Problems
2.1 Introduction
All physical phenomena in our Universe derive from four fundamental forces. Gravity binds stars and creates the tides. The strong force binds baryons and mesons and controls nuclear reactions. The weak force mediates neutrino interactions and changes the flavor of quarks. The fourth force, the Coulomb-Lorentz force, animates a particle with chargeqand velocityυin the presence of an electric fieldEand a magnetic fieldB:
F=q(E+υ×B). (2.1)
The subject we callelectromagnetismconcerns the origin and behavior of the fieldsE(r, t) andB(r, t) responsible for the force (2.1).
We will come to learn that the electric and magnetic fields are very closely related. However, like many siblings, thetime-independentquantitiesE(r) andB(r) do not look alike and do not interact.
Static electric fields require charge separation. The largest such separations (∼105m) are associated with electrostatic discharges in the upper atmosphere. Static magnetic fields (apart from magnetic matter) require only charge in steady motion. As far as is known, the maximum size of magnetic field patterns may approach cosmic dimensions (∼1020m).
Time-dependentEandBfields are more like a newly married couple. Initially, they remain close to their sources. Then, in a moment of subtle reorganization, they break free and—in the form of electromagnetic radiation—race away to an independent, intertwined existence. The two fields are inextricably bound together in the X-rays which reveal the atomic-scale structure of DNA, in the microwaves which facilitate contemporary telecommunications, and in the radio waves which reveal the large-scale structure of the Universe.
The full story of these matters is neither short nor simple. In this chapter, we begin with the primitive concepts of charge and current. A brief review of the history of electromagnetism leads to definitions forE(r, t) andB(r, t) and to the Maxwell equations which relate the fields to sources of charge and current. We then turn to the relationship between microscopic electromagnetism and macroscopic electromagnetism. This includes a discussion of spatial averaging and a derivation of the matching conditions required by the macroscopic theory. Two short sections discuss the limits of validity of the classical theory and the SI system of units used in this book. The chapter concludes with a heuristic
“derivation” of the Maxwell equations.
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30 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
2.1.1 Electric Charge
The wordelectricderives from the Greek word for amber (ηλεκτρoν), a substance which attracts bits of chaff when rubbed. Sporadic and often contradictory reports of this peculiar phenomenon appeared for centuries. Then, in 1600, William Gilbert dismissed all of them as “esoteric, miracle-mongering, abstruse, recondite, and mystical”. His own careful experiments showed that many materials, when suitably prepared, produced an “electric force” like amber.
Electrical research was revolutionized in 1751 when Benjamin Franklin postulated that rubbing transfers a tangible electric “fluid” from one body to another, leaving one with a surplus and the other with a deficit. When word of the American polymath’s proposal reached Europe, Franz Aepinus realized that theelectric chargeQwas a variable that could be assigned to an electric body. He used it to express Franklin’s law of conservation of charge in algebraic form. He also pointed out that the electric force was proportional toQ. Today, we understand charge to be an intrinsic property of matter, like mass. Moreover, no known particle possesses a charge which is not an integer multiple of the minimum value of the electron charge1
e=1.602 177 33(49)×10−19C. (2.2)
Typically, a neutral atom, molecule, or macroscopic body acquires a net charge only through the gain or loss of electrons, each of which possesses a charge−e.
Despite the fundamental discreteness of charge implied by its quantization, electromagnetic theory develops most naturally if we define a continuous charge per unit volume or volumecharge density, ρ(r). By construction,dQ=ρ(r)d3ris the amount of charge contained in an infinitesimal volume d3r. The total chargeQin a finite volumeV is
Q=
V
d3r ρ(r). (2.3)
Classically, this is a straightforward definition whenρ(r) is a continuous function of the usual sort.
It is equally straightforward in quantum mechanics because the charge density is defined in terms of continuous wave functions. For example, the charge density for a system ofN indistinguishable particles (each with chargeq) described by the many-particle wave function(r1,r2, . . . ,rN) is
ρ(r)=q
d3r2
d3r3· · ·d3rN|(r,r2, . . . ,rN)|2. (2.4) It is often useful to imagine continuous distributions of charge which are confined to infinitesimally thin surface layers. This suggests we identifyrS as a point centered on an infinitesimal element of surfacedSand define a charge per unit area, or surface charge density,σ(rS) sodQ=σ(rS)dS.The total charge associated with a finite surfaceSis
Q=
S
dS σ(rS). (2.5)
A charge per unit length, or linear charge densityλ(),plays a similar role for continuous distributions confined to a one-dimensional filament. In that case,dQ=λ()d, wherepoints from the origin to to the line elementd.
Finally, we return to (2.2) and use the absence of data to establish a finite size for the electron as motivation to define a classicalpoint charge as a vanishingly small object which carries a finite
1 We omit quarks, particles with fractional charge, because they cannot be separated from the hadrons they constitute.
Equation (2.2) defines the Coulomb (C) as the charge carried by 6.2415096×1018electrons.
2.1 Introduction 31
(a) (b)
S dS
d
C K
j nˆ
Figure 2.1: (a) The currentIis the integral overSof the projection of the volume current densityjonto the area elementdS; (b) the currentIis the integral overCof the projection ofK×nˆ(the cross product of the surface current densityKand the surface normal ˆn) onto the line elementd.
amount of charge.2The mathematical properties of the delta function are well suited to this task and we represent the charge density ofNpoint chargesqklocated at positionsrkas
ρ(r)= N k=1
qkδ(r−rk). (2.6)
Substituting (2.6) into (2.3) yields the correct total charge. However, one should be alert for possible (unphysical) divergences in other quantities associated with the singular nature of (2.6).
2.1.2 Electric Current
Electric charge in organized motion is calledelectric current. The identification of electric current as electric charge in continuous motion around a closed path is due to Alessandro Volta. Volta was stimulated by Luigi Galvani and his observation that the touch of a metal electrode induces the dramatic contraction of a frog’s leg. A skillful experimenter himself, Volta discovered that the contraction was associated exclusively with the passage of electric charge through the frog’s leg.
Subsequent experiments designed to generate large electric currents led him to invent the battery in 1800.
We make the concept of electric current quantitative using Figure 2.1(a). By analogy with fluid flow, let ˆnbe the local unit normal to an element of surfacedSand letdS=dSn. We define aˆ current densityj(r, t) sodI=j·dS=j·ndSˆ is the rate at which charge passes throughdS. The total current that passes through a finite surfaceSis
I = dQ dt =
S
dS·j. (2.7)
We can write an explicit formula forj(r, t) when a velocity fieldυ(r, t) characterizes the motion of a charge densityρ(r, t). In that case, the fluid analogy suggests that the current density is
j=ρυ. (2.8)
If the charge is entirely confined to a two-dimensional surface, it is appropriate to replace (2.8) by a surface current density
K=συ. (2.9)
2 Electron-positron scattering experiments judge the electron to be a structure-less object with a charge radius less than 1.2×10−19m. See A. Bajo, I. Dymnikova, A. Sakharov,et al., inQuantum Electrodynamics and Physics of the Vacuum, edited by G. Cantatore, AIP Conference Proceeding, volume 564 (AIP, Woodbury, NY, 2001), pp. 255-262.
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32 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
Figure 2.1(b) shows that the current which flows past a curveCon a surface can be expressed in terms of the local surface normal ˆnusing
I=
C
d·K×nˆ=
C
K·( ˆn×d). (2.10)
The second integral in (2.10) makes it clear that only the projection ofKonto the normal to the line elementd(in the plane of the surface) contributes toI[cf. (2.7)].
2.1.3 Conservation of Charge
As far as we know, electric charge is absolutely conserved by all known physical processes. The only way to change the net charge in a finite volume is to move charged particles into or out of that volume.
Chemical reactions create and destroy chemical species, and quantum processes create and destroy elementary particles, but the total charge before and after any of these events is always the same.
Indeed, the most stringent tests of thisprinciple of charge conservationsearch for the spontaneous decay of the electron into neutral particles like photons and neutrinos. If this occurs at all, the mean time for decay exceeds 1024years (Belliet al.1999).
For our purposes, the most useful statement of charge conservation begins with the surface integral representation of the currentIin (2.7). If we choose the surfaceSto be closed, the divergence theorem (Section 1.4.2) permits us to expressIas an integral over the enclosed volumeV:
I=
V
d3r∇ ·j. (2.11)
Because the vectordSin (2.7) points outward fromV, (2.11) is the rate at which the total chargeQ decreasesin the volumeV. An explicit expression for the latter is
−dQ dt = −d
dt
V
d3r ρ= −
V
d3r∂ρ
∂t. (2.12)
Equating (2.11) and (2.12) for an arbitrary volume yields a local statement of charge conservation called thecontinuity equation,
∂ρ
∂t + ∇ ·j=0. (2.13)
The continuity equation says that the total charge in any infinitesimal volume is constant unless there is a net flow of pre-existing charge into or out of the volume through its surface.