the distribution) according to Coulomb's law. The second term (n=1) corresponds to the fieldcreated by an electric dipole momentP, as discussed elsewhere on this site in full details (including non-static cases).
Q1(u) = u . P
(2008-04-03) A steady current produces a steady magnetic field.
Electricity and magnetism were known asseparate phenomena for centuries.
In 1752,Benjamin Franklin (1706-1790)performed his famous (and dangerous)electric kite experimentwhich established firmly that lightning is an electrical discharge. Franklin himself never wrote about the story but he proofread the account whichJoseph Priestley(1733-1804) gave 15 years after the event. Priestley concludes that report with the comment: "This happened in June 1752,a month after the electricians in France had verifiedthe same theory, but before he heard of anything they had done."
It's unclear who those "electricians in France" are, but the following textbyLouis-GuillaumeLe Monnier appears (inFrench) in the Encyclopédieof Diderot and d'Alembert (71818 articles in 35 volumes, the first 28of which were edited by Diderot himself and published between 1751 and 1766).
Indeed, many people must have wonderedwhy the needle of a compass goes haywire near a bolt of lightning. However, the havoc brought about by lightningmay have precluded the proper investigation of this comparatively delicate aspect.
In 1802, the Italian jurist DomenicoRomagnosi (1761-1835) experimented with a voltaic pile to charge capacitors. He observed that their sudden discharges would deflect a nearby magnetic needle. This raw observation was reported in newspapers. Although Romagnosi didn'texplicitly mention the connection between magnetism and electric current, at least two othersdid it for him when they described his experiments:
Essai théorique et expérimental sur le Galvanisme (1804) p. 340 by Giovanni Aldini (1762-1834).
Manuel du Galvanisme (1805) by Joseph Izarn (1766-1847).
The crucial fact that a steady electric current does produce magnetism was finally established,by a Danish scholar, who became famous for that:
On April 21, 1820, the Danish physicistHans Christian Oersted(1777-1851) was preparing demonstrations for one of his lecturesat the University of Copenhagen. He noticed that a compass needle was deflected when a large electrical current wasflowing in a nearby wire. This precise instant marks the birth of electromagnetism, the study of the interrelated phenomena of electricity and magnetism.
Contrary to popular belief,the discovery of Ørsted was not entirely a chanceaccident (R.C. Stauffer,1953). As early as 1812, Ørsted had published speculations that electricity and magnetism wereconnected. So, when the experimental evidence came to him, he was prepared to make the best of it.
(2008-01-04) The magnetic field produced by a static distribution of electric currents.
Experimentally,Ørstedhad found that a given current in a straight wire creates in its immediate vicinity a magnetic field which seems inversely proportionalto the distance from the wire. The French physicists Jean-Baptiste Biot and Félix Savart proposed that thecontribution of each piece of the wire actuallyvaries inversely as the square of the distance to the observer. Over the entire length of thewire, such contributions do add up to a total field which varies inversely as thedistance from the wire. The Biot-Savart law can be precisely stated as follows:
Contribution to the Magnetostatic Field at the Origin ofa Current ElementIat Positionr.
B =
orI
4 r 3
In this, I is the quantity (current multiplied by thesmall length it travels) which results fromintegrating the current density j (current per unit of surface) over a smallelement of volume. In particular, for a thin wire circuit whose length element s is traversed by a total current I (counted positively in the direction of s ) we have I = I s.
Note that we're using the vector r which goes from the location of interest to the sources. This is a convenient viewpoint for practical computations which seek to obtain a magnetic fieldat a specific point from remote distributions of current. However, many authorstake the opposite viewpoint (opposite sign ofr) to describe thefield produced at a remote location by currents located at the origin.
In many practical applications, the magnetic field is known to have a simplesymmetry and Ampère's Law (below) may yield the valueof the magnetic field throughout space without tedious integrations (just like thetheorem of Gausseasily yields the electrostatic field in caseswith spherical, planar or cylindrical symmetries). One example where no such shortcut is available is that of themagnetic induction on the axis of a circular current loop:
In that case, all radial contributions cancel out, so theresulting magnetic induction B is oriented along the axis (B = Bz ).
Because of the similarity of the relevant triangles, the contribution Bz is R/d times what's given by theabove law:
Bz = (R / d) B = (R / d) (oI/ 4d2) s
As the elements s simply add up to the circumference (2R) we obtain:
Bz = (R / d) (oI/ 4d2) (2R) = oI R2 / d3
In particular, the field at the center of the loop (d = R) is: Bz = o I / 2R.
Helmholtz Coil
Consider two coils (or two loops) like the above,sharing the same vertical axis. Let their respective altitudes be +a and -a. By the previous result, the magnetic induction B (on the axis) at altitude z is:
B = oI R2 [ R2 + (a-z)2] -3/2 + [ R2 + (a+z)2] -3/2
The second derivative of this expression with respect to z at z = 0 is:
B'' (0) = 3oI R2 [ 4a2 - R2 ] ( R2 +a2) -7/2
The value a = ½ R is thus the largest for which the magnetic induction has a single maximum along thevertical axis, in the center of the apparatus (for larger values of a, B'' is positive at the center z = 0, which indicates a minimum there).
This configuration where the separation between the twoloops is equal to their radius (2a = R) is known as a Helmholtz coil. It yields a magnetic induction which is almost uniform near the center of the coil. Namely:
B = (4/5) 3/2oI / R = 0.71554... oI / R
(2008-05-12) Amultivalued functionwhose gradient is themagnetostatic induction.
In a current-free region of space, a scalar potential can be defined (called the magnetic scalar potential ) whose negative gradient is the magnetostatic inductiongiven by the Biot-Savart law.
For a simply-connected region, such a potential is well-defined (up to a uniform additive constant). Otherwise, an essential ambiguity arises whenever the region containsloops which are interlocked with loops of outside current. In that case a continuous potential can only be defined modulo a certainnumber of discrete quantities (each of which corresponds to one interlocking outside current).
The magnetic scalar potential V for the induction B created by a loop of thin wire is simply proportional to thecurrent I in that loop and to the solid angle subtended by the south side of that loopat the location of the observer :
B
=
grad V
V
=
o I
4
is defined modulo 4, which is consistent with the aforementioned "ambiguity". The sign convention is such that the south side of a small loop is seen at a solid anglewhich exceeds a multiple of 4 by a small positive quantity.
This is just a nice way to express the Biot-Savart law whilemaking it clear that, in static distributions,all currents must circulate in closed loops (div j = 0). Neither this approachnor the Biot-Savart law itself can deal with dynamic distributionswhere local electric charges may vary according to the inbound flux of current.
(2008-03-10) (Peregrinus, 1269) The magnetic field (magnetic inductionB) has vanishing divergence.
It's a simple matter to establish with elementary methodsthat the aboveBiot-Savart lawdescribes a field with zero divergence: First, we can verify directly (using Cartesian expressions) that the divergence of the Biot-Savart field vanishes at any nonzerodistance from its source I.
Then, we may check that B has zero flux through any tiny sphere centered on I (this is true because of a trivialsymmetry argument). Thus, the divergence of the Biot-Savart field is identically zero,even at the very location of a source!
The magnetic field may well have sources other than electrical currents (including the dipole moments related to the intrinsicspins of point particles which are part of the modern quantum picture). Nevertheless, all sources ever observed yield magnetic fields with no divergence. Like all scientific facts, this can be stated as a law which holds until disproved by experiment:
In the vocabulary ofmultipoles,only monopole fields have nonzero divergence (in particular,anydipolar field is divergence free). Thus, the vanishing divergence of B is often expressed by stating that there are no magnetic monopoles.
This was first stated in 1269 by the French scholar Peter Peregrinus (Pierre Pèlerin de Maricourt) whofirst described magnetic poles and observed that a magneticpole could not be isolated (they always come in opposite pairs).
This law has survived all modern experimental tests so far andit is postulated to remain valid in the general nonstatic case. It is arguably the oldest of thefour equations of Maxwell. Unfortunately, unlike the other three (Gauss's Law, Faraday's Law, Ampère-Maxwell Law) it has no universally-accepted name...
It's very often referred to as the "magnetic Gauss law",which is rather awkward. Calling it the "Gauss-Weber Law" would seem acceptablebecause the name of Gauss is universally associatedwith the electric counterpart of the law while the magnetic flux so governed (see next paragraph) is closely associated with the name of Wilhelm Eduard Weber (1804-1891) a younger colleague of Gauss after whom the SI unit of magneticflux (Wb) is named.
I argue that the law ought to be called Pèlerin's law (or the Law of Peregrinus ). The relation itself is often called Maxwell-Thomson equation. I'm jumping on the bandwagon, although I don'tthink I ever heard the term as a student.
Because of that law, the magnetic flux () enclosed by a given oriented loopis well-defined as the flux of the magnetic induction B through any surface which is bordered (andoriented) by that loop.
Searching for magnetic monopoles
A famous argument byPaul Dirac (1931) shows that the existence of even a single true magnetic monopole in theUniverse would imply a quantization of electric charge everywhere (as observed). Many physicists do not yet rule out the existence of magnetic monopoles (like any proper physical law,Pèlerin's law only holds until proven wrong experimentally).
A true magnetic monopole would be completely surrounded by a closed surfacetraversed by a nonzero total magnetic flux. The two ends of a thinflux tube do not qualify as monopoles, because the return flux through the cross-section of the tube balances exactly thenonzero flux traversing the rest of any closed surfaceenclosing one pole (but not the other). For example, the magnetic flux which flowsfrom north to southoutside a long bar magnetis exactly balanced by the flux of the strong field which flows from south to north inside the magnet itself.
Mathematically, we may envision an ideal flux tube (often dubbed a Dirac string ) as the infinitely thin version of the above, namelya line carrying, within itself, a finite magnetic flux from one of its extremities (the south pole) to the other (the north pole). The total magnetic flux () through a cross-section is constant along such a Dirac string.
In the Summer of 2009, two independent teams found that actual flux tubes insome so-called spin ices could have cross-sections small enough to fit in the spaces between theatoms of the crystal. Such tubes behave like the ideal Dirac strings presented above. The whole thing looks as though some of the cells in thecrystal contain a magnetic monopolewhile an opposite monopole is found nearby, possibly several cells away...
Those exciting discoveries do not violate Pèlerin's law (magnetic poles still come only in pairs, connected bythin flux tubes). Unfortunately, they were heralded in press releases, review articles and popular magazines as a "discovery of magnetic monopoles". So, a newurban legend was bornwhich makes it slightly more difficult to teach basic science...
(2008-04-25) The magnetic circulation is o times the enclosed current.
WhatGauss did in 1813 for theCoulomb law of 1785,André-Marie Ampère (1775-1836)did in 1825 for theBiot-Savart law of 1820. Unlike the law of Gauss, Ampére's law only holds in the static case. It had to be amended by Maxwell in 1861 for the dynamic case. Here's Ampère's static law (1825) in differential form:
rot B = o j
By theKelvin-Stokes formula,the circulation of a vector around an oriented loop is equal to the flux ofits rotational (curl) through any smooth oriented surfacebordered by that loop. This yields Ampère's law in integral form :
o I oSj . dS = SB . dr
The simplest (and most fundamental) direct application of Ampère's law isto retrieve the experimental fact which prompted theformulation of theBiot-savart lawto begin with, namelythat the magnetic induction B due to a long straight wireis inversely proportional to the distance from that wire:
Indeed, consider a circular loop of radius r whose axis is astraight wire carrying a current I. For reasons of symmetry, the magnetic induction B on that loop is tangent to it. Its projection on the oriented tangent is a constant B (seesign conventions). The magnetic circulationis 2r B and Ampère's law gives:
2r B = o I or, equivalently: B = o I/ 2r
Another popular (and important) application of Ampère's law yieldsthe magnetic field due to an infinitely long solenoid (of arbitrary cross-section) : For a long solenoid consisting of n loops of wire per unit of height (each carrying the same current I) the magnetic induction vanishes outside andhas the following value inside the solenoid:
B = o n I
This can be established by noticing first that the direction of the magnetic induction B must be everywhere vertical (i.e., parallel to theaxis of a solenoid with horizontal cross-section). That is so because the horizontal contribution of each element of current isexactly canceled by the horizontal contribution from its mirror image with respectto the horizontal plane of the observer.
We may then apply Ampère's law to any rectangular loop with two vertical sidesand two horizontal ones (on which the circulation of B is zero,because it's perpendicular to the line element). This establishes that the magnetic field is constant inside the solenoid and constant outsideof it, with the difference between the two equal to the value advertised above. (The fact that the constant value of the induction outside of the solenoid must be zero isjust common sense, or else the magnetic energy of the solenoid per unit of height would be infinite.)
Sneak Preview :
In 1861, Maxwell was able to amend the static law of Ampèreinto the following generalization, which holds in all cases (including changing charge distributions).
Ampère-Maxwell Law (1861)
rot B
1
E
= oj
c
t
We shall postpone thediscussion of this crowning achievement (which made the entire structure of electromagnetism consistent) so we can present first a key breakthroughmade by Faraday on August 29, 1831 (when James Clerk Maxwell was 2 months old): Thelaw of magnetic induction.
(2005-07-19) On the electric circulation induced arounda varying magnetic flux.
MichaelFaraday (1791-1867) was the son of a blacksmith,and a bookbinder by trade. Effectively, he would remain mathematically illiterate, but he became an exceptionally brilliant experimental scientist who would laythe conceptual foundations that occupiedseveral generations of mathematical minds. In 1810, Faraday started attending the lectures thatHumphry Davy (1778-1829) had been giving atthe Royal Institution of London since 1801.
In December 1811, Faraday became an assistant of Davy,whom he would eventually surpass in knowledge and influence. Faraday was elected to the Royal Society in 1824,in spite of the jealous opposition of Sir HumphryDavy (who was its president from 1820 to 1827). In February 1833,Faraday became the first Fullerian Professor of Chemistry at the Royal Institution The chair was endowed by his mentor and supporter John "Mad Jack" Fuller(1757-1834).
Arguably, the greatest of Faraday's many scientific contributions wasthe Law of Induction which he formulated in 1831. After explaining the 1820 observation ofØrsted in terms of whatwe now call the magnetic field, Faraday did much more than inventthe electric motor. Eventually, he opened entirely new vistas for physics. He proposed that light itself was an electromagneticphenomenon and lived to be proven right mathematically by his young friend,James Clerk Maxwell.
Faraday's Law (1831)
rot E +
B
= 0
t
Heinrich Friedrich "Emil Khristianovich"Lenz (1804-1865). Lenz's Law (1833).
The magnetic flux... =B . d =dB . + B .d First term = Magnetic Induction. Second Term = Lorentz Force.
(2008-04-02) On the electric induction produced in a circuit by its own magnetic field.
The American physicist Joseph Henry (1797-1878)discovered the law of induction independently ofFaraday. Henry went on to remark that the magnetic field created by a changingcurrent in any circuit induces in that circuit itself an electromotive force which tends to oppose the change in current.
(2008-04-30) The Ampère-Maxwell law holds even with changing charge distributions.
A simple way to show that theabove static version of Ampère's law fails in the presence of changing electric fields is to consider how acapacitorbreaks the flux of current it receives from a conducting wire: An open flat surface between the capacitor's two plateshas no current flowing through it, unlike a surface with the same border which the wire happens to penetrate.
In 1861, Maxwell realized that, since electric charge is conserved,a difference in the flux of currentthrough two surfaces sharing the same border must imply a changein the total electric charge q contained in the volume between those two surfaces.
ByGauss's theorem, this translates intoa changing flux of the electric field throughthe closed surface formed by the two aforementioned open surfaces. More precisely, and remarkably, the "missing" flux of the current density j is exactly balanced by the flux of the vector o E/t.
Maxwell identified this as the density of a quantityhe called displacement current. He saw that the sum of the actual current and the displacementcurrent was divergence-free. This made that sum a prime candidatefor taking on the role played by the ordinary density of current inthe static version ofAmpère's law. Therefore,Maxwell proposed that Ampère's law should be amended accordingly:
rot B = o (j + o E/t )
Putting the fields on one side and the sources on the the other, we obtain:
Ampère-Maxwell Law (1861)
rot B
1
E
= oj
c
t
At this point, we merely define c as a convenient constant satisfying:
oo c 2 = 1
The paramount fact that c turns out to be thespeed of light will be seen to be aconsequence of putting all of Maxwell's equations together...
(2005-07-18) The 4 basic laws of electricity and magnetism, discovered one by one.
Gauss's Magnetic Law = Maxwell-Thomson equation = Pélerin's Law (1269). Gauss' Electric Law = Coulomb's Law = Poisson's equation. Faraday's Law of Induction. Ampère's Law (became Maxwell-Ampère equation).
(2005-07-09) They predictedelectromagnetic waves before Hertz demonstrated them.
I have also a paper afloat, with an electromagnetic theory of light, which,till I am convinced to the contrary, I hold to be great guns. James Clerk Maxwell (1831-1879) [ letter toCharles H.Cay (1841-1869) dated January 5, 1865 ]
Maxwell's equations govern the electromagneticquantities definedabove:
The electric fieldE (in V/m or N/C).
The magnetic inductionB (in teslas; T or Wb/m 2).
The density of electric charge (in C/m3)
The density of electric currentj (in A/m2)
Maxwell's Equations (1864) in modern vectorial form:
rot E +
B
= 0
div E =
t
o
rot B
1
E
= oj
div B = 0
c
t
The three electromagnetic constants involved are tied by one equation:
o o c 2 = 1
o is the electric permittivity of the vacuum (in F/m)
(2005-07-09) (1746) The continuity equation expresses the conservation of electric charge.
A direct consequence of Maxwell's equations is the following relation,which expresses the conservation of electric charge (: div rot Bvanishes). This conclusion holds if and only if the 3 aforementionedelectromagnetic constants are related as advertised above.
Continuity Equation
divj +
= 0
t
Historically, the relation is reversed: The conservation of electric charge had been formulated before 1746, independently byBenjamin Franklin (1706-1790) and WilliamWatson (1715-1787). This was more than a century before Maxwell used it togeneralize Ampère's lawinto the proper equation which made the whole theoretical structure perfect.
(2005-07-09) Electromagnetic fields propagate at the speed of light (c).
Using theidentityrot rot V =grad divVV when = 0 and j = 0, Maxwell's equations imply that any electromagnetic component verifies:
1
y
=
c
t
This wave equation shows thatelectromagnetism propagates at celerity c in a vacuum. Thus, Maxwell's equations support the electromagnetic theory of light whichMichael Faradayhad proposed well before all the evidence was in. (He engaged in such speculations in 1846, at the end of one of his famous lecturesat the Royal Institution, because he had run out of things to say that particular Friday night!)
In 1883, the Irish physicistGeorge FitzGerald (1851-1901) remarked that an oscillating current ought to generateelectromagnetic radiation (radio waves). FitzGerald isalsoremembered for his 1889 hypothesis that all moving objects areforeshortened in the direction of motion (the relativisticFitzGerald-Lorentz contraction).
The propagation of radio waves was firstdemonstrated experimentally in 1888, by Heinrich RudolfHertz (1857-1894).
(2005-07-15) TheLorentz force transfers energy between thefield and the charges.
The power F.v of theLorentz force is q E.v. Thus, the power received by the electric charges per unit ofvolume is E.j. The charge carriers may then convert the power so received from the local electromagneticfield into other forms of energy (including the kinetic energy of particles).
The quantity E.j may be expressed in terms of the electromagneticfields by dotting into E/o both sides of theAmpère-Maxwell equation:
Plugging that into the previous equation, we obtain an important relation:
Electromagnetic Balance of Energy Density : Poynting Theorem (1884)
div (
EB
) +
(
oE 2 + B 2/o
) = E . j
o
t
2
This is due to a pupil of Maxwell, JohnHenry Poynting (1852-1914). S = EB / o is the Poynting vector.
In the above, the right-hand side is the opposite of the power delivered by the field to the sources, per unit of volume. So, it's the density of the power released by the sources to the field. The left-hand side is thus consistent with the followingenergy for the electromagnetic field:
Electromagnetic Energy Density
1/2o ( E 2 + c2B 2 )
The above Poynting theorem states that, the variation of thisenergy in a given volume comes from power that is either delivered directly by inside sourcesor radiated through the surface, as the flux of the Poynting vector.
In the context of Classical Field Theory, the above is the Hamiltonian density, whereasthe Lagrangian density of the electromagneticfield is a Lorentz scalar (a merepseudo-scalar likeE.B won't do) namely:
Lagrangian Density
1/2o ( E 2 c2B 2 )
Identifying the above with the usual formulas for the Hamiltonian (H=T+U) and the Lagrangian (L=T-U) we may think of the square of E as a kinetic term (T) and the square of B as a potential term (U). The analogy is more compelling when a special gauge is used which makes the electrostaticpotential () vanish everywhere,as is the case for the standardLorenz gauge in the particular case ofa crystal of magnetic dipoles. For in such cases, the electric field consists entirely oftime-derivatives of A...
The above is for the electromagnetic field by itself. In the presence of charges which interact with the field in theform of a density of Lorentz forces, the corresponding Lagrangian densityof interaction should be added:
Lagrangian Density
1/2o ( E 2 c2B 2 ) ( j.A )
Still missing are all the non-electromagnetic terms needed to determine correct expressionsof the conjugate momenta and Hamiltonian density...
(2005-07-15) (Progressive Waves) The simplest solutions to Maxwell's equations,away from all sources.
In the absence of electromagnetic sources ( = 0, j =0 ) we may look forelectromagnetic fields whose valuesdo not dependon the y and z cartesian coordinates. A solution of this type is called a progressive planar wave and it may be establisheddirectly from theaboveequations of Maxwell,without invoking the electromagneticpotentials introducedbelow.
Indeed, when all derivatives with respect to y or z vanish,the 8scalar relations which express Maxwell's equationsin cartesian coordinates become:
Bx
=
0
Ex
=
0
x
x
0
=
1
Ex
0
=
Bx
c 2
t
t
Bz
=
1
Ey
Ez
=
By
x
c 2
t
x
t
By
=
1
Ez
Ey
=
Bz
x
c 2
t
x
t
To solve this, we introduce the new variables u = t - x/c and v = t + x/c For any quantity , the two expressions of thedifferential form d yield the expressions of the partial derivatives with respect to the new variables :
d =
dt +
dx =
du +
dv
t
x
u
v
dt =1/2 ( dv + du ) and dx =c/2 ( dv - du )
Therefore,
=
1
c
u
2
t
x
=
1
+ c
v
2
t
x
We may apply this back and forth when is one ofthe cartesian components of E or B, using the above relations between those. For example:
y
=
1
y
c
y
=
c2
z
+
c
z
= c
z
u
2
t
2
x
2
x
2
t
u
Thus, Ey c Bz doesn't depend on u. Likewise, neither does Ez + c By Similarly, both Ey + c Bz and Ez c By do not depend on v.
(2009-12-13) Electromagnetic waves (or stationary fields) exert a mechanical pressure.
In 1871,Maxwell himself predicted this as a consequence ofhis ownequations. In 1876, Adolfo Bartoli (1851-1896) remarked that the existence of radiation pressure is also an unavoidable consequenceofthermodynamics. (Radiation pressure is thus sometimes called Maxwell-Bartoli pressure.) Maxwell-Bartoli pressure was first demonstrated experimentally byPyotr Lebedev in 1899.
The first proper measurement of radiation pressure was made in 1899 by Pyotr Lebedev (1866-1912). In 1901, the pressure of light was measuredat Dartmouth byNichols andHull to an accuracy of about 0.6% (the original Nichols radiometer is at theSmithsonian). To avoid the aforementioned effect (dominant inCrookes radiometers) a Nichols radiometer must operate in a high vacuum.
(2005-07-13) Devised by LudwigLorenz in 1867 [whenH.A. Lorentz was only 14].
SinceMaxwell's equations assert thatthe divergence of B vanishes,there is necessarily a vector potentialA of which B is the rotational (or curl).
B = rot A
Faraday's law now reads rot [E +A/t ] = 0 . The square bracket is the gradient of a scalar potential, called - for consistency withelectrostatics:
E = gradA/t
These two equations do not uniquely determine the potentials,as the same fields are obtained with the following substitutions of thepotentials, valid for any smooth scalar field .
AA + grad t
This leeway can be used to make sure the following equation is satisfied,as proposed by Ludwig Lorenz in 1867. (Watch the spelling... There's no "t".)
The Lorenz Gauge (1867)
divA +
1
= 0
c2
t
The Lorenz Gauge doesn't eliminate the above type of leeway. It restricts it to a free field propagating at celerity c, according to the wave equation :
1
y
= 0
c
t
The two Maxwell equations which don't involve electromagnetic sources are equivalent to theabove definitions of E andB in terms ofelectromagnetic potentials. Using the Lorenz Gauge, the other two equations reduce to the following relationsbetween the electromagnetic sources and the potentials:
D'Alembert's Equations
1
=
c2
t
o
A
1
A
= oj
c2
t
Without the Lorenz Gauge, more complicated relations would hold:
1
=
( divA +
1
)
c2
t
o
t
c2
t
A
1
A
=
o j + grad
( divA +
1
)
c2
t
c2
t
Formerly viewed as a mere mathematical convenience (which Maxwell himself didn't like at all) the Lorenz gauge is now considered fundamental,becausequantum theory assigns a physical significance to the potentials.
The Lorenz gauge is relativistically covariant (if it's true in one frame of reference it's true in all of them). This isn't the case for other popular gauges, including the Coulomb gauge (div A = 0) once favored by Maxwell. Such putative gauges are thus incompatible with the objectivity of potentials.
The expressions of the Lagrangian, Hamiltonian and canonical momentum of a charged particle in an electromagnetic field do depend explicitly on the potentials,although the classicalLorentz force derived from them does not depend on the choice of a gauge (seeelsewhere on this site for a proof).
Canonical momentum of a particle of mass m, charge qand velocityv
p =
qA +
mv
1 -v2/c2
Lagrangian of a charged particle :
L = q (A.v ) m c 2
1v2 / c2
(2005-07-15) General solutions of Maxwell's equations using the Lorenz gauge.
As shownabove, the miraculous effect ofthe Lorenz gauge is thatit effectively decouples electricity and magnetismto turn Maxwell equations into parallel differential equations that can formally be solved using standard techniques (thed'Alembert equations are named afterJean-le-Rond d'Alembert,who solved the related homogeneouswave equation). One relation equates second derivatives of the electric potential to the electric density . The other [vectorial] relationequates the same derivatives of each component ofthe vector potential A to the corresponding component ofthe density of current j. The mathematical solution for each component (and, therefore,for the whole thing) can be expressed as the sum of three terms said to be, respectively,retarded,advanced andfree :
= (1) + + + o A = (1)A + A+ + Ao
Usually, only = 0 is considered, for the causality reasonsdiscussed below.
= 1 isan alternate choice which reverses the arrow of time. In 1945,Wheeler& Feynman fantasized about the possibility of = ½.
The free terms (superscripted o) are exactly what we havealready encounteredas the remaining degrees of freedom after imposingthe Lorenz gauge. They correspond mathematically tosolutions of the homogeneous differential equations (zero charges and currents characterize free space). Happily, the fact that they appear again here means that the choice of thatgauge really involved no loss of generality. (This is not coincidental but we may pretend it is.)
The retarded terms are given by the followingexpressions, proposed byAlfred-MarieLiénard (1869-1958;X1887) in 1898and byEmil Wiechert (1861-1928) in 1900. They're known as the Liénard-Wiechert potentials.
Electrodynamic Retarded PotentialsAand
(t,r)
=
( t ||rs|| / c , s )
d3s
4o||r - s ||
A(t,r)
=
oj ( t ||rs|| / c , s )
d3s
4||r - s ||
This is similar to the expressions obtained in the static cases (electrostatics, magnetostatics) except that the fields we observe here and now depend on a prior state of the sources. The influence of the sources is delayed by the time it takesfor the "news" of their motions to be broadcasted at speed c.
The so-called advanced potentials ( A+ and+ ) are formally obtained by making c negative in the above retarded expressions (or equivalently by reversing the arrow of time). This is just like what we've already encounteredin the case ofplanar waves,with two possible directions of travel. However, the physical interpretation is not nearly as easy now that we'redealing with some causality relationship between the field and its"sources".
Advanced potentials make the situation here and now (potentials and/or fields) depend on the future state of remote "sources". Such a thing may be summarily dismissed as "unphysical" but this failsto make the issue go away. Indeed, quantum treatments of electromagnetic fields (photons in Quantum Field Theory ) imply that a field can create some of its sources in the form of chargedparticle-antiparticle pairs. What seems to be lacking is the coherence of such creations because of statistical and/or thermodynamicalconsiderations (which feature a pronounced arrow of time). I don't understand this. Nobody does...
What's clear, however, is that the distinction between past and future vanishesin stationary cases. This makes advanced potentials relevant and/or necessary,without the need for mind-boggling philosophical considerations.
We've only shown (admittedly skipping the mathematical details) that potentials that obey the Lorenz gauge would necessarily be given by theabove formulas (possibly adding advanced and free components). Conversely, we ought to determine now what restrictions, if any, (pertaining to the sources and j) would make the above solutions verify the assumed Lorenz gauge. However, we shallpostpone this discussionto present first aclarification of the physics...
(2005-08-21) An expression derived from theLiénard-Wiechertretarded potentials.
Let and j denote ( t R / c , s ) and j ( t R / c , s ).
As always, R = ||rs || is the distance from a source (located at s) to the observer (at r). The following expressions of the fields then hold:
Electrodynamic fields obtained from retarded potentials :
E(t,r)
=
1
[
(rs )
+
( / t )(rs )
j /t
d3s
4o
R 3
cR 2
c 2R
B(t,r)
=
o
[
j (rs )
+
(j /t ) (rs )
d3s
4
R 3
cR 2
In the static case, only the first term of either expression subsistsand we retrieve either theCoulomb lawof electrostatics or theBiot-Savart lawof magnetostatics.
A changing distribution of charges and currents generates the additionalterms whose amplitudes dominate at large distancesbecause they only decrease as 1/R. This is what makes radio transmission practical!
I find those expressions for the electromagnetic fields caused by dynamic sources very enlightening. Personally, I discovered them after establishing the dipolarsolutions of Maxwell's equations, which strongly suggest such formulas. They are now known as Jefimenko's equations, in honor of Oleg D. Jefimenko (1922-2009). They were probably discovered privately many times. According toKirk T. McDonald (1997)the first textbook which mentions them is the second editionof Panofsky and Phillips (1962).
Here's an outline of how those formulas can be derived from the well-knownintegrals giving the retarded potentials. In either of those integrals, t is a constant and so are the coordinates x,y,z of r. Differentiation with respect to x,y,z or t is thus performed bydifferentiating the integrand,which involves only numerical expressions of thefollowing type (using the notations introduced at the outset):
k(R) f ( t-R/c ,s )
In this, k(R) is simply proportional to 1/R (but we may treat it like some unspecified function of R ). Both factors depend on x,y,z only because R does. The function f depends on time; k doesn't. Thechain rule yields:
f
=
f
( t
R
)
=
1
R
f
x
t
x
c
c
x
t
R / x is obtained by differentiating R2 = (r-s)2 . Namely:
R dR = ( x-sx) dx + ( y-sy) dy + ( z-sz) dz
f
=
x - sx
f
x
c R
t
From this basic relation, and its counterparts along y and z, we obtain:
gradf =
f
r -s
t
c R
The same relations applied to the components fx fy fz of a vector F yield:
rotF =
F
r -s
t
c R
Another relation (needed only in thenext section) involves a dot product :
div F =
F
r -s
t
c R
Handling the scaling part introduced above as k(R) is similarbut less tricky conceptually, because k is simply ascalar function of a single argument (the distance R between source and observer) with a straight derivative k'. (As k is proportional to 1/R, we have k'(R) = -k/R.)
The conclusion follows from two general identities ofvector calculus and onetrivial equation (expressing that k is time-independent) namely:
rot (kF) = grad k F + krot F
grad (kf ) = fgrad k kgradf
t (kF) = kFt
The first line yields the expression of B, the sum of the last two gives E.
(2010-12-06) An expression derived from theLiénard-Wiechertadvanced potentials.
Let's now forget the aura of mystery traditionally associated with advanced solutions. Reversing the direction of time simply reverses causality. Bluntly, when the photons kick the electrons,the values of the fields are related to the values of theso-called "sources" at a later time (the sources are not the causes in this case; their name is misleading).
Now, and j denote ( t + R / c , s ) and j ( t + R / c , s ).
Compare this formally to thesimilar expressionsfor retarded potential and notice the changesof sign that occur in the second column but not the third! Thoses changes can be traced down to the beginning of the proof outlined abovefor retarded potentials, since for a function f ( t+R/c , s ) :
f
=
f
( t +
R
)
=
1
R
f
x
t
x
c
c
x
t
The corresponding change of sign (compared to retarded potentials) applies to the dynamical parts of grad or rot A but does not formally affectthe A/t component of E.
One important consequence of such changes of signs is that it affects thedistant fields in a way which reversesthe sign of Larmor's formula. In other words, contrary to popular belief, an accelerated or deceleratedcharge need not radiate electromagnetic energy away. It does so only when the change of its motion is the cause of changingfields, not when it's the result of such changing fields. Electromagnetic energy always flows from cause to effect.
We can combine the methods and the preliminary specific equations established in that sectionwith another generalidentity of vector calculus:
div (kF) = F . grad k + k divF
(2005-08-11) Accelerated [bound] charges radiate energy away, or do they?
Consider thedipolar solutions to Maxwell'sequation (retarded spherical waves) presentedelsewhere on this site. At a large distance, the dominant field components are proportionalto the second derivatives p'' or m''. For an electric dipole, the dominant far-field component of the Poynting vector ( EB / o ) is thus in the radial direction of the normed vector u:
o
u
d2p
2
u
(4 r)2c
dt 2
This is a radial vector whose length is proportional to sin2 = 1 - cos2 (where is the angle between p'' and the direction of u). Its flux through the surface of the sphere of radius r is thetotal power radiated away:
o
d2p
2
(1 - cos2 ) (2 r 2sin ) d =
o
||p''|| 2
(4 r)2c
dt 2
0
6 c
Likewise, the total power radiated by a magnetic dipole is :
( o / 6 c3 ) || m''|| 2
Let's use asubterfuge to compute the power radiated away by a single charge q near the origin: Place a charge -q (a "witness") at the origin itself. At large distances, the resulting variable dipole p = q r(t) would produce essentially the same dynamic field (at time t+r/c) as the lone moving charge q (as long as its acceleration does not vanishand its distance to the origin remains small). This translates into the following so-called Larmor formula (derived in 1897 by Joseph Larmor, 1857-1942):
Power radiated by a charge q
o q 2
d2r
2
6 c
dt 2
The above argument skirts near-field difficulties, but it seems inadequate wheneverthe moving charge is not confined to the immediate vicinity ofthe artificial "witness" charge. In particular, we don't obtain a clear picture of what happens,in the long run, when a charge is subjected toa constant acceleration... It has been argued that no power would be lost away in this case,which (according toGeneral Relativity) is equivalent to a motionless chargein a constant gravitational field. Even so, a varying gravity ought to make charges radiate(classically, at least).
A promising way out of that dilemma (2006-10-16)is to consider the thermal nature of the above exchange of energy,allowing the formula to hold, in some statistical way, as the classical counterpartof a quantum effect... Indeed, in 1976,W.G. Unruh foundthat an acceleration g (or, equivalently, a gravitational field) entails a heat bath whose temperature T is proportional to it :
Unruh's Temperature T (1976)
k T =
h
g
4 2 c
(2005-08-09) Classical Theory of the Electron. Strange inertia of charged particles.
The motion of an electron (point particle of charge q) submitted to a force F has been described in terms of the following 4-dimensional equation,where (primed) derivatives of the position R are withrespect to the particle's proper time [ defined via: (c d)2 = (c dt)2-(dx)2-(dy)2-(dz)2 ].
Lorentz-Dirac Equation (1938)
mR'' = F +
o q 2
[ R''' +
|R' ><R' |
R''' ]
6c
c2
|R' ><R' | is a squaretensor (the product of the 4D velocity and its dual). The bracketed sum is only relevant for a point particle of nonzero charge. Its nature has been highly controversial since 1892, when H.A. Lorentz first proposed a Theory of the Electron derived microscopically fromMaxwell's equations and fromthe expression of theelectromagnetic force now named after him. Lorentz would only consider the electromagnetic part of the rest mass m (i.e., 3m/4). In 1938, Paul Dirac derived the above covariantly,for the total mass m.
Physically, the initial value of the acceleration (R'' )in this third-order equation cannot be freely chosen (so the overall constraints are comparable to those of an ordinary newtonian equation). Almost all mathematical solutions are unphysical ones,which are dubbed self-accelerating or runaway because they would makethe particle's energy grow indefinitely, even if no force was applied.
However, more than one initial value of the acceleration could make physicalsense. The wholly classical Lorentz-Dirac equationthus allows a nondeterministic behavior more often associated withquantum mechanics.
The Lorentz-Dirac equation has other weird features, including the needfor a so-called preacceleration contradicting causality, since the equation would require an electron to anticipate any impending pulse of force...
