Download Multipole radiation fields from the Jefimenko equation for the

Multipole radiation fields from the Jefimenko equation for the magnetic
field and the Panofsky-Phillips equation for the electric field
R. de Melo e Souza,a兲 M. V. Cougo-Pinto, and C. Farina
Universidade Federal do Rio de Janeiro, Instituto de Fisica, Rio de Janeiro, RJ 21945-970, Brazil
M. Moriconi
Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ 24210-340, Brazil
共Received 12 February 2008; accepted 8 September 2008兲
We show how to obtain the first multipole contributions to the electromagnetic radiation emitted by
an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the
Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation
of the electromagnetic potentials. © 2009 American Association of Physics Teachers.
关DOI: 10.1119/1.2990666兴
I. INTRODUCTION
The derivation found in most textbooks of the electromagnetic fields generated by arbitrary sources in vacuum starts
by calculating the corresponding electromagnetic potentials
共see, for instance, Ref. 1 and 2, or 3兲. After the retarded
potentials are obtained 共assuming the Lorenz gauge兲 the
electromagnetic fields are calculated with the aid of the relations E = −⵱⌽ − 共1 / c兲共⳵A / ⳵t兲 and B = ⵱ ⫻ A 共we use Gaussian units兲. The resulting expressions for the fields are usually
called Jefimenko’s equations because they appeared for the
first time in the textbook by Jefimenko.4 Jefimenko’s equations are obtained in Ref. 1 from the retarded potentials and
are obtained directly from Maxwell’s equations in Ref. 5.
共Heras6 had already derived Jefimenko’s equations directly
from Maxwell’s equations.兲 References 8 and 7 obtain a less
common form of Jefimenko’s equations for the electric field,
but this form is more convenient for studying radiation.
Griffiths and Heald9 illustrate Jefimenko’s equations by
obtaining the standard Liénard-Wiechert fields for a point
charge. Ton10 provides an alternative derivation of Jefimenko’s equations and three applications, including that of a
point charge in arbitrary motion. Heras has generalized Jefimenko’s equations to include magnetic monopoles and obtained the electric and magnetic fields of a particle with both
electric and magnetic charge in arbitrary motion.11 He has
also discussed Jefimenko’s equations in material media to
obtain the electric and magnetic fields of a dipole in arbitrary
motion12 and has derived Jefimenko’s equations from Maxwell’s equations using the retarded Green function of the
wave equation.6
The main purpose of this paper is to enlarge the list of
problems that are solved directly from Jefimenko’s equations
共or the equivalent兲. Our procedure avoids completely the use
of electromagnetic potentials. Specifically, we shall obtain
the electric dipole, the magnetic dipole, and the electric
quadrupole terms of the multipole expansion due to the radiation fields of an arbitrary localized source.
II. JEFIMENKO’S EQUATIONS FROM MAXWELL’S
EQUATIONS
In this section we present three methods of calculating
Jefimenko’s equations directly from Maxwell’s equations.
The first method closely follows Ref. 5. The second method
makes use of a Fourier transformation as discussed by Ref.
16. For our purposes it suffices to do a Fourier transforma67
Am. J. Phys. 77 共1兲, January 2009
http://aapt.org/ajp
tion only in the temporal coordinate. We shall see that this
method, although longer than the previous one, avoids any
possibility of misleading manipulations with retarded quantities. The subtleties in calculations involving retarded quantities have been discussed in Refs. 13–15. We then present an
alternative method that is a variation of the first one; the
main difference is the order of integration.
A. Direct calculation using the retarded Green function
The following approach is similar to that in Refs. 5, 6, and
13. Maxwell’s equations with sources in vacuum are given
by
⵱ · E = 4␲␳ ,
共1兲
⵱ · B = 0,
共2兲
⵱⫻E=−
⵱⫻B=
1 ⳵B
,
c ⳵t
共3兲
4␲
1 ⳵E
.
J+
c
c ⳵t
共4兲
If we take the curl of Eq. 共3兲 and the time derivative of Eq.
共4兲, we obtain
冉
⵱2 −
冊
冉
冊
1 ⳵2
1 ⳵J
,
2 2 E = 4␲ ⵱␳ + 2
c ⳵t
c ⳵t
共5兲
where we have used ⵱ ⫻ 共⵱ ⫻ F兲 = ⵱共⵱ · F兲 − ⵱2F, with ⵱2F
= 共⵱2Fx兲x̂ + 共⵱2Fy兲ŷ + 共⵱2Fz兲ẑ with F an arbitrary function.
Similarly, we obtain for the magnetic field
冉
⵱2 −
冊
1 ⳵2
4␲
J.
2 2 B=−
c ⳵t
c
共6兲
The solutions of Eqs. 共5兲 and 共6兲 can be obtained with the aid
of the retarded Green function Gret共x , t ; x⬘ , t⬘兲, which satisfies the inhomogeneous differential equation
冉
⵱2 −
冊
1 ⳵2
Gret共x,t;x⬘,t⬘兲 = ␦共x − x⬘兲␦共t − t⬘兲,
c2 ⳵t2
共7兲
and is zero for t − t⬘ ⬍ 0. The solution for Gret for t − t⬘ ⬎ 0 is
given by3
© 2009 American Association of Physics Teachers
67
Gret共x,t;x⬘,t⬘兲 = −
=−
1 ␦共t⬘ − 共t − 兩x − x⬘兩/c兲兲
兩x − x⬘兩
4␲
1 ␦共t⬘ − 共t − R/c兲兲
,
R
4␲
共8兲
where R ⬅ 兩R兩 = 兩x − x⬘兩. With the help of Eq. 共8兲 the solution
of Eq. 共5兲 can be written as
冕 ⬘冕
E共x,t兲 = −
dx
R
冉
dt⬘␦共t⬘ − 共t − R/c兲兲
1 ⳵J共x⬘,t⬘兲
⫻ ⵱⬘␳共x⬘,t⬘兲 + 2
c
⳵t⬘
=−
冕
dx⬘
关⵱⬘␳兴
−
R
冕
dx⬘
=
冕
冕
dx⬘
dx⬘
⵱ ⬘关 ␳ 兴
+
R
关␳兴R̂
+
R2
冕
冕
冊
共9兲
关J̇兴
,
c 2R
dx⬘
共10兲
dx⬘
关␳˙ 兴R̂
−
cR
关␳˙ 兴R̂
−
cR
冕
冕
dx⬘
dx⬘
关J̇兴
c 2R
关J̇兴
.
c 2R
共11兲
共12兲
1
c
=
1
c
冕 ⬘冕 ⬘
冕 ⬘ ⬘
dx
R
dt ␦共t⬘ − 共t − R/c兲兲⵱⬘ ⫻ J共x⬘,t⬘兲
dx
关⵱ ⫻ J兴.
R
共13兲
冕
1
dx⬘
⵱⬘ ⫻ 关J兴 − 2
R
c
冕
R̂ ⫻ 关J̇兴
.
dx⬘
R
We integrate by parts and obtain
68
−
1
c2
dx⬘
冕
Am. J. Phys., Vol. 77, No. 1, January 2009
1
关J兴
+
R
c
dx⬘关J兴 ⫻ ⵱⬘
R̂ ⫻ 关J̇兴
.
R
冉冊
1
R
共15兲
dx⬘
冋
册
关J兴 ⫻ R̂ 关J̇兴 ⫻ R̂
+
.
cR2
c 2R
共16兲
In Sec. II A we showed how to obtain Jefimenko’s equations by a careful treatment of the derivatives of retarded
quantities. This point is crucial—spatial derivatives cannot
be commuted with retarding the functions, because the retarded function depends on the coordinates in its time argument. A simple way to circumvent this difficulty is to use
Fourier transforms and factor out the time dependence of the
functions so these subtleties are not encountered. We will
show how to obtain the electric field and will leave the derivation of the magnetic field as an exercise for the interested
reader 共see, for example, Ref. 16兲.
We start with the electric field given by Eq. 共9兲. The term
involving the current density takes the form after integration
over time,
冕 ⬘冕 ⬘
冕 ⬘
dx
R
dt ␦共t⬘ − 共t − R/c兲兲
=−
dx
1 ⳵J
共x⬘,t⬘兲
c2 ⳵t⬘
关J̇兴
.
c 2R
共17兲
We introduce the Fourier transformation ˜␳共x⬘ , ␻兲 of ␳共x⬘ , t⬘兲
as ␳共x⬘ , t⬘兲 = 兰˜␳共x⬘ , ␻兲e−i␻t⬘d␻ and express the remaining
contribution to the electric field in Eq. 共10兲 as
−
冕 ⬘冕 ⬘ ⬘
冕 ⬘冕 ⬘ ⬘
冕 ⬘ ⬘
冕 冕 ⬘
dx
R
dt ␦共t − 共t − R/c兲兲⵱⬘␳共x⬘,t⬘兲
=−
⫻
=−
共14兲
冕
冉 冊 冕
B. Fourier method
dx
R
dt ␦共t − 共t − R/c兲兲
d␻⵱ ˜␳共x , ␻兲e−i␻t⬘
d␻e−i␻t
=−
If we use the relation ⵱⬘ ⫻ 关J兴 = 关⵱⬘ ⫻ J兴 + 共R̂ / c兲 ⫻ 关J̇兴 共see
Ref. 5兲, Eq. 共13兲 takes the form
1
B共x,t兲 =
c
dx⬘⵱⬘ ⫻
The first term on the right-hand side of Eq. 共15兲 is the retarded Biot-Savart term; the transverse radiation field is contained in the last term. Equations 共12兲 and 共16兲 are the Jefimenko equations.
−
In the last step we integrated by parts and discarded surface
terms, because the charge distribution is localized. Equation
共12兲 is one of the Jefimenko’s equations.4,5 Note the retarded
character of the electric field. The first term on the right-hand
side of Eq. 共12兲 is the retarded Coulomb term.
As shown by Panofsky and Phillips,7 there is an equivalent
way of deriving Eq. 共12兲 for the electric field which manifestly shows the transverse character of the radiation field. A
discussion of this point can be found in Ref. 8; we will
discuss this point in Sec. III.
To obtain the desired expression for the magnetic field, we
use the retarded Green function in Eq. 共8兲 to rewrite Eq. 共6兲
as
B共x,t兲 =
冕
The first term on the right-hand side of Eq. 共15兲 vanishes
because the current distribution is localized in space. We use
the relation ⵱⬘共1 / R兲 = −R̂ / R2 to obtain
B共x,t兲 =
where the notation 关¯兴 means that the quantity inside the
brackets is a function of x⬘ and is evaluated at the retarded
time t⬘ = t − 兩x − x⬘兩 / c.
At this point much care must be taken, because ⵱⬘关␳兴
⫽ 关⵱⬘␳兴. Due to a bad choice of notation, the author in Ref.
13 incorrectly used the quantity 关⵱⬘␳兴 as if it were ⵱⬘关␳兴
and, after an integration by parts, an incomplete result for the
electric field was found, as pointed out in Ref. 14. The correct relation is given by ⵱⬘关␳兴 = 关⵱⬘␳兴 + R̂关␳˙ 兴 / c 共see, for instance, Ref. 5兲. If we use the correct relation, Eq. 共10兲 becomes
E共x,t兲 = −
1
c
B共x,t兲 =
冕
d␻e−i␻t
− ˜␳共x⬘, ␻兲
冉
dx
冕
共18a兲
eikR
⵱⬘˜␳共x⬘, ␻兲
R
再
dx⬘ ⵱⬘
冊
冉
eikR
˜␳共x⬘, ␻兲
R
R̂ ikR
R̂
e
2 − ik
R
R
冎
共18b兲
冊
de Melo e Souza et al.
共18c兲
68
=
冕 ⬘ 冕
冕 ⬘ 冕
+
=
R̂
R2
dx
冕
dx
dx⬘
III. MULTIPOLE RADIATION
VIA JEFIMENKO’S EQUATIONS
˜ 共x⬘, ␻兲e−i␻共t−R/c兲
d␻␳
R̂
cR
R̂关␳兴
+
R2
˜ 共x⬘, ␻兲共− i␻兲e−i␻共t−R/c兲
d␻␳
共18d兲
冕
共18e兲
dx⬘
R̂关␳˙ 兴
.
cR
To obtain Eq. 共18c兲 we integrated by parts; to obtain Eq.
共18d兲 we discarded the surface term.
We combine Eq. 共9兲 with Eqs. 共17兲 and 共18兲 and obtain the
electric field generated by arbitrary 共but localized兲 sources,
namely, Eq. 共12兲. On the right-hand side of Eq. 共18a兲 it is
obvious that ⵱⬘ acts only on ˜␳共x⬘ , ␻兲. Hence, in Eq. 共18b兲
there is no possibility of thinking that ⵱⬘ also acts on the
exponential eikR. The subtleties of dealing with retarded
quantities are circumvented by the Fourier method.
The main purpose of this section is to add to the list of
problems that can be handled directly with Jefimenko’s equations by calculating the first multipole contributions to the
radiation fields of an arbitrary localized source. We shall obtain the first three contributions, namely, the electric dipole,
the magnetic dipole, and the electric quadrupole terms. Most
textbooks treat this problem by calculating first the electromagnetic potentials.
Although Eq. 共12兲 for the electric field gives the correct
expression for the electric field of moving charges, it is preferable for our purposes to write it in an equivalent form as
given in Refs. 7 and 8:
E共x,t兲 =
C. Postponing the delta-function integration
When we are faced with integrals involving delta functions, we are usually tempted to do them first, because they
are so easy. To derive Jefimenko’s equation for the electric
field given by Eq. 共12兲, this procedure is not optimum. Let us
start with Eq. 共9兲:
E共x,t兲 = −
冕 冕
冉
dx⬘
R
dt⬘␦共t⬘ − 共t − R/c兲兲
冊
dx
⫻
冉
再
dt⬘ ␦⬘共t⬘ − 共t − R/c兲兲
1
1
␳共x⬘,t⬘兲⵱⬘R + 2 J共x⬘,t⬘兲
Rc
Rc
冊
冎
冉 冊
+ ␦共t⬘ − 共t − R/c兲兲 ⵱⬘
冕 ⬘冕
dx
+
再
1
␳共x⬘,t⬘兲 .
R
冉
dt⬘ ␦⬘共t⬘ − 共t − R/c兲兲 −
冊
共关J兴 · R̂兲R̂ + 共关J兴 ⫻ R̂兲 ⫻ R̂
cR2
+
dx
共关J̇兴 ⫻ R̂兲 ⫻ R̂
.
c 2R
R̂
␳共x⬘,t⬘兲
Rc
冎
R̂
1
2 J共x⬘,t⬘兲 + ␦共t⬘ − 共t − R/c兲兲␳共x⬘,t⬘兲 2 .
Rc
R
共21兲
Am. J. Phys., Vol. 77, No. 1, January 2009
冉
1
⫻ − ⳵i⬘R
c
冊
=关⵱⬘ · J兴 + 关J̇i兴
共22兲
⳵
兩Ji共x⬘,t⬘兲兩t⬘=t−R/c
⳵t⬘
共23a兲
冉 冊
R̂
Xi
= − 关␳˙ 兴 + 关J̇兴 · ,
cR
c
共23b兲
where Xi ⬅ xi − xi⬘ and in the last step we used the continuity
equation. From Eq. 共23b兲 we have 关␳˙ 兴 = −⵱⬘ · 关J兴 + 关J̇兴 · R̂ / c,
which can be substituted into the second term on the righthand side of Eq. 共12兲 to yield
共20兲
By using the properties of the Dirac delta function, we can
easily perform the integration over t⬘ to obtain Eq. 共12兲. An
analogous calculation provides an expression for the magnetic field.
69
dx
We obtain Eq. 共22兲 starting with Jefimenko’s equation for the
electric field in Eq. 共12兲. Our derivation will follow the one
in Ref. 8. Note that 共the Einstein convention of implicit summation over repeated indices is assumed兲
1
c
If we use the relations ⵱⬘R = −R̂ and ⵱⬘1 / R = R̂ / R2, we obtain
E共x,t兲 =
关␳兴R̂
R2
+
共19兲
Instead of first performing the time integration using the
Dirac delta function, we integrate by parts on both terms on
the right-hand side of Eq. 共19兲 so that it takes the form
冕 ⬘冕
dx
⵱⬘ · 关J兴 = 兩⳵i⬘Ji共x⬘,t⬘兲兩t⬘=t−R/c +
1 ⳵J
⫻ ⵱⬘␳共x⬘,t⬘兲 + 2 共x⬘,t⬘兲 .
c ⳵t⬘
E共x,t兲 =
冕 ⬘
冕 ⬘
冕 ⬘
冕
1
关␳˙ 兴R̂
dx⬘ =
R
c
冕
+
1
c2
共⵱⬘ · 关J兴兲R̂
dx⬘
R
冕
共关J̇兴 · R̂兲R̂
dx⬘ .
R
共24兲
We show that the first term on the right-hand side of Eq. 共24兲
is proportional to 1 / R2, so that it does not contribute to the
radiation field. Observe that
−
1
c
冕
共⵱⬘ · 关J兴兲R̂
dx⬘
R
=−
êk
c
=−
êk
c
冕⬘
冕 ⬘冉 冊
共⳵i 关Ji兴兲
⳵i 关Ji兴
Xk
dx⬘
R2
共25a兲
êk
Xk
dx⬘ +
R2
c
冕 ⬘冉 冊
关Ji兴⳵i
Xk
dx⬘
R2
共25b兲
de Melo e Souza et al.
69
=0 +
=
1
c
1
=
c
冕
êk
c
再
关Ji兴 Xk
冎
2 Xi ␦ki
−
dx⬘
R3 R R2
共25c兲
2共关J兴 · R̂兲R̂ − 关J兴
dx⬘
R2
冕
共关J兴 · R̂兲R̂ + 共关J兴 ⫻ R̂兲 ⫻ R̂
dx⬘ ,
R2
共25d兲
冕
共25e兲
冊 册
x̂ · x⬘
⫻ x̂ ⫻ x̂,
c
where t0 = t − r / c is the retarded time of the origin. A comparison of Eqs. 共28兲 and 共37兲 leads to the relation
冕
冕
1
c
+
1
c2
共关J兴 · R̂兲R̂ + 共关J兴 ⫻ R̂兲 ⫻ R̂
dx⬘
R2
1
共关J̇兴 · R̂兲R̂
dx⬘ − 2
R
c
冕
关J̇兴
dx⬘
R
共26a兲
关␳兴R̂
R2
+
共关J兴 · R̂兲R̂ + 共关J兴 ⫻ R̂兲 ⫻ R̂
dx
cR2
+
共关J̇兴 ⫻ R̂兲 ⫻ R̂
,
dx
c 2R
1
c
Brad共x,t兲 =
冕
冕
dx⬘
1
c 2r
共关J̇兴 ⫻ R̂兲 ⫻ R̂
.
c 2R
dx J̇ x ,t0 +
再冕
dx⬘J̇共x⬘,t0兲 = ei
共26b兲
关J̇兴 ⫻ R̂
,
Rc
冕 ⬘冉⬘
1
c 2r
=
ei
It can be shown17 that for time-varying sources in motion,
Eq. 共27兲 gives the radiation fields plus additional nonradiative terms of order O共1 / R2兲. For instance, if a Hertz dipole is
accelerated, integration of Eq. 共27兲 yields radiation fields
plus nonradiative terms of order O共1 / R2兲 which are induced
by the dipole motion 共see Ref. 17 for details兲.
In the radiation zone we can write R̂ ⯝ x̂, 1 / R ⯝ 1 / r, and
R ⯝ r − x̂ · x⬘, where we defined r = 兩x兩. If we substitute these
approximations into Eq. 共27兲, we obtain
Brad共x,t兲 ⯝
共1兲
共x,t兲 =
Brad
冕
共27兲
dx⬘
Now we are ready to calculate the first multipole contributions for the radiation fields. We need to calculate only one
of the radiation fields because the other is readily obtained
by Eq. 共30兲, which also shows that the fields in the radiation
zone are mutually orthogonal. We start by calculating the
electric dipole term and then consider the next order contribution given by both the magnetic dipole and electric quadrupole terms.
冎
dx⬘J̇共x⬘,t0兲 ⫻ x̂.
共31兲
We write the unit vectors of the Cartesian basis as êi = ⵱⬘xi⬘,
共i = 1 , 2 , 3兲, and write any vector v as v = êivi = êi共v · êi兲. The
integral in Eq. 共31兲 can be expressed as
冕 ⬘
冕 ⬘
冕 ⬘
dx
共30兲
The lowest order contribution to the radiation fields comes
from the electric dipole term. For simplicity, we calculate the
lowest order contribution to the radiation magnetic field,
共1兲
,
which we denote by Brad
which is Eq. 共22兲. If we consider arbitrary time-varying
sources at rest and use Eqs. 共16兲 and 共22兲, the 共transverse兲
magnetic and electric radiation fields are given by
70
dx J̇ x ,t0 +
A. The electric dipole contribution
关␳兴R̂
dx⬘
R2
+
Erad共x,t兲 =
冕 ⬘冋 冉 ⬘
Erad共x,t兲 = Brad共x,t兲 ⫻ x̂.
where we have used the identity 共a ⫻ b兲 ⫻ c = 共a · c兲b
− 共b · c兲a, and the fact that the surface term appearing in Eq.
共25b兲 vanishes because the sources are localized. We substitute Eq. 共25兲 into Eq. 共24兲 and insert the result into Eq. 共12兲
to obtain
=
1
c 2r
共29兲
冕
E共x,t兲 =
Erad共x,t兲 ⯝
冊
x̂ · x⬘
⫻ x̂,
c
Am. J. Phys., Vol. 77, No. 1, January 2009
=ei
冕 ⬘ ⬘
冕 ⬘ ⬘
冕 ⬘⬘ ⬘
冕 ⬘⬘ ⬘
− ei
dx J̇共x ,t0兲 · ei
dx J̇共x ,t0兲 · ⵜ⬘xi⬘
共32a兲
dx ⵱ · 共xi J̇共x⬘,t0兲兲
dx xi ⵜ · J̇共x⬘,t0兲,
共32b兲
where in the last step we integrated by parts. Because we are
considering localized sources, the first integral on the righthand side of Eq. 共32a兲 vanishes 共after the use of Gauss’
theorem this integral is converted to a zero surface term兲.
The remaining integral may be cast into a convenient form if
we use the relation
⵱⬘ · J̇共x⬘,t0兲 = −
⳵2␳共x⬘,t0兲
,
⳵t2
共33兲
which is a direct consequence of the continuity equation. To
obtain Eq. 共33兲 we used the relation ⳵␳共x⬘ , t0兲 / ⳵t
= ⳵␳共x⬘ , t0兲 / ⳵t0 共see Ref. 18, note 5兲. We substitute Eqs. 共33兲
and 共32a兲 into Eq. 共31兲 and obtain
共28兲
共34兲
de Melo e Souza et al.
70
where p共t0兲 is the electric dipole moment of the distribution
at the retarded time t0. Now it is clear why this first term is
called the electric dipole term. The radiation electric field is
readily obtained from Eq. 共30兲
共1兲
共x,t兲 =
Erad
关p̈共t0兲 ⫻ x̂兴 ⫻ x̂
.
c 2r
where we have identified the second time derivative of the
magnetic dipole moment of the charge distribution at the
retarded time as m̈共t0兲. Hence, the antisymmetric contribution for the radiation magnetic field is given by
共35兲
Equations 共31兲 and 共35兲 are the radiation fields of the electric dipole term, that is, the first-order contribution to the
multipole expansion. These expressions are valid for arbitrary but localized, sources in vacuum such as an oscillating
electric dipole.
共2a兲
共x,t兲 =
Brad
冉
J̇ x⬘,t0 +
共2a兲
共x,t兲 =
Erad
冊
x̂ · x⬘
x̂ · x⬘
J̈共x⬘,t0兲.
⬇ J̇共x⬘,t0兲 +
c
c
共36兲
We substitute Eq. 共36兲 into Eq. 共28兲 and identify the next
共2兲
order contribution to the radiation magnetic field, Brad
, given
by
共2兲
共x,t兲 =
Brad
1
c 3r
冉冕
冊
dx⬘J̈共x⬘,t0兲共x̂ · x⬘兲 ⫻ x̂.
x̂ ⫻ m̈共t0兲
.
c 2r
共2s兲
共x,t兲 =
Brad
1 1
c2r 2c
再
冕
dx⬘关J̈共x⬘,t0兲共x̂ · x⬘兲
冎
再
冕
dx⬘关J̈共x⬘,t0兲共x̂ · x⬘兲
冕
dx⬘J̈共x⬘,t0兲共x̂ · x⬘兲 = ei
冕
dx⬘共x̂ · x⬘兲J̈共x⬘,t0兲 · ei
=ei
冕
dx⬘共x̂ · x⬘兲J̈共x⬘,t0兲 · ⵜ⬘xi⬘ . 共43b兲
冎
共38兲
dx⬘关J̈共x⬘,t0兲x̂ · x⬘
− 共J̈共x⬘,t0兲 · x̂兲x⬘兴 ⫻ x̂
=
71
1
c 2r
再 冉冕
dx⬘
冊
x⬘ ⫻ J̈共x⬘,t0兲
⫻ x̂
2c
m̈共t0兲
Am. J. Phys., Vol. 77, No. 1, January 2009
共43a兲
dx⬘J̈共x⬘,t0兲共x̂ · x⬘兲
= − ei
For pedagogical reasons we shall treat the magnetic dipole
and electric quadrupole cases separately.
Consider the antisymmetric term of Eq. 共38兲. We denote
共2a兲
, use 共a ⫻ b兲 ⫻ c = 共c · a兲b − 共c · b兲a,
this contribution by Brad
and write it in the following suggestive way:
冕
共42兲
We manipulate the first integral on the right-hand side of Eq.
共42兲 in the same way as before. We have
冕
+ 共J̈共x⬘,t0兲 · x̂兲x⬘兴 ⫻ x̂.
1 1
共2a兲
共x,t兲 = 2
Brad
c r 2c
dx⬘兵J̈共x⬘,t0兲共x̂ · x⬘兲
If we integrate by parts and remember that the surface term
vanishes, we obtain
− 共J̈共x⬘,t0兲 · x̂兲x⬘兴 ⫻ x̂
1 1
+ 2
c r 2c
冕
+ 共J̈共x⬘,t0兲 · x̂兲x⬘其 ⫻ x̂.
⫻共x̂ · x⬘兲 and sum and subtract 21 关J̈共x⬘ , t0兲 · x̂兴x⬘ in the integrand of the right-hand side of Eq. 共37兲, we obtain
1 1
共2兲
共x,t兲 = 2
Brad
c r 2c
共41兲
Expressions 共40兲 and 共41兲 are valid for an arbitrary, but localized, time-varying source at rest in vacuum such as an
oscillating magnetic dipole.
共2s兲
. We
We denote the symmetric term of Eq. 共38兲 as Brad
have
共37兲
For reasons that will become clear we will split the integral
in Eq. 共37兲 into antisymmetric and symmetric contributions
under the exchange of J and x⬘. This rearrangement will give
rise to the magnetic dipole and electric quadrupole terms of
the multipole expansion for the radiation fields.
If we write J̈共x⬘ , t0兲共x̂ · x⬘兲 as 21 J̈共x⬘ , t0兲共x̂ · x⬘兲 + 21 J̈共x⬘ , t0兲
共40兲
The appearance of the magnetic dipole moment of the distribution justifies the name given to this contribution. The corresponding radiation electric field is readily given by
B. Next order contribution
To calculate the next order term we need to take into account the second term of the expansion
关m̈共t0兲 ⫻ x̂兴 ⫻ x̂
.
c 2r
冎
共39a兲
=ei
冕
共39b兲
dx⬘xi⬘⵱⬘ · 关共x̂ · x⬘兲J̈共x⬘,t0兲兴
共44a兲
dx⬘xi⬘关共ⵜ⬘共x̂ · x⬘兲兲 · J̈共x⬘,t0兲
+ 共x̂ · x⬘兲ⵜ⬘ · J̈共x⬘,t0兲兴.
共44b兲
We use ⵱⬘ · J̈共x⬘ , t0兲 = −⳵3␳共x⬘ , t0兲 / ⳵t3, the relation18
⳵␳共x⬘ , t0兲 / ⳵t = ⳵␳共x⬘ , t0兲 / ⳵t0兲, and ⵱⬘共x̂ · x⬘兲 = x̂ to obtain
冕
dx⬘J̈共x⬘,t0兲共x̂ · x⬘兲 = − ei
+
ei
⫻ x̂,
冕
⳵3
⳵t3
冕
冕
dx⬘x̂ · J̈共x⬘,t0兲xi⬘
dx⬘共x̂ · x⬘兲xi⬘␳共x⬘,t0兲,
共45兲
which implies
de Melo e Souza et al.
71
冕
dx⬘关J̈共x⬘,t0兲共x̂ · x⬘兲 + 共J̈共x⬘,t0兲 · x̂兲x⬘兴
=
⳵3
⳵t3
冕
dx⬘共x̂ · x⬘兲x⬘␳共x⬘,t0兲.
共46兲
given by Eqs. 共40兲 and 共41兲, are the first corrections to the
leading order term given by Eqs. 共34兲 and 共35兲. In this sense,
we can write the first multipole contributions to the multipole
expansion for the radiation fields of a completely arbitrary,
but localized, time-varying source at rest in vacuum as
The left-hand side of Eq. 共46兲 is the integral in Eq. 共42兲. We
write the factor 1 / 2 in Eq. 共42兲 as 3 / 6 and obtain
共2s兲
共x,t兲 =
Brad
=
再
1
⳵3
3
6c r ⳵t3
再
1
⳵3
3
6c r ⳵t3
冕
冕
冎
冎
dx⬘共3x̂ · x⬘兲x⬘␳共x⬘,t0兲 ⫻ x̂
共47a兲
共47b兲
dx⬘关3共␰ · x⬘兲x⬘ + r⬘2␰兴␳共x⬘,t兲.
共48兲
1 ត
Q共x̂,t0兲 ⫻ x̂.
6c3r
共49兲
The corresponding symmetric contribution for the radiating
electric field is obtained from Eq. 共30兲:
共2s兲
共x,t兲 =
Erad
1 ត
关Q共x̂,t0兲 ⫻ x̂兴 ⫻ x̂.
6c3r
共50兲
In order to interpret the above term, let us define a linear
operator Qt, for a fixed instant t, by Qt共␰兲 ⬅ Q共␰ , t兲. Note
that Qt is a linear operator so that Qt共␣1␰1 + ␣2␰2兲
= ␣1Qt共␰1兲 + ␣2Qt共␰2兲, for ␣1, ␣2 苸 R. The linear operator Qt
is called the electric quadrupole operator. Because Qt共êi兲 is a
vector in R3, we can write it as a linear combination of the
basis vectors, namely
3
Q 共êi兲 = 兺 Qtjiê j
t
共i = 1,2,3兲.
共51兲
j=1
The coefficients Qtji are the Cartesian elements of the electric
quadrupole tensor 共a second rank tensor兲 of the distribution
at instant t. That is why we interpret Eqs. 共49兲 and 共50兲 as the
electric quadrupole contribution for the radiation fields.
The electric quadrupole radiation fields given by Eqs. 共49兲
and 共50兲, together with the magnetic dipole radiation fields
72
Erad共x,t兲 = Brad共x,t兲 ⫻ x̂.
共52兲
共53兲
ACKNOWLEDGMENTS
The authors are indebted to the referees for their valuable
comments and suggestions. R.S.M. and C.F. would like to
thank CNPq for partial financial support and M.M. would
like to thank FAPERJ for financial support.
a兲
Electronic mail: [email protected]
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1
This transformation takes a vector and an instant of time
共␰ , t兲 and transforms it into the vector Q共␰ , t兲, given by Eq.
共48兲. If we use the definition 共48兲, Eq. 共47a兲 for the antisym共2兲
becomes
metric contribution for Brad
共2s兲
共x,t兲 =
Brad
冎
1 ᠮ
Q共x̂,t0兲 ⫻ x̂ + ¯ ,
3c
With these radiation fields, we can calculate the corresponding Poynting vector and the power radiated by the source.
where in the last step we included in the integrand the term
r⬘2x̂␳共x⬘ , t0兲, where r⬘ = 兩r⬘兩, which gives a vanishing contribution to the result because x̂ ⫻ x̂ = 0. Now, we define the
transformation Q such that
冕
再
1
p̈共t0兲 ⫻ x̂ + 关m̈共t0兲 ⫻ x̂兴 ⫻ x̂
c 2r
+
dx⬘共3共x̂ · x⬘兲x⬘ + r⬘2x̂兲
⫻␳共x⬘,t0兲 ⫻ x̂,
Q共␰,t兲 =
Brad共x,t兲 =
Am. J. Phys., Vol. 77, No. 1, January 2009
de Melo e Souza et al.
72