Table Of ContentSingular behaviour of the electromagnetic field (the
0 static case revisited)
1
0
2
C. Vrejoiu‡ and R. Zus §
n
a University of Bucharest, Department of Physics,
J PO Box MG - 11, Bucharest-Magurele,RO - 077125,Romania
3
2
Abstract. Thesingularitiesoftheelectromagneticfieldarederivedtoincludeallthe
] point-like multipoles representing an electric charge and current distribution. Partial
h
p results obtained in a previous paper [1] are completed to represent accurately all the
- terms included in these singularities.
s
s
a
l
c
.
s 1. Introduction
c
i
s
y In the cases of electrostatic and magnetostatic fields of point-like dipoles, one has
h
the well-known procedure of introducing Dirac δ-function terms for obtaining correct
p
[ expressions of the electric and magnetic fields defined on the entire space. The
1 corresponding field expressions take the following form [2]:
v
4 1 1 3(ν ·p)ν −p 1
E (r) = − pδ(r)+ = − pδ(r)+(E) , (1)
1 p 3ε 4πε r3 3ε r6=0
1 0 0 0
4 where ν = r/r, and
.
1
2µ µ 3(ν ·m)ν −m 2µ
0 B (r) = 0 mδ(r)+ 0 = 0 mδ(r)+(B) . (2)
0 m 3 4π r3 3 r6=0
1
: In these equations, by (...) we understand an expression in which the derivatives
v r6=0
i are calculated supposing r 6= 0, representing some well-known expressions of the fields.
X
The expressions from equations (1) and (2) are introduced in Ref. [2] as conditions of
r
a
compatibility with the average value of the electric or magnetic field over a spherical
domain containing all the charges or currents inside. Another procedure for introducing
equations (1) and (2) is based on an extension of the derivative ∂ ∂ /(1/r) to the entire
i j
space [3]:
1 4π 3ν ν − δ
∂ ∂ = − δ δ(r) + i j ij. (3)
i jr 3 ij r3
A more pedagogical and suitable approach for understanding the origin of the difference
between the electric and magnetic cases is done in Ref. [4]. Refs. [5] and [6] contain
‡ E-mail : vrejoiu@fizica.unibuc.ro
§ E-mail: roxana.zus@fizica.unibuc.ro
Singular behaviour of the electromagnetic field (the static case revisited) 2
generalizations of the equations (1) and (2) to the dynamic case for oscillating electric
and magnetic dipoles.
The objective of the present paper is to establish the singularities of the
electromagnetic field associated to a system of electric charges and currents assimilated
with a point-like multipolar system. These singularities are established for an arbitrary
multipolar order in the static case.
In section 2, the procedure of separating the δ-form singularities of the multiple
partial derivatives of r−1 is described. In sections 3 and 4, we separate the δ-
form singularities of the fields corresponding to the point-like equivalent multipole
distributions, based on the multipole expansions of the fields associated to an electric
charge or current distribution confined in a domain D.
We point out that the formalism presented in this paper has as mathematical basis
the properties of the irreducible tensorial representations of the proper rotations group
[7]. For pedagogical and larger accessibility reasons, we give an explicit calculation
based on the properties of the tensor contractions such that the procedure has a simple
algebraic character. A counterpart of the procedure used in the present paper could be
represented by the technique of the spherical function expansions and the cited issues
can be a basis for such an approach. However, the use of the spherical coordinates can
leadtotheomission oftheδ-formsingularity contributions, asarguedintheconclusions.
2. Some delta-function identities
The treatment of some delta-function identities in Ref. [3] can be easily generalized to
obtain the necessary identities for higher order derivatives of 1/r. In the points different
from the origin O, the function 1/r is a solution of the Laplace equation:
1
∆ = 0, r 6= 0 . (4)
r
It is well-known how one can extend ∆(1/r) as a distribution to the entire space, as
solution of the Poisson equation:
1
∆ = −4πδ(r) . (5)
r
Equation(3)representsthenextstepofgeneralizationandcanbecontinuedforarbitrary
multiple partial derivatives of 1/r i.e. for ∂ ...∂ (1/r) for arbitrary n. For r 6= 0, this
i1 in
multiple partial derivative is given by the formula
1 1
(n)
∂ ...∂ = C . (6)
i1 in r rn+1 i1...in
(cid:18) (cid:19)r6=0
In this equation, the coefficient C is fully symmetric with respect to i ...i and can be
1 n
written as
[n]
2
C(n) = (−1)k(2n−2k −1)!!δ ...δ ν ...ν . (7)
i1...in {i1i2 i2k−1i2k 2k+1 in}
k
X
In the last equation, [β] is the integer part of β and by A we understand the sum
{i1...in}
over all the permutations of the symbols i giving distinct terms. From equation (5), it
q
Singular behaviour of the electromagnetic field (the static case revisited) 3
is obvious that we have to consider the derivatives of 1/r extended over the entire space
as distributions (generalized functions) which contain singular distributions (δ-functions
for example) as separate terms.
Let be a function F(r) and suppose the existence of the integral of the product
F(r)φ(r), with φ(r) an arbitrary smooth function (a test function from the domain of
the distributions), on the spherical region D , with arbitrary radius R, delimited by the
R
spherical surface Σ with the center in O:
R
d3xF(r)φ(r) = lim d3xF(r)φ(r) . (8)
ZDR ε→0Zε<r<R
This integral can be expressed excluding from the domain D a spherical domain of
R
radius ε centered in O. Writing this last limit of integrals, we can interpret the function
F(r) as a distribution defined by
(F(r)) , φ(r) = lim d3x(F(r)) φ(r) .
r6=0 r6=0
D E ε→0ZDR\Dε
This distribution can be extended such that its support includes the point O. A new
term θ(ε−r)F(r) can be naturally introduced by the identity
F (r) = θ(ε−r)F(r,t)+θ(r −ε)F(r),
ε
associated with the extension of the integral to the entire domain D :
R
hF(r), φ(r)i = lim d3xF(r)φ(r)+ d3xF(r)φ(r) . (9)
ε→0"ZDε ZDR\Dε #
Moreover, we suppose the existence of the integral (8) for the partial derivatives of
F. Let us consider the partial derivative ∂ F(r,t) and the problem of extending this
i
function as a distribution. The definition (9) becomes:
hF(r,t), φ(r)i = lim dSν F(r,t)φ(r)− d3xF(r,t)∂ φ(r)
i i
ε→0(cid:20)IΣε ZDε
+ d3x∂ F(r,t)φ(r) , (10)
i
ZDR\Dε #
where Σ is the sphere of radius ε centered in O and the Gauss theorem was employed.
ε
Let us apply this definition to the derivative ∂ ∂ (1/r) and, for simplifying the notation,
i j
let
1
D (r,t) = ∂ ...∂ ,
i1...in i1 in r
such that we can write
1 1 1
(D , φ) = lim dSν ∂ φ(r)− d3x∂ ∂ φ(r) + d3x(∂ ∂ )φ(r) .(11)
ij i j j i i j
ε→0(cid:20)IΣε r ZDε r ZDR\Dε r #
Since the last integral on the domain D \ D represents the distributions associated
R ε
with the F- expressions for r 6= 0 and, in the case of the electromagnetic field, they will
be the well-known expressions of the multipole expansions, in the following, we consider
only that part of hD i containing singular distributions with point-like support i.e.,
ij
Singular behaviour of the electromagnetic field (the static case revisited) 4
actually, the difference
1
(D ) , φ = hD , φi−lim d3x(∂ ∂ )φ(r)
D ij (0) E ij ε→0ZDR\Dε i jr
1 1
= lim dSν ∂ φ(r)− d3x∂ ∂ φ(r) . (12)
i j j i
ε→0(cid:20)IΣε r ZDε r (cid:21)
By D we denote a distribution having as support the point given by the vector r .
(r0) 0
The surface integral,
1 1
lim dSν ∂ φ(r) = −lim dS ν ν φ(r) ,
ε→0IΣε i jr ε→0IΣε r2 i j
after inserting the Taylor series of the function φ(r) and since on the sphere r = ε,
becomes [3],
1
lim dSν ∂ φ(r) = −lim dΩ(ν)ν ν [φ(0)+εν (∂ φ) +...] (.13)
ε→0IΣε i jr ε→0 Z i j k k 0
Let us introduce the angular average:
1
hg(ν)i = g(ν)dΩ(ν) . (14)
4π
Z
Particularly, we have the well-known formula [8]:
0, n = 2k +1,
hν ...ν i = (15)
i1 in 1 δ ...δ , n = 2k, k = 0,1,...
(n+1)!! {i1i2 in−1in}
Excepting the term containing φ(0), all the terms in equation (14) are proportional to
positive powers of ε and, consequently, vanish with ε → 0, such that
1 4π 4π
lim dSν ∂ φ(r) = −4πhν ν i φ(0) = − δ φ(0) = − δ hδ(r), φ(r)i .
i j i j ij ij
ε→0IΣε r 3 3
Considering the second integral in the right-hand side of equation (11), we can write
1 ε 1
lim d3x ∂ ∂ φ(r) = −lim r2dr dΩ(ν)ν [(∂ φ) +rν (∂ ∂ φ) +...] = 0 ,
ε→0ZDε jr i ε→0Z0 Z jr2 i 0 k i k 0
such that finally
1 4π
∂ ∂ = − δ δ(r) , (16)
i j ij
r 3
(cid:18) (cid:19)(0)
i.e. the delta-singularity from equation (3) of Ref. [3].
Let us consider the distribution D . Considering only the part having O as
ijk
support,
1 1
(D ) , φ = lim dSν ∂ ∂ φ(r)− d3x(∂ ∂ )∂ φ(r) . (17)
(cid:16) ijk (0) (cid:17) ε→0(cid:20)IΣε i j kr ZDε j kr i (cid:21)
The surface integral becomes:
1 1
lim dSν ∂ ∂ φ(r) = lim dSν (3ν ν −δ )φ(r)
ε→0IΣε i j kr ε→0IΣε ir3 j k jk
and, introducing the Taylor series for φ(r),
1 1
lim dSν ∂ ∂ φ(r) = 4π lim (3ν ν ν −ν δ )[φ(0)+εν (∂ φ) +...] , (18)
ε→0IΣε i j kr ε→0(cid:28) ε i j k i jk l l 0 (cid:29)
Singular behaviour of the electromagnetic field (the static case revisited) 5
and
1
lim dSν ∂ ∂ φ(r)
i j k
ε→0IΣε r
1 1
= 4πh3ν ν ν ν −ν ν δ i(∂ φ) = 4π δ δ − δ δ (∂ φ) . (19)
i j k l i l jk l 0 5 {ij kl} 3 il jk l 0
(cid:18) (cid:19)
Concerning the integral on D from equation (17), we have to observe that, beginning
ε
from this derivative order, there is a non-zero contribution for ε → 0 [3]. Indeed,
introducing equation (16) in equation (17) and noticing that the term (∂ ∂ (f(τ)/r))
j k r6=0
gives a null contribution to the limit for ε → 0, we can write
1 4π 4π
−lim d3x∂ ∂ ∂ φ(r) = d3xδ δ(r)∂ φ(r) = δ (∂ φ) .
j k i jk i jk i 0
ε→0ZDε r 3 ZDε 3
Finally, equations (17), (18) and (19) give
4π
(D ) = − δ ∂ δ(r) , (20)
ijk (0) {ij k}
5
i.e. the delta-singularity from equation (4) of Ref. [3].
Obviously, this procedure becomes very complicated for higher order derivatives.
Fortunately, for the electromagnetic field, some invariance properties allow a
considerable simplification of such calculations.
3. Singularities of the electrostatic field
Let us consider the multipole expansions of the electrostatic field. Given an electric
charge distribution with support included in the domain D, the scalar potential is
expressed in the exterior of a sphere containing this domain by the following multipolar
series:
1 (−1)n P 1 (−1)n P(n)
Φ(r) = ∂ ...∂ i1...in = ∇n|| . (21)
4πε n! i1 in r 4πε n! r
0 n≥0 0 n≥0
X X
In this expansion, the coordinate system origin O is supposed in D and P(n) is the n-th
order electric multipolar moment defined by the Cartesian components in the general
dynamic case:
P (t) = d3x x ...x ρ(r,t) : P(n)(t) = d3x rnρ(r,t). (22)
i1...in i1 in
ZD Z
D
In equation (21) we employed the following notation for tensorial contractions:
A B , n > m
i1···in−mj1···jm j1···jm
(A(n)||B(m)) = A B , n = m . (23)
i1···i|n−m| Aj1···jnBj1···jn , n < m
j1···jn j1···jni1···im−n
For the multipole expansion of the electric field E(r) = −∇Φ(r), we can write
1 (−1)n−1 P(n) 1 (−1)n−1 P
E(r) = ∇n+1|| = e ∂ ∂ ...∂ i1...in, (24)
4πε n! r 4πε i n! i i1 in r
0 n≥1 0 n≥1
X X
where, for simplicity, the electric charged system is considered neutral (Q = 0). We have
to search the singularities of E given by equations (24). It appears that cumbersome
Singular behaviour of the electromagnetic field (the static case revisited) 6
calculationsareinvolved forhighernifweapplytheformulaeforhigherorderderivatives
of 1/r as in the previous section. However, we can employ an invariance property
of the electrostatic field to the substitutions of all moments P(n), for all n, by their
corresponding symmetric and trace-free STF projections P(n) [9, 10, 11]. Retaining the
notation p for the first order moment, this invariance stands for the invariance of the
multipole expansion of the electrostatic field to the following substitutions:
p, P(2), P(3), ... → p, P(2), P(3)... ;
(25)
These STF tensors can be expressed by the following formula:
(−1)n 1
P = d3xρ(r)r2n+1∂ ...∂ , (26)
i1...in (2n−1)!! i1 inr
ZD
which, actually, differ from the projections by numerical factors. Here,
(λ)
∂ = ∂ ...∂ ∂ ...∂ .
i1...in i1 iλ−1 iλ+1 in
Some care is necessary when one considers this invariance property when we have to
establish the delta-type singularities of the electromagnetic field, since this property is
true only for r 6= 0. Indeed, as we firstly see in the electrostatic case, the substitutions
P(n) → P(n) in equation (24) give additional terms containing ∆(1/r) and their
derivatives, which, in the case r 6= 0, can be eliminated. But, extending the expressions
associated to the multipole expansions to the entire space, including the origin O, these
additional terms give notable contributions. This process of searching the delta-form
singularities of the electromagnetic field is lost in the previous paper [1] and here, we
try to correct it.
Let us consider the delta-singularity corresponding to the electric dipolar field:
1 p 1
E(1) = e ∂ ∂ j = − pδ(r), (27)
(0) 4πε i i j r 3ε
0 0
a result obtained by directly applying equation (16).
For the 4-polar term from E, we firstly consider the expansion (24) expressed by
the primitive moments P(n):
P
1
E(2)(r) = − e ∂ ∂ ∂ jk . (28)
i i j k
8πε r
0
With the help of equation (20), we obtain:
1 1
E(2) = e P δ ∂ δ(r) = e P (δ ∂ δ(r)+δ ∂ δ(r)+δ ∂ δ(r))
(0) 10ε i jk {ij k} 10ε i jk ij k ik j jk i
0 0
1 1 1
= e (2P ∂ δ(r)+P ∂ δ(r)) = e P ∂ δ(r)+ P ∂ δ(r) . (29)
i ij j jj i i ij j jj i
10ε 5ε 2
0 0 (cid:18) (cid:19)
Note that P(2) is symmetric. This result suggests the possibility to facilitate the
calculation of the delta-type singularities of the electromagnetic field if instead of the
“primitive” tensors P(n) we can employ the STF projections P(n). For the present
case, it will be simpler to calculate the contractions of the last type of tensors with the
coefficients C(n) from equation (6).
Singular behaviour of the electromagnetic field (the static case revisited) 7
For n = 2, we search the trace-free part of the symmetric tensor P(2) as
P = P +δ Λ . (30)
ij ij ij
The parameter Λ is determined such that P = 0. One obtains
ii
1
Λ = P . (31)
ii
3
Let us introduce equation (30) in equation (28):
1 1 Λ 1
E(2)(r) = − e P ∂ ∂ ∂ − e ∂ ∆ .
i jk i j k i i
8πε r 8πε r
0 0 (cid:18) (cid:19)
For r 6= 0, the last term vanishes and, indeed, E is invariant to the substitution
P(2) → P(2). Extending this field to the entire space, this last term contributes with a
delta-singularity such that we must write
1 1 1
E(2)(r) = − e P ∂ ∂ ∂ + Λ∇δ(r) . (32)
i jk i j k
8πε r 2ε
0 0
Inserting equation (20) in equation (32), and retaining only the delta-singularities, we
can write
1 1
E(2)(r) = e P ∂ δ(r)+ Λ∇δ(r) . (33)
(0) 5ε i ij j 2ε
0 0
Substituting equation (30) in equation (33), one obtains equation (29).
The advantage of employing the STF moments P(n) instead of the primitive
moments P(n) is manifest for higher order terms from the multipolar expansion. Even
from n = 3, the contraction of the angular average of a product of more then six factors
ν with a primitive moment represented by a tensor which is only symmetric becomes
cumbersome.
Though, maybe, only of theoretical interest, let us consider the general case of
arbitrary n. The STF projection T (P(n)) of the symmetric tensor P(n) is defined, up
to a numerical factor, by the equation
T (P(n)) ≡ P = P −δ Λ , (34)
i1...in i1...in i1...in {i1i2 i3...in}
where Λ(n−2) is a symmetric tensor and is defined by the conditions of the trace-free
character of P(n). For low values of n (the ones of practical interest), the components
Λ
can be calculated directly from the equation system representing the vanishing of
i1...in
all the traces of the tensor P(n). For higher orders n, there is a general formula known
in literature [8], [12] which, with the notation from the present paper, is written as
[n/2] (−1)m(2n−1−2m)!!
T [P(n)] = δ ...δ P(n:m) . (35)
i1...in (2n−1)!! {i1i2 i2m−1i2m i2m+1...in}
h i mX=0
P(n:m) denotes the components of the (n−2m)-th order tensor obtained from P(n)
i2m+1...in
by contracting m pairs of symbols i. This equation is known as the detracer theorem
[12]. As a consequence of this theorem, the components of the tensor Λ(n−2) are written
as
[n/2−1] (−1)m[2n−1−2(m+1)]!!
Λ [P(n)] = δ ...δ P(n:m+1) . (36)
i1...in−2 (m+1)(2n−1)!! {i1i2 i2m−1i2m i2m+1...in−2}
m=0
X
Singular behaviour of the electromagnetic field (the static case revisited) 8
Inserting equation (34) in the expression of E(n)(r) given by equation (24), we obtain
(−1)n−1 1 (−1)n−1 1
E(n)(r) = P(n)||∇n+1 + e δ Λ ∂ ∂ ...∂ .
4πε n! r 4πε n! i {i1i2 i3...in} i i1 inr
0 0
and, since the contraction with the fully symmetric tensor Λ(n−2) produces C2 =
n
n(n−1)/2 identical terms,
(−1)n−1 1 (−1)n−1 1
E(n)(r) = P(n)||∇n+1 + e Λ ∂ ∂ ...∂ ∆
4πε n! r 8πε (n−2)! i i1...in−2 i i1 in−2 r
0 0
(−1)n−1 1 (−1)n
= P(n)||∇n+1 + Λn−2||∇n−1δ(r) . (37)
4πε n! r 2(n−2)!ε
0 0
Searching the extension of E(n) to the entire space, for establishing the δ-type
singularities, we have to calculate the limit
(−1)n−1 1
E(n), φ = lim d3x P(n)||∇n+1 φ(r)
D (0) E 4πε0n! ε→0 ZDε (cid:18) r(cid:19)
(−1)n
+ lim d3xΛ(n−2)||∇n−1δ(r)φ(r) ,
2(n−2)!ε0 ε→0ZDε
or
(−1)n−1 1
E(n), φ = lim d3x P(n)||∇n+1 φ(r)
D (0) E 4πε0n! ε→0 ZDε (cid:18) r(cid:19)
1
− Λ((n−2)|| ∇n−1φ . (38)
2(n−2)!ε 0
0
(cid:16) (cid:17)
As done in Ref. [1], the limit of the remaining integral from equation (38) can be easily
expressed for arbitrary n. Applying the Gauss theorem, we can write
1
lim d3x P(n)||∇n+1 φ(r)
ε→0ZDε (cid:18) r(cid:19)
1 1
= lime dSν P ∂ ...∂ φ(r)− d3x P ∂ ...∂ ∂ φ(r) .(39)
ε→0 i(cid:20)IΣε i (cid:18) i1...in i1 inr(cid:19) ZDε (cid:18) i1...in i1 inr(cid:19) i (cid:21)
Introducing the Taylor series for the function φ(r) in the surface integral from the last
equation and since r = ε on the sphere Σ , we can write
ε
1
lime dSν P ∂ ...∂ φ(r)
ε→0 iIΣε i (cid:18) i1...in i1 inr(cid:19)
∞ εα−n+1
= 4πe lim P ν C(n) ν ...ν ∂ ...∂ φ . (40)
iε→0 α! i1...in i i1...in in+1 in+α in+1 in+α 0
αX=0 D E (cid:16) (cid:17)
Let us evaluate the tensorial contraction which is present of the general term in the
series from the previous equation:
(n)
P ν C ν ...ν .
i1...in i i1...in in+1 in+α
D E
From equation (7) we can easily see that all the terms containing at least a symbol δ
iqis
with 1 ≤ q, s ≤ n give null results by contraction with the traceless tensor P(n). Only
the term corresponding to k = 0 in equation (7) can give results different from zero for
Singular behaviour of the electromagnetic field (the static case revisited) 9
this contraction. Moreover, a result different from zero can be obtained if and only if
α+1 ≥ n. Therefore, equation (42) can be written as
1
lime dSν P ∂ ...∂ φ(r) = 4πe (−1)n(2n−1)!!
ε→0 iIΣε i (cid:18) i1...in i1 inr(cid:19) i
∞ εα−n+1
×lim P ν ...ν ν ...ν ν ∂ ...∂ φ . (41)
ε→0 α! i1...in i1 in in+1 in+α i in+1 in+α 0
α=Xn−1 D E (cid:16) (cid:17)
For α > n−1, the corresponding terms from the series in equation (41) contain positive
powers of ε. Consequently, the corresponding limits for ε → 0 vanish. From this series
only the term for which
α = n−1 (42)
can be different from zero and the result of the limit in equation (41) is given by
1 4π(−1)n(2n−1)!!
lime dSν P ∂ ...∂ φ(r) =
ε→0 iIΣε i (cid:18) i1...in i1 inr(cid:19) (n−1)!
×e P ν ...ν ν ...ν ν ∂ ...∂ φ
i i1...in i1 in in+1 i2n−1 i in+1 i2n−1 0
4π(−1)n(2nD−1)!! E (cid:16) (cid:17)
= P(n)||ν2n || ∇n−1φ . (43)
(n−1)! 0
D E (cid:16) (cid:17)
Let us evaluate now the volume integral in equation (39):
P(n)
lim d3x ∇n|| ∇φ(r)
ε→0ZDε r !
P(n) P(n)
= lim dSν|| ∇n−1|| ∇φ(r)− d3x ∇n−1|| ∇2φ(r) .(44)
ε→0"IΣε r ! ZDε r ! #
The part corresponding to the surface integral can be written as
P(n) 1
lim dSν|| ∇n−1|| ∇φ(r) = e P lim dSν ∂ ...∂ ∂ φ(r) .
ε→0IΣε r ! i i1...in ε→0IΣε in (cid:18) i1 in−1r(cid:19) i
(45)
Introducing the Taylor series for φ(r) and standing out the average over ν, we obtain
P(n)
lim dSν|| ∇n−1|| ∇φ(r)
ε→0IΣε r !
∞ εα−n+2
= 4πe lim P C(n−1,n−1)ν ...ν × ∂ ...∂ ∂ φ . (46)
iε→0 α! i1...in i1...in−1 in in+α in+1 in+α i 0
αX=0 D E (cid:16) (cid:17)
Analogously to the reasoning from the previous case, we can see that the limit is zero
since for α = n−2
(n−1,n−1)
P C ν ν ...ν = 0.
i1...in i1...in−1 in in+1 i2n−2
D E
Therefore,
P(n) P(n)
lim d3x ∇n|| ∇φ(r) = −lim d3x ∇n−1|| ∇2φ(r) , (47)
ε→0ZDε r ! ε→0ZDε r !
Singular behaviour of the electromagnetic field (the static case revisited) 10
and by repeatedly applying the procedure, all terms cancel. Finally, only the surface
integral from equation (39) gives a limit different from zero and
1 4π(−1)n(2n−1)!!
lim d3x P(n)||∇n+1 φ(r) = P(n)||ν2n || ∇n−1φ . (48)
ε→0ZDε (cid:18) r(cid:19) (n−1)! D E (cid:16) (cid:17)0
Let us consider the contraction
P(n)||ν2n = P hν ...ν ν ...ν i .
i1...in i1 in j1 jn
D E
To this contraction contribute only the terms from the average of the ν-product not
containing factors δ , with 1 ≤ k, l ≤ n. According to equation (15), the terms giving
ikil
non-zero contributions are of the form
1
δ ...δ
(2n+1)!! i1j1 injn
and all such terms are obtained considering the n! permutations of the indices j ...j
1 n
in this product. Therefore, the final expression in equation (43) is given by
1 4π(−1)nn
lim d3x P(n)||∇n+1 φ(r) = P(n)|| ∇n−1φ . (49)
ε→0ZDε (cid:18) r(cid:19) 2n+1 (cid:16) (cid:17)0
Equation (38), with equation (49) inserted in, yields
1 1
E(n), φ = − P(n)|| ∇n−1φ − Λ(n−2)|| ∇n−1φ .(50)
(0) (n−1)!(2n+1)ε 0 2(n−2)!ε 0
0 0
D E (cid:16) (cid:17) (cid:16) (cid:17)
Therefore, the delta-form distribution associated to the 2n-polar electric field is given
by
(−1)n (−1)n
E(n) = P(n)||∇n−1δ(r)+ Λ(n−2||∇n−1δ(r) . (51)
(0) (n−1)!(2n+1)ε 2(n−2)!ε
0 0
One can easily see that the result (32) for n = 2 is, indeed, a particular case of the
formula (51). In the case n = 3, equation (51) becomes:
1 1
E(3) = − P(3)||∇2δ(r)− Λ(1)||∇2δ(r) .
(0) 14ε 2ε
0 0
The result can be expressed in terms of primitive tensors P(n) introducing equation (34)
in equation (51):
(−1)n (−1)n
E(n) = P(n)||∇n−1δ(r)+ Λ(n−2||∇n−1δ(r)
(0) (n−1)!(2n+1)ε 2(n−2)!ε
0 0
(−1)n
− e δ Λ ∂ ...∂ δ(r) .
(n−1)!(2n+1)ε i {ii1 i2...in−1} i1 in−1
0
In the last term from this equation, there are n−1 identical terms of the form
e Λ ∂ ∂ ...∂ δ(r) = Λ(n−2)||∇n−1δ(r)
i i2...in−1 i i2 in−1
and C2 = (n−1)(n−2)/2 terms of the form
n−1
e Λ ∂ ...∂ δ(r) = Λ(n−2)||∇n−3∆δ(r) .
i ii3...in−1 i3 in−1
