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Note of explanation: J“ “m P The reader may be startled to ﬁnd (in all but the earliest printings) the association
of Danish physicist Ludvig V. Lorenz’s name instead of Dutch physicist Hendrik A. L0—
rentz’s with the relation (6.14) between the scalar and vector potentials. Yet it is a fact
that in 1867 Lorenz, in a paper entitled “On the identity of the vibrations of light with
electrical currents,” (op. cit.) exploited the retarded solutions for the potentials, derived
(6.14) and equations equivalent to wave equations for the electric ﬁeld, and discussed the
characteristics of light propagation in conductors and transparent media, contemporane
ously with Maxwell. H. A. Lorentz has ample recognition in physics terminology without
the misattribution of (6.14) to him (by others, beginning around 1900). As Van Bladel*
observes, it is up to textbook authors to accord Lorenz his due.‘ *1. Van Bladel, IEEE Antennas and Propagation Magazine 33, No. 2, 69 (April 1991). ‘ TAn earlier author who deplored the lack of recognition of Lorenz’s contributions is A. O’Rahilly,
Electromagnetic Theory, Dover Publications, New York (1965) [originally published as Electromag—
netics, Longman Green and Cork University Press (1938)], footnote, p. 184. (a) 240 Chapter 6 Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws—SI form of the Maxwell equations. Then the inhomogeneous equations in (6.6) can
be written in terms of the potentials as .51
V26) + 0—: (V  A) = —p."€0 (6.10)
1 62A 1 ad)
2 e 4 —— —  + — f : — .
V A C, ar2 V(V A C, at) uOJ (611) We have now reduced the set of four Maxwell equations to two equations. But they are still coupled equations. The uncoupling can be accomplished by exploit~
ing the arbitrariness involved in the deﬁnition of the potentials. Since B is deﬁned through (6.7) in terms of A, the vector potential is arbitrary to the extent that
the gradient of some scalar function A can be added. Thus B is left unchanged by the transformation,
A —> A' = A + VA (6.12) For the electric ﬁeld (6.9) to be unchanged as well, the scalar potential must be
simultaneously transformed, A
(D—><I)'=LD—(L (6.13)
at
The freedom implied by (6.12) and (6.13) means that we can choose a set of potentials (A, (I!) to satisfy the Lorenz condition (1867),* 16(1)
VA+—ﬁ:0 6.14 This will uncouple the pair of equations (6.10) and (6.11) and leave two inho
mogeneous wave equations, one for <1) and one for A: 1 62¢) V26 — E W : —p/e0 (6.15)
1 62A VZA — E y = _,aUJ (6.16) Equations (6.15) and (6.16), plus (6.14), form a set of equations equivalent in all
respects to the Maxwell equations in vacuum, as observed by Lorenz and others. 6.3 Gauge Transformations, Lorenz Gauge, Coulomb Gauge The transformation (6.12) and (6.13) is called a gauge transformation, and the
invariance of the ﬁelds under such transformations is called gauge invariance. To
see that potentials can always be found to satisfy the Lorenz condition, suppose
that the potentials A, (I) that satisfy (6.10) and (6.11) do not satisfy (6.14). Then
let us make a gauge transformation to potentials A’, (13' and demand that A’, (IN satisfy the Lorenz condition: 16(1)’ 1 1 2
VA’+—,—=02V~A+;Q+V2Ai—Za—% (6.17)
C a: C a: c at *L. V. Lorenz, Phil. Mag. Ser. 3, 34, 287 (1867). See also p. 294. 180 . Classical Electrodynamics The deﬁnition of B and E in terms of the potentials A and (1) according to
(6.29) and (6.31) satisﬁes identically the two homogeneous Maxwell’s
equations. The dynamic behavior of A and (I) will be determined by the
two inhomogeneous equations in (6.28). At this stage it is convenient to restrict our considerations to the
microscopic form of Maxwell’s equations. Then the inhomogeneous
equations in (6.28) can be written in terms of the potentials as V26) + igwA) = —47'rp (6.32)
2
wit—$37?— (VA+%—a§)=—4:wI (6.33) We have now reduced the set of four Maxwell’s equations to two equations.
But they are still coupled equations. The uncoupling can be accomplished
by exploiting the arbitrariness involved in the deﬁnition of the potentials.
Since 13 is deﬁned through (6.29) in terms of A, the vector potential is
arbitrary to the extent that the gradient of some scalar function A can be
added. Thus B is left unchanged by the transformation, A —+ A’ = A + VA (6.34) In order that the electric ﬁeld (6.31) be unchanged as well, the scalar
potential must be simultaneously transformed, (D—>(I)'=<D—l% (6.35)
c at The freedom implied by (6.34) and (6.35) means that we can choose a set of potentials (A, (D) such that
1 6(1) VA+——=0 (6.36)
car This will uncouple the pair of equations (6.32) and (6.33) and leave two
inhomogeneous wave equations, one for (D and one for A: 1 62(1) V26) — c—2 g = —4m'p (6.37)
1 32A 4n VzA—F—=——J 6.38
c2 312 c ( ) Equations (6.37) and (6.38), plus (6.36), form a set of equations equivalent
in all respects to Maxwell‘s equations. E1 [Sect. 6.5] TimeVarying Fields, Maxwell’s Equations, Conservation Laws 181 6.5 Gauge Transformations; Lorentz Gauge; Coulomb Gauge The transformation (6.34) and (6.35) is called a gauge transformation,
and the invariance of the ﬁelds under such transformations is called gauge
invariance. The relation (6.36) between A and (I) is called the Lorentz
condition. To see that potentials can always be found to satisfy the
Lorentz condition, suppose that the potentials A, (I) which satisfy (6.32)
and (6.33) do not satisfy (6.36). Then let us make a gauge transformation
to potentials A', (If and demand that A’, (13’ satisfy the Lorentz condition: lady 13(1) 132A
VA'  =0=VA — V2A——— 6.39
+cat +c6t+ 62812 ( ) Thus, provided a gauge function A can be found to satisfy 132A 13¢)
VzA————=—(V'A ——) 6.4
c2 at2 + c at ( 0) the new potentials A’, (D’ will satisfy the Lorentz condition and the wave
equations (6.37) and (638). Even for potentials which satisfy the Lorentz condition (6.36) there is
arbitrariness. Evidently the restricted gauge transformation, AaA +VA
6.41
(Daub—1% ( )
c 3r
1 3%
h VzA———=0  6.42
w ere 02 812 ( ) preserves the Lorentz condition, provided A, (I) satisfy it initially. All
potentials in this restricted class are said to belong to the Lorentz gauge.
The Lorentz gauge is commonly used, ﬁrst because it leads to the wave
equations (6.37) and (6.38) which treat (D and A on equivalent footings,
and second because it is a concept which is independent of the coordinate
system chosen and so fits naturally into the considerations of special
relativity (see Section 11.9). Another useful gauge for the potentials is the socalled Coulomb or
transverse gouge. This is the gauge in which V  A = o (6.43) ...
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