Problems
Application 2.1 Moving Point Charges
2.2 The Maxwell Equations in Vacuum
Since∇ ·υk=0, this can be written in the form
∂ρ/∂t = −∇ ·
k
qkυkδ(r−rk). (2.16)
This formula is consistent with the continuity equation (2.13) if j(r, t)=
N k=1
qkυkδ(r−rk). (2.17)
2.2 The Maxwell Equations in Vacuum
The unification of electricity, magnetism, and optics was achieved by James Clerk Maxwell with the publication of his monumentalTreatise on Electricity and Magnetism(1873). Maxwell characterized his theory as an attempt to “mathematize” the results of many different experimental investigations of electric and magnetic phenomena. A generation later, Heinrich Hertz famously remarked that
“Maxwell’s theory is Maxwell’s system of equations.” He made this statement because he “not always felt quite certain of having grasped the physical significance” of the arguments given by Maxwell in theTreatise. The four “Maxwell equations” Hertz had in mind (independently proposed by Heaviside and Hertz) are actually a concise version of twelve equations offered by Maxwell.
It is traditional to develop the Maxwell equations through a review of the experiments that motivated their construction. We do this here (briefly) for the simple reason that every physicist should know some of the history of this subject. As an alternative, Section 2.7 offers a heuristic “derivation” of the Maxwell equations based on symmetry arguments and minimal experimental input.
2.2.1 Electrostatics
Experimental work by Priestley, Cavendish, and Coulomb at the end of the 18th century established the nature of the force between stationary charged objects. Extrapolated to the case of point charges, the force on a chargeqat the pointrdue toNpoint chargesqkat pointsrkis given by the inverse-square Coulomb’s law:
F= 1 4π 0
N k=1
qqk
r−rk
|r−rk|3. (2.18)
The pre-factor 1/4π 0reflects our choice of SI units (see Section 2.6). Using the point charge density (2.6), we can restate Coulomb’s law in the form
F=qE(r), (2.19)
where the vector fieldE(r) is called theelectric field:
E(r)= 1 4π 0
d3rρ(r) r−r
|r−r|3. (2.20)
Generalizing, we define (2.20) to be the electric field for any choice ofρ(r).This definition makes theprinciple of superpositionexplicit: the electric field produced by an arbitrary charge distribution is the vector sum of the electric fields produced by each of its constituent pieces. Given (2.20), the mathematical identities
∇ 1
|r−r|= − r−r
|r−r|3 and ∇2 1
|r−r|= −4π δ(r−r) (2.21)
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34 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
are sufficient to show that
∇ ·E=ρ/0 and ∇ ×E=0. (2.22)
The first equation of (2.22) isGauss’ law. It is the first of the four Maxwell equations. The second equation of (2.22) is valid for electrostatics only.
2.2.2 The Field Concept
An important conceptual shift occurs when we pass from (2.18) to (2.19). The first of these conjures up the picture of a non-local force which acts between pairs of charges over arbitrarily large distances.
By contrast, Coulomb’s law in the form (2.19) suggests the rather different picture of an electricfield which pervades all of space.3A particle experiences a force determined by the local value of the field at the position of the particle. For static problems, the “action-at-a-distance” and field points of view are completely equivalent.
The true superiority of the field approach becomes evident only when we turn to time-dependent problems. In that context, we will see that fields can exist quite independently of the presence or absence of charged particles. We will endow them with properties like energy, linear momentum, and angular momentum and treat them as dynamical objects with the same mechanical status as the particles.
2.2.3 Magnetostatics
William Gilbert was the first person to conduct systematic experiments on the nature of magnetism.4In de Magnete(1600), Gilbert correctly concluded that the Earth behaves like a giant permanent magnet.
But it was not until 1750 that John Michell showed that the force between the ends (“poles”) of two rod-like permanent magnets follows an inverse-square law with attraction (repulsion) between unlike (like) poles. Somewhat later, Coulomb and Gauss confirmed these results and Poisson eventually developed a theory of magnetic “charge” and force in complete analogy with electrostatics.
In 1820, Oersted made the dramatic discovery that a current-carrying wire produces effects quali- tatively similar to those of a permanent magnet. Biot, Savart, and Amp`ere followed up quickly with quantitative experiments. A watershed moment occurred when Amp`ere published his calculation of the force on a closed loop carrying a currentIdue to the presence ofNother loops carrying currents Ik(see Figure 2.2). Ifrpoints to the line elementdof loopIandrkpoints to the elementdkof the kthloop, Amp`ere’s formula for the force onI is5
F= −μ0
4π
I d· N k=1
Ikdk
r−rk
|r−rk|3. (2.23)
The pre-factorμ0/4πreflects our choice of SI units (see Section 2.6).
With a bit of manipulation, (2.23) can be recast in the form F=
I d×B(r), (2.24)
3 The concept of the field, if not its mathematical expression, is generally credited to Michael Faraday. See McMullin (2002) in Sources, References, and Additional Reading.
4 See the first paragraph of Section 2.1.1. The wordmagneticderives from the proper name Magnesia. This is a district of central Greece rich in the naturally magnetic mineral lodestone.
5 The scalar product in (2.23) isd·dk.
2.2 The Maxwell Equations in Vacuum 35
d dk
I
Ik
r rk
O
Figure 2.2: Two filamentary loops carry currentIandIk. The vectorsrandrkpoint to the line elementsdand dk, respectively.
where themagnetic fieldB(r) has a form first determined by Biot and Savart in 1820:
B(r)= μ0
4π N k=1
Ikdk× r−rk
|r−rk|3. (2.25)
Later (Section 9.3.1), we will learn that the substitutionI d→
d3rjtransforms formulae valid for linear circuits into formulae valid for volume distributions of current. Accordingly, we generalize (2.25) anddefinethe magnetic field produced by any time-independent current density as
B(r)= μ0 4π
d3rj(r)×(r−r)
|r−r|3 . (2.26)
The principle of superposition is again paramount: the magnetic field produced by a steady current distribution is the vector sum of the magnetic fields produced by each of its constituent pieces.
In 1851, William Thomson (later Lord Kelvin) used an equivalence between current loops and permanent magnets due to Amp`ere to show that the magnetic field produced by both types of sources satisfies
∇ ·B=0 and ∇ ×B=μ0j. (2.27)
It is a worthwhile exercise to confirm that (2.26) is consistent with both equations in (2.27) as long as the current density satisfies thesteady-currentcondition [cf. (2.13)],
∇ ·j=0. (2.28)
The first equation of (2.27) is the second of the Maxwell equations. It has no commonly agreed-upon name. The equation on the right side of (2.27), which is valid for magnetostatics only, is often called Amp`ere’s law.
2.2.4 Faraday’s Law
Michael Faraday was the greatest experimental scientist of the 19th century.6Of particular importance was Faraday’s discovery that a transient electric current flows through a circuit whenever the magnetic flux through that circuitchanges(Figure 2.3). Starting from surprisingly different points of view, the mathematical expression of this fact was achieved by Neumann, Helmholtz, Thomson, Weber, and Maxwell. In modern notation, Faraday’s observation applied to a circuit with resistanceRmeans that
−d dt
S
dS·B=I R. (2.29)
6 Every physicist should at least glance through Faraday’sDiaryor hisExperimental Researches in Electricity.Both are wonderfully readable chronicles of over 40 years (1820-1862) of experimental work.
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36 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
N S υ
N S υ
S C S
I I
Figure 2.3: A typical experiment which reveals Faraday’s law. Current flows in opposite directions in the filamentary wireCwhen the permanent magnet moves upward or downward. The areaSis bounded by the wireC.
The domain of integrationS is any surface whose boundary curve coincides with the circuit. Our convention is that the right-hand rule relates the direction of current flow to the direction ofdS. In that case, the minus sign in (2.29) reflectsLenz’ law: the current creates a magnetic field which opposes the original change in magnetic flux.
To complete the story, we need only recognize that scientists of the 19th century understood Ohm’s law for current flow in a closed circuitCto mean that
I R=
C
d·E. (2.30)
For our application, the pathCin (2.30) bounds the surfaceSin (2.29). Therefore, after setting (2.30) equal to (2.29), Stokes’ theorem (Section 1.4.4) yields the differential form ofFaraday’s law, the third Maxwell equation:
∇ ×E= −∂B
∂t. (2.31)
2.2.5 The Displacement Current
Thedisplacement currentis Maxwell’s transcendent contribution to the theory of electromagnetism.
Writing in 1862, Maxwell used an elaborate mechanical model of rotating “magnetic” vortices with interposed “electric” ball bearings (see Figure 2.4) to argue that the current densityjin Amp`ere’s law must be supplemented by another term when the electric field varies in time. This is the displacement current,jD=0∂E/∂t. Inserting this into (2.27), we get the fourth and final Maxwell equation, usually called theAmp`ere-Maxwell law,
∇ ×B=μ0j+ 1 c2
∂E
∂t. (2.32)
Today, it is usual to say that the displacement current is absolutelyrequired if Amp`ere’s law and Gauss’ law are to be consistent with the continuity equation (2.13). This argument was unavailable to Maxwell because he did not associate electric current with electric charge in motion.7
Maxwell dispensed entirely with mechanical models when he wrote hisTreatise. Instead, he intro- ducedjD without motivation, remarking only that it is “one of the chief peculiarities” of the theory.
7 Maxwell (who worked long before the discovery of the electron) did not regard charge as an intrinsic property of matter subject to a law of conservation. To him, Gauss’ law did not mean that charge was the source of an electric field. It meant that spatial variations of an electric field were a source of charge. Maxwell’s conception of current is not easily summarized. See Buchwald (1985) in Sources, References, and Additional Reading.
2.2 The Maxwell Equations in Vacuum 37
Figure 2.4: Maxwell’s sketch of a mechanical model of rotating hexagonal vortices with interposed ball bearings. He inferred the existence of the displacement current from a study of the kinematics of this device.
Figure from Maxwell (1861).
Presumably, the fact that the displacement current was essential for wave solutions which propagate at the speed of light had convinced him of its basic correctness. The rest of the world became convinced in 1888 when Hertz discovered electromagnetic waves (in the microwave range) with all the properties predicted by Maxwell’s theory.
2.2.6 Putting It All Together
Classical electromagnetism summarizes a vast body of experimental information using the concepts of charge densityρ(r, t), current densityj(r, t), electric fieldE(r, t), and magnetic fieldB(r, t). We have seen in this chapter that partial differential equations connect the fields to the sources and vice versa. Two of these are the explicitly time-dependent curl equations (2.31) and (2.32). Two others are the divergence equations in (2.22) and (2.27), which survive the transition from static fields to time-dependent fields without change. Taken together, we get the foundational equations of our subject and the main result of this chapter, theMaxwell equations:
∇ ·E= ρ 0
∇ ·B=0 (2.33)
∇ ×E= −∂B
∂t ∇ ×B=μ0j+ 1
c2
∂E
∂t. (2.34)
The direct connection to experience comes when we specify the force whichρ(r, t) andj(r, t) exert on a charge densityρ(r, t) and current densityj(r, t):
F(t)=
d3r[ρ(r, t)E(r, t)+j(r, t)×B(r, t)]. (2.35) The electric part of (2.35) generalizes Coulomb’s law of electrostatics to time-dependent situations.
The magnetic part of (2.35) was derived by Oliver Heaviside in 1889 by supplementing the Maxwell equations with a postulated expression for the energy of interaction between a current loop and an external magnetic field. An essentially similar derivation was presented somewhat later by Lorentz.
Following tradition, we will call (2.35) the “Coulomb-Lorentz” force law.
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(b) (a)
Figure 2.5: Contour plot of the valence charge density of crystalline silicon: (a)ρ(r) extracted from X-ray diffraction data; (b)ρ(r) computed from quantum mechanical calculations and the Maxwell equations. Periodic repetition of either plot in the vertical and horizontal directions generates one plane of the crystal. The rectangular contours reflect “bond” charge between the atoms (black circles). The white regions of very low valence charge density reflect nodes in thesp3wave functions. Figure from Tanaka, Takata, and Sakata (2002).