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23 November
1998
PHYSICS
LETTERS
A
Physics Letters A 249 ( 1998) 1-9
ELXWER
Electric and magnetic fields of a toroidal dipole
in arbitrary motion
Jo& A. Heras
lnstituto de Fkica, Universidad National Autdnoma de MPxico, Apartado Postal 20-364. OlooO M.kxico, D.F:. Mexico
Received I5 August 1998; accepted for publication 10 September
Communicated by V.M. Agranovich
1998
Abstract
Electric and magnetic fields of an arbitrarily moving particle possessing a constant toroidal moment -r are derived from
the general solution of Maxwell’s equations for electric and magnetic fields of a toroidal moment density. The fields divide
themselves naturally into five parts: the very near fields which vary as l/p
and depend on I,@; the near fields which
vary as i/R” and depend on T, /3 and 8; the intermediate fields which vary as l/R2 and depend on T, 0, fi and & the far
...
fields which vary as 1/R and depend on 7, fi, p, p and p; and the delta-terms which represent the fields inside the source.
The total power radiated by this dipole is then calculated for the special case in which the velocity and its derivatives are
parallel. @ 1998 Elsevier Science B.V.
PACS: 03.5O.De: 41.20.Bt;
4l.lO.H~
1. Introduction
In 1967 Dubovik and Cheshkov [ l] discovered a third family of moments in the classical electrodynamics,
the so-called toroidal moments which are independent of the electric and magnetic moments. Since then
the interest in this new class of moments has increased considerably
(an early review on toroidal moments
was presented by Dubovik and Tosunyan [2] and a more recent review by Dubovik and Tugushev [ 31).
Ginzburg and Tsytovich 141 were apparently the first in investigating the fields of a point toroid dipole in
uniform motion. They discussed the Cherenkov radiation emitted by a toroidal dipole. The fields of a dipole
at rest possessing a time-varying toroidal moment were discussed by Dubovik and Shabanov [ 5 J . Afanasiev
and Stepanovsky
[ 61 have investigated the radiation fields of toroidal-like time-dependent
sources. Explicit
expressions for the radiation fields of a toroidal dipole in arbitrary motion and formulas for the radiated power
and the radiation reaction force of a nonrelativistic
toroidal dipole have recently been derived by the present
author [7]. The angular momentum loss by a radiating toroidal dipole has recently been discussed by Radescu
and Vlad [ 81.
In this paper a more general treatment of the theory of moving toroidal dipoles is presented. Specifically,
the general solution of Maxwell’s equations for electric and magnetic fields of a toroidal moment density in
vacuum is obtained by means of a time-dependent
extension of Helmholtz’s theorem [9,10]. The solution
037%9601/98/$
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2
J.A. Heras/Physics
Letters A 249 (1998) 1-9
shows that both the electric field and the magnetic field are composed of two terms: a retarded-integral
term
representing the corresponding field outside the toroidal source plus a contact term (that evaluated at the field
point and the present time) describing the field inside the source. It is shown that the magnetostatic field of a
time-independent
toroidal source is entirely confined. The general solution is used first to derive the fields of a
point dipole at rest possessing a time-varying toroidal moment. In a second application, general expressions for
the electric and magnetic fields of a moving particle possessing a time-varying toroidal moment are derived.
These fields are written in terms of conventional parameters (n, B, B, . ..) when the toroidal moment is constant.
These fields divide themselves naturally into five parts: the expected l/R, l/R*, 1/R3 parts; the novel l/R”
part and the delta-function
part. The total power radiated by this toroidal dipole is then calculated when the
velocity and its derivatives are parallel.
2. Time-dependent
extension of Helmholtz’s
theorem
The classical Helmholtz theorem of vector analysis can be formulated for a time-dependent
vector field
F(x, t) which propagates hyperbolically
in vacuum and vanishes at infinity [9,10]. This extension of the
theorem states that F is determined by specifying its divergence, curl and time derivative. An expression for F
is given by [ lo]
F(x,r)
=
&
SS(
{v’.F(x’,t’))V’G(x,t;x’,t’)
+ $G(x,t;x’,i’)-&{
+ (V’ x F’(x’J’)}
x V’G(x,t;x’,t’)
dF(a:;‘r’)})d3x’dr’,
(1)
where the time integration is from -KJ to +oc and the spatial integration is over all space. G( x, t; x’, t’) =
S( t’ + R/c - t) /R is the free-space retarded Green function of the wave equation, 6 being the delta function,
R =) x - x’ 1, x the field point, x’ the source point and c the speed of light.
3. The electric and magnetic fields of a toroidal source
As pointed out by Dubovik and Shabanov [ 51, in classical electrodynamics
there exist toroidal sources apart
from the magnetization
and polarization sources. The effective current associated with the toroidal moment
density T(x, t) (which might be called “toroidization”)
is given by Jeff = cV x (V x T). Thus, Maxwell’s
equations for electric and magnetic fields of a toroidal sources in vacuum and Gaussian units take the form
V.E=O,
(2a)
V.B=O,
(2b)
VxE+;y=O,
(2c)
VXB-;~
1 37E
=&rVx(VxT).
These equations can be integrated by applying Eq. ( 1). In fact, if the quantities
specified by means of Eqs. (2a), (2~) and (2d), then IQ. ( 1) yields the result
(2d)
V . E, V x E and dE/dt
are
J.A. Heras/Physics
Similarly,
B=-
the use of Eqs. (2b)-(2d)
Letters A 249 (1998) l-9
into Eq. ( 1) yields the expression
J.I
V’G x [V’ x (V’ x T)] d3xrdt’.
(3b)
Eqs. (3) constitute the solution of Eqs. (2) subject to the conditions that the fields E and B, and their
derivatives, vanish at infinity and the source T is confined.
The spatial derivatives in Eqs. (3) can be transformed into time derivatives. To do this the following results
are required,
d as(u)
- cR -g-
a/G = $6(u)
3nsni
a”af’G =
(
_
pi
(da)
f$ 6”‘s (x -
-
R3
,
x’) ) S( u )
(3”“5,@)
-
(u = t’ + R/c - t, (n = R/R)’ = n’ and S’j is the Kronecker
can be transformed into the convenient form
E=-
ss(
3n(n
+
R3c
IZXT
-+R=c=
B=
* F) - F
nxT
3n(n
* T) - i;
R=c=
aTt4
I
az,“,jd,
(g)
(4b)
delta). With the aid of these expressions,
Eqs. (3)
...
+ ’ ’ ‘,“,:
S(u) d3x’dt’ - g
T,
$
,
(5a)
>
...
RC3 >
S(u) d3x’ dt’ + 47rV x T ,
(5b)
where a overdot means differentiation
with respect to t’. In obtaining Eqs. (5), the boundary conditions have
been used. It is interesting to note that the terms -( 8~/3c) (aT/at)
and 497V x T in Eqs. (5) are vector
functions evaluated at the field point and the present time - without the presence of these terms, Eqs. (5) do
not strictly satisfy Maxwell’s equations (2). These “contact” terms represent the values of the fields inside the
toroidal source. The values of the fields outside the toroidal source are given by the corresponding
retarded
integrals appearing in Eqs. (5).
Evidently, Eqs. (5) reduce to
B=4rVxT,
(6)
for a time-independent
toroidal source. Since T is confined, the field B in Eq. (6) is entirely confined. This is
a remarkable property of the magnetostatic field of a toroidal source. The magnetostatic field of a magnetized
source does not exhibit this property [ 111.
4. The oscillating toroidal dipole
As a first application of Eqs. (5) consider a particle at rest (at the point xc) with a toroidal moment
oscillating in time r = r(t) e, where r(t) is the magnitude of the toroidal moment which is a periodic function
of time and e is its direction. The associated toroidization vector is given by T(x, t) = eT( t) S(x - x0). With
this specific source, Eqs. (5) are integrated. The result is
E=
B=
_ T(t’){3(e*n)n
R2c2
_+(t’){3(e.n)n-e}
R3c
fqt’> e x II
R2C2
+
‘i’(t’)e
Rc3
x n
- 47n-(t)e
-e}
-
?(t’)n
x VS{x -x0},
x (n x e)
Rc3
--- 81r dr e&x
3c dt
-
~0)
,
(74
(7b)
4
J.A. Hews/Physics
Letters A 249 (1998) 1-9
where now n = (x - xo)/{x - x0\ and the overdot means differentiation
with respect to t’ = t - R/c with
R = Ix - x01. Eqs. (7) are similar to those given by Dubovik and Shabanov in Ref. [ 51. Eq. (7a) (without
the delta term) was also considered in Ref. [ 81.
5. A toroidal dipole in arbitrary
motion
The problem of finding the fields of a toroidal dipole in motion is considerably more complicated than that
of computing the fields of a toroidal dipole at rest. The relativistic transformation of the toroidization demands
that, in the same sense that a moving magnetization has an associated polarization, a moving toroidization has
necessarily an associated axial vector density which should appear on the right-hand side of Eq. (2a). However,
this problem may be avoided to a certain extent by assuming that the dipole is observed in a frame where there
is only toroidization and it is given by T(x, t) = 7(t) S{x - r(t)}.
After an integration by parts, Eqs. (5) can be written in the equivalent form
(W
(8b)
It is should be noted that the 1/R parts of these equations,
(94
(9b)
have been derived previously by the author [ 71. For point toroidal sources, Eqs. (9) yield the associated
radiation fields: if the time-dependent
toroidal source is at rest, as the oscillating toroidal dipole, then Eqs. (9)
give only radiation fields, that is, E1iR = E,d and Bl/R = Brad. But if the source is moving in arbitrary manner
then Eqs. (9) yield radiation fields plus nonradiative terms, that is, in general E~IR = Erad+ nonradiative terms
and Bl JR = B,,j + nonradiative
terms.
With the source T(x, t) = 7(t) S{x - r(t)}, Eqs. (8) yield the electric and magnetic fields of an arbitrarily
moving dipole possessing a time-varying toroidal moment,
3n(n
R2(1-n.@)c2
-=j-f{x
87r dr
r(t)} - ~$S{X
ltX7
R2(1 -n
*7)-7
1 [
1 [
ret
d3
dt3
I1x (nx7)
R(l-n./?)c3
1
ret
(104
-r(r)},
nxr
./3)c2 Etf dt3 R(1 - n~fi)c3
d3
_-
+47rVS{x
-r(t)}
x 7,
(lob)
where now n = (X - r (t’) )/IX - r( t’) ( and the square brackets with the subscript “ret” means that the bracketed
with R = Ix - r( t’) I. Although Eqs. (10)
quantity is to be evaluated at the retarded time t’ = t - R(t’)/c
have a relatively simple form, they do not exhibit explicitly the useful separation of the fields into their l/R,
/.A. Hems/Physics
Letters A 249 (1998) 1-9
5
1/R2, . . . parts. Such a separation of the fields, however, can be accomplished by performing all the specified
time derivatives in Eqs. (10). This task, although straightforward, is extremely laborious. It involves long and
complicated vector manipulations
and the full expressions obtained for the fields turn out to be very lengthy.
By performing the specified derivatives in Eqs. (10) and making dT/dt = 0 and K = 1 - n - /?, one obtains
E=-
{3n(n.7)
-T}
[
{:s(X&)+~~(&)}
3 dn
+cdix
fnx
(lla)
B=
1 d3n 1
>gRK+Fp
1 d2n
(1 lb)
are given by [ 111
where the explicit time-derivatives
1 dn
--_=
c dt
1 d2n
--=
c= dt2
n x (n x p)
RK
’
nx{(n-P)XP}+nx(nx8)(1-P2)
RK3c
ld
1
4
-c dt ( RK ) =z+
n-P
=jgg+
Id
1
_ n*P
-_
c dt ( R3K )
R3 K3c
ld=
1
-= 3(n.P)=
c= dt* ( RK >
RK5c2
Pm (n-P>
R=K3
P.(n-P>
R3K3
I P*(n-P>
pK3
+
n-i)
- n(nx/?)2_nx(nxp)
R=K=
R2K3
’
’
n-/3
+m’
+2n.p
R4K2 ’
+ 2{2P*
RK4c2
(n -P)
- (n x f02}(n.j?)
R2KSc
+{nX(nX8)-2P+n}.B+P.(n-P){28.(n-8)-(nxP)*}
R* K4c
R=K
R3K5
_ (n x p>=
R3K4
(120
J.A. Heras/Physics Letters A 249 (1998) l-9
3(?24)2
R2 K5c2
+ {n x (n xp)
+
r2.S
-2P+n}.p+3(n.p)(n.b)
R3 K4c
+ P*(n-P>{W*(n-PI
-(n
R4K5
The third time derivatives
1 d3n
--_=__
c3 dt3
1 d2
212P.
R2K4c2 +
(n - P) - (n X B2}(n
R3 K=c
+
.i?)
n-b
R3 K3c
x P12} _ (n x/3>2+3n./3{/3.(n-/3)}
R4K4
(1%)
in Eqs. ( 11) are much lengthier. They can be written as
i,
nx
c2 dt2 ( RK >
(nxP>l
,f~(~)[(n.P)t~+n(~~.B)+n(n.~~)+~~]
)
( 13a)
.
( 13b)
The explicit form of Eqs. ( 13) follows from using Eqs. ( 12a)-( 12g) and
1dP
b
c dr
Kc’
--=a
1 d2/3
-----_c2 dt2
ldp
--=c dt
(14a)
p
K2c2
jj
Kc’
+
(14b)
( 14c)
(14d)
J.A. Heras/Physics
3n.p
7
Letters A 249 (1998) 1-9
2(nxP)2-&(n-j3)
t
R2K5c
(nxB2-2P.(n-P)
R3K5
t 14f)
2(n x p)’
=--
RK4c
8(n.B)2
K6c3
+
14e)
(14g)
’
16(n x b)2(n.
+ 2(n - S)
K5C’-
2nx cnx P) *P-d(n X p>
RK5c2
p)
RK6c’
* (n X
p) +
8(n x /3)4 _ 6(n x P)2(n
R2K6c
R2K5c
.p)
( 14h)
In Eqs. ( 11) and ( 13) the notation [ dF/dt],,
means dF( t’) /dt and not dF( t’)/dt’, that is, the “ret” outside
the square brackets applies to the arguments of the functions inside and not to the variable of differentiation.
Eqs. ( 11) are the electric and magnetic fields of an arbitrarily moving particle possessing a constant toroidal
moment. A detailed interpretation of these fields is complicated. However, there are some points that are relevant
for an interpretation of these equation:
(i) Very neal; near; intermediate and fur jields. When Eqs. (12)-( 14) are used in Eqs. ( 1 1 ), the fields
separate naturally into five parts: the very near fields Every near and B,, nex, which vary as 1/R4 and depend
on p and r; the near fields E,,, and B,,,, which vary as l/R3 and depend on j3, b and r; the intermediate
fields Eint and Bi,t, which vary as I/R2 and depend on p, fi, $ and r; the far fields Ef, and Bf,, which vary
as 1/R and depend on /?, /% j), b and r; and the delta-fields
source. Thus, the complete fields read
E = Eve, near + Enem+ Einr+ Far + EM
B = &ry
mill
+
E de1and Bdel which represent
the fields within the
3
t
&em+ Bint+ Bfx + Bdel.
15a)
(1%)
It should be emphasized that the delta fields E &I = -(8rrr/3c)d[6{x
- r(t)}]/dt
and &jet = 4rrVS{x r(t)} x T are evaluated at the field point and the present time; they are essential for achieving the consistency
of Eqs. ( 11) with the Maxwell equations (2). The novelty in Eqs. (15) is the presence of the very near fields,
which are not present in the fields of electric and magnetic dipoles.
(ii) Static limit. When the velocity and its derivatives are zero, that is, when the toroidal dipole is at absolute
rest, Eq. ( 1 la) becomes E = 0 and Eq. (1 lb) reduces to the static form: B = 47rV6{x - x0} x 7, where x0
is the point where the dipole is at rest. As may be seen, the field B is completely confined to a point.
(iii) Uniform motion. It follows from Eqs. ( 11) that the fields of a toroidal dipole in uniform motion
(/I = B = 2 = 0) exhibit an exceedingly complicated form. The electric field is given by the l/R” part of
Eq. ( 1 la) plus the associated contact term and the magnetic field by the l/R” part of Eq. ( 1 lb) plus the delta
term, that is, E = Every nem + E&l and B = Bvery near+ Bdel.
(iv) RudiutionJields.
The far fields derived from Eqs. ( I 1),
( 16b)
are radiation fields. These fields depend on j?,&&
to assume p = 0 and B = 0 at least instantaneously.
and p, which are in general independent.
In this case Eqs. ( 16) reduce to
This allows one
J.A. Heras/Physics
Letters A 249 (1998) 1-9
...
= [nlret x &ad.
&ad
1 ret ’
(174
Therefore, even when both the velocity and the acceleration of a toroidal dipole are instantaneously
equal to
zero (at the retarded time), the dipole can still produce a radiation field on account of the second derivative of
its acceleration.
6. Radiated power by a toroidal dipole
The radiated power P( t’) expressed
dP(t’)
dLI
IREI [ 1 -
= ;
in terms of the dipole’s own time is given by
n * j?],,
(18)
.
By using Eq. ( 16a), one obtains
..
dP( t’)
-=
da
A@
[g
(:2y$j3
- (n. T)2]
+ 3oqy)4;;,;B)
[
...
+ 2O(n-jQ(n.i))(n./Lq
+ 30(?24)3(n4)
(1 -n.P)‘O
(I -n.j3)”
In order to find the total radiated
power P(t’)
...
(n*P12
+ (1 --n*P)9
+ 1oo~;wm&~)2
II
(19)
*et’
at a fixed time t’, it is necessary
to specify the direction
of
the vectors 7, p,p, p and j. The simplest example is one in which the vectors /3,& fi and p are parallel.
Thus, consider a dipole with toroidal moment in direction of the z-axis and moving along this axis. Therefore,
p = tp,
jj = e+,
fl = ijj,
n = f(sin8cos~$)
P(t’)
jj = ij
+ j(sinesin+)
225r2
= y&
i
=
p”
s
an d r = ir. With these specific values and with dR = sin 8 dr9d4
+ tcos0, Eq. (19) is first integrated over 4,
sin3 8 co@ 8
delret+
y
)J
delret
1
sin3 ec0s4e
(I -pcose)l*
K
...2
,;",";;deo
+
de,ret+
0
77
J
ret
0
j j
(;$;;;;;,2
0
0
5opD2+15&
lo+
[$bJ
(1 -pcos8)'"
7r
+ g
and
6
sin3 ecos2e
(1 -pcosep
0
de
1
(20)
ret
The integrals over 0 are computed directly and the resulting expressions can be written in terms of powers of
y= (1 -p2>- ‘/2 . After some laborious calculation one obtains an expression for the total power radiated by a
moving toroidal dipole when the velocity and its derivatives are parallel,
r2 lop6
PC+>= -$
77
(26880~~~ - 64512~~’ + 52416~‘~ - 16016~‘~ + 1287~‘~)
[
+ 16O@fi
231
(4032~~’ - 6496~‘~ + 2926~‘~ - 297~‘~)
...
(896~‘~ - 1624~‘~ + 836~‘~ - 99~‘~)
>
J.A. Heras/Physics
. ..
$
WPPP
I-9
. ..2
..’
(56~‘~ - 56~‘~ + 9~‘~) + $
63
Letter.7 A 249 (1998)
(120~‘~ - 140$* + 27~“)
9
.
1ret
(21)
Finally, consider the low velocity limit of Eq. (21). The approximation
/3 < 1 implies y M 1 and thereby
/3y = ,/m
M 0. This approximation also implies that the effect of retardation becomes unimportant
[ 121.
...
Therefore, by writing b = a/c, B = h/c and P = a/c, Eq. (21) for low velocities reduces to the expression
p=-
50r2a6
7c”
which
40?a2b2
+
7c9
has been previously
1272a3ii
+
7c9
272ii2
~
+
15c7 ’
(22)
derived in Ref. [7].
Acknowledgement
The author thanks Dr. Octavia Novaro for his valuable support and Dr. Karo Michaelian
of the manuscript.
for a careful reading
References
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(2 1 V.M. Dubovik, L.A. Tosunyan, Sov. Part. Nucl. 14 ( 1983) 504.
[ 31 V.M. Dubovik. V.V. Tugushev, Phys. Rep. 187 (1990) 147.
14 I V.L. Ginzburg, V.N. Tsytovich, Sov. Phys. JETP 61 (1985) 48.
]S] V.M. Dubovik, S.V. Shabanov in: Essays on the Formal Aspects of Electromagnetic Theory, A. Lakhtakia, ed. (World Scientific,
Singapore, 1993) pp. 399474.
]6] G.N. Afanassiev, Y.P. Stepanovsky, J. Phys. A 28 (1995) 4565.
]7] J.A. Hems, Phys. Lea. A 237 ( 1998) 343.
[S] E.E. Radescu, D.H. Vlad, Phys. Rev. E 57 (1998) 6030.
[ 91 J.A. Hems, Am. J. Phys. 62 ( 1994). 525.
[IO] J.A. Hems, Am. J. Phys. 63 (1995), 928.
[ I I I J.A. Hems, Phys. Rev. E, to be published.
[ 121L. Eiges, The Classical Electromagnetic Field (Dover, New York. 1972) p. 287.