Problems
Application 2.1 Moving Point Charges
2.7 A Heuristic Derivation
To make further progress, we putρ=j=0 and take the curl of one of the two equations in (2.78).
We then use the vector identity∇ ×(∇ ×F)= −∇2F+ ∇(∇ ·F) and substitute in from the other curl equation in (2.78). When these steps are carried out, separately, for each equation in (2.78), the result is
∇2E−km
ke
∂2E
∂t2 =0 and ∇2B−km
ke
∂2B
∂t2 =0. (2.79)
These equations have wave-like solutions which propagate at the speed of lightc(as seen in experiment) if
ke
km
=c2. (2.80)
Until 1983, the value ofcwas determined from experiment. Since that time, the speed of light has been defined (by the General Conference on Weights and Measures) as exactly
c=299 792 458 m/s. (2.81)
This legislation demotes the meter to a derived unit. It is the distance traveled by light in 1/299792458 seconds.
We have said that theraison d’ˆetreof the SI system is to make the joule (J=N·m) the natural unit of electrical energy, just as it is for mechanical energy. In 1901, Giorgi pointed out that this will be the case if the constant in (2.75) is chosen to be
km=10−7N
A2. (2.82)
This choice fixes the definition of the ampere. It is the constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross section, produces a force of 2×10−7N per meter of length when the two wires are separated by a distance of 1 m.
Finally, SI eliminates (“rationalizes”) the factors of 4πin the Maxwell equations by introducing a magnetic constantμ0where
μ0=4π km=4π×10−7 N
A2, (2.83)
and, using (2.80), an electric constant0where
0=1/μ0c2=1/4π ke. (2.84)
With these definitions, (2.77) and (2.78) take the forms quoted in (2.33) and (2.34). No particular physical meaning attaches to eitherμ0or0.
The polarizationPhas dimensions of a volume density of electric dipole moment. The magnetization Mhas dimensions of a volume density of magnetic dipole moment. This is enough information to see that (2.51) and (2.52) are dimensionally correct. Dimensional consistency similarly dictates the appearance of the factors0andμ0in (2.53) and (2.54). This leads to the form of the in-matter Maxwell equations written in (2.55) and (2.56). On the other hand, these definitions imply that in vacuum there is a fieldDand a fieldEwhich describe exactly the same physical state, but which are measured in different units. The same is true forBandHin vacuum.
2.7 A Heuristic Derivation
We have emphasized that the principles of electromagnetism can only be revealed by experiment.
Nevertheless, it is an interesting intellectual exercise to try to deduce the Maxwell equations using only symmetry principles, minimal theoretical assumptions, and (relatively) minimal input from experiment.
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52 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
The most profound discussions of this type exploit the symmetries of special relativity.22 Here, we use a heuristic argument based on inversion and rotational symmetry, translational invariance, and four pieces of experimental information: the existence of the Lorentz force, charge conservation, superposition of fields, and the existence of electromagnetic waves.
We begin with experiment and infer the existence of the electric fieldE(r, t) and the magnetic field B(r, t) from trajectory measurements on charged particles. These reveal the Lorentz force,
F=q(E+υ×B). (2.85)
We will assume that electromagnetism respects the symmetry operations of rotation and inversion.
This means that (2.85) must be unchanged when these orthogonal transformations (Section 1.7) are performed.
Rotational invariance is guaranteed by the fact thatF,E, andυ×Bare all first-rank tensors (vectors).
As for inversion, we deduce from the discussion in Section 1.8.1 thatF,E, andυ×Bmust all be polar vectors or they must all be axial vectors.23The position vectorris a polar vector. Therefore, sinceυ=dr/dtandF=d(mυ)/dt, the forceFis also a polar vector. This means thatEandυ×B must be polar vectors as well. We have just seen thatυis a polar vector. Hence, (1.162) shows thatB must be an axial vector.
Our task now is to guess an equation of motion for each field which respects the same symmetries.
For the moment, we neglect sources of charge and current. The experimental fact of superposition of fields restricts us to linear equations and, for simplicity, we consider only first derivatives of space and time. The most straightforward guesses, that∂E ∂t∝Eand∂B ∂t∝B,cannot be correct because they lead to (unphysical) exponential growth or decay of the fields as a function of time.24The more interesting guesses, that ∂E ∂t∝B and∂B ∂t∝E,cannot be accepted either because they mix polar vectors and axial vectors on different sides of the same equation. The guesses∂E ∂t∝r×B and∂B ∂t∝r×Erepair this problem, but violate the reasonable requirement that the dynamical equations should not change if r→r+cwherecis a constant vector. That is, the field equations should be invariant to uniform translations.
The gradient operator∇ ≡∂ ∂rchanges sign under inversion, but is unchanged whenr→r+c.
This suggests the use of∇ ·Band∇ ·E. Unfortunately, these are scalars and thus unacceptable for the right side of a vector equation of motion. On the other hand, ifk1andk2are constants, viable candidate equations of motion which respect the symmetry operations of rotation, inversion, and translation are
∂E
∂t =k1∇ ×B and ∂B
∂t =k2∇ ×E. (2.86)
We now conduct an experiment which monitors charge and field in two adjacent, infinitesimal cubical volumes of space (Figure 2.11). Our first observation, att=0,finds both boxes empty. The next observation, att=dt,reveals equal and opposite static charges in the boxes and a negatively directed electric field everywhere on their common wall. No magnetic field is detected. By charge conservation, an electric current must have flowed through the wall att= 12dtand separated charges which were spatially coincident att=0.This current is the only possible source of the electric field because the charges and the field appeared simultaneously att=dt.A logical inference is25
∂E
∂t =k3j at t= 12dt. (2.87)
22 See, for example, L.D. Landau and E.M. Lifshitz,The Classical Theory of Fields, 2nd edition (Addison-Wesley, Reading, MA, 1962), Chapter 4.
23 We assume that the electric chargeqis unchanged by spatial inversion.
24 Alternatively, we may demand that both sides of the equation transform indentically under time-reversal. See Table 15.1 of Section 15.1.
25 The alternativeE∝ −jcontradicts the static experimental results att=dt.
Sources, References, and Additional Reading 53
t = 0
– +
t = dt
–
– + +
+ + +
+
– –
– –
E
Figure 2.11: A charge-separation experiment. Att=0 no charge or fields are detected in two adjacent volume elements. Att=dtpositive charge appears in the right box, negative charge appears in the left box, and an electric field is detected on their common face.
Combining (2.87) with (2.86) yields
∂E
∂t =k1∇ ×B+k3j and ∂B
∂t =k2∇ ×E. (2.88)
These are the desired equations of motion.
As a final step, take the divergence of both members of (2.88) and use the continuity equation (2.13).
This gives
∂
∂t(∇ ·B)=0 and ∂
∂t(∇ ·E+k3ρ)=0. (2.89) If we take as “initial conditions” the vanishing of∇ ·Band∇ ·E+k2ρ, the two equations in (2.89) guarantee that these conditions will remain in place for all time:
∇ ·E= −k3ρ and ∇ ·B=0. (2.90)
The structures of (2.88) and (2.90) are now completely determined. It remains only to determine the constants. This depends on the choice of units, as discussed in Section 2.6.
Sources, References, and Additional Reading
The quotation at the beginning of the chapter is from Chapter 2 of
C. Domb,Clerk Maxwell and Modern Science(Athlone Press, Bristol, 1963).
Section 2.1 For more on nearly static atmospheric electric fields and nearly static cosmic magnetic fields, see E.R. Williams, “Sprites, elves, and glow discharge tubes”,Physics Today, November 2001.
R.M. Kulsrud and E.G. Zweibel, “On the origin of cosmic magnetic fields”,Reports on Progress in Physics71, 046901 (2008).
The experimental test of conservation of charge mentioned in the text is
P. Belli, R. Bernabei, C.J. Dai,et al., “New experimental limit on electron stability”,Physics Letters B460, 236 (1999).
Section 2.2 Two well-regarded histories of electromagnetism are
E.T. Whittaker,A History of the Theories of Aether and Electricity(Philosophical Library, New York, 1951).
O. Darrigol,Electrodynamics from Amp `ere to Einstein(University Press, Oxford, 2000).
A thought-provoking essay by a philosopher and historian of science is
E. McMullin, “The origins of the field concept in physics”,Physics in Perspective4, 13 (2002).
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54 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
To learn how Maxwell and his contemporaries thought about electromagnetism in the years before the discovery of the electron, see
M.S. Longair,Theoretical Concepts in Physics(University Press, Cambridge, 1984).
J.Z. Buchwald,From Maxwell to Microphysics(University Press, Chicago, 1985).
B.J. Hunt,The Maxwellians(University Press, Cornell, 1991).
Figure 2.4 was taken from the first reference below. The second reference is Maxwell’sTreatise:
J.C. Maxwell, “On physical lines of force”,Philosophical Magazine21, 281 (1861).
J.C. Maxwell,A Treatise on Electricity and Magnetism(Clarendon, Oxford, 1873).
Our discussion of duality and magnetic charge was inspired by Section 6.11 of J.D. Jackson,Classical Electrodynamics,3rdedition (Wiley, New York, 1999).
Section 2.3 Figure 2.5 was taken from
H. Tanaka, M. Takata, and M. Sakata, “Experimental observation of valence electron density by maximum entropy method”,Journal of the Physical Society of Japan71, 2595 (2002).
An excellent discussion of the passage from microscopic electromagnetism to macroscopic electromagnetism (and much else) is contained in
F.N.H. Robinson,Macroscopic Electromagnetism(Pergamon, Oxford, 1973).
A formal treatment of Lorentz averaging which uses the cell-averaging method of the text is
C. Brouder and S. Rossano, “Microscopic calculation of the constitutive relations”,European Physical Journal B45, 19 (2005).
Microscopic electromagnetic fields near surfaces are discussed in A. Zangwill,Physics at Surfaces(University Press, Cambridge, 1988).
A. Liebsch,Electronic Excitations at Metal Surfaces(Plenum, New York, 1997).
Figure 2.7 was taken from
H. Ishida and A. Liebsch, “Static and quasistatic response of Ag surfaces to a uniform electric field”,Physical Review B66, 155413 (2002).
The delta function approach to deriving the field matching relations has been re-invented many times over the years. A particularly complete treatment is
V. Namias, “Discontinuity of the electromagnetic fields, potentials, and currents at fixed and moving bound- aries",American Journal of Physics56, 898 (1988).
Section 2.4 An interesting discussion of the origin and use of non-local dielectric functions like the one which appears in Section 2.4.2 is
U. Ritschel, L. Wilets, J.J. Rehr, and M. Grabiak, “Non-Local dielectric functions in classical electrostatics and QCD models”,Journal of Physics G: Nuclear and Particle Physics18, 1889 (1992).
Section 2.5 An excellent introduction to non-classical light, quantum optics, and semi-classical radiation theory is
R. Loudon,The Quantum Theory of Light,3rdedition (University Press, Oxford, 2000).
The original proposal to use correlation measurements to distinguish classical from non-classical light was R. Glauber, “The quantum theory of optical coherence”,Physical Review 130, 2529 (1963).
Figure 2.9 was taken from
G.T. Foster, S.L. Mielke, and L.A. Orozco, “Intensity correlations in cavity QED”,Physical Review A61, 53821 (2000).
Electromagnetic tests for “new physics” are the subject of
L.-C. Tu, J. Luo, and G.T. Gilles, “The mass of the photon”, Reports on Progress in Physics 68, 77 (2005).
Q. Bailey and A. Kostelecky, “Lorentz-violating electrostatics and magnetostatics”,Physical Review D70, 76006 (2004).
A.S. Goldhaber and W.P. Trower, “Magnetic monopoles”,American Journal of Physics58, 429 (1990).
Problems 55
Section 2.6 The superiority of either the Gaussian or the SI unit system is self-evident to the passionate advocates of each. We draw the reader’s attention to two articles which inject some levity into this dreary debate:
W.F. Brown, Jr., “Tutorial paper on dimensions and units”,IEEE Transactions on Magnetics20, 112 (1984).
H.B.G. Casimir, “Electromagnetic units”,Helvetica Physica Acta41, 741 (1968).
Section 2.7 This section was constructed from arguments given in
A.B. Midgal,Qualitative Methods in Quantum Theory(W.A. Benjamin, Reading, MA, 1977).
P.B. Visscher,Fields and Electrodynamics(Wiley, New York, 1988).
Problems
2.1 Measuring B LetF1andF2be the instantaneous forces that act on a particle with chargeqwhen it moves through a magnetic fieldB(r) with velocitiesυ1andυ2, respectively. Without choosing a coordinate system, show thatB(r) can be determined from the observablesυ1×F1andυ2×F2ifυ1andυ2are appropriately oriented.
2.2 The Coulomb and Biot-Savart Laws The electric and magnetic fields for time-independent distributions