On the field energy of two charges with application to electric dipoles
CHRISTOPHER G. PROVATIDIS
School of Mechanical Engineering
National Technical University of Athens
9 Iroon Polytechniou, 157 80 Zografou
GREECE
Abstract: - This paper revisits the topic of electrostatic field energy due to a pair of the electric charges. Not
only point-like charges but also charged spheres of radius 𝑅 are studied. The self-energies as well as the
interaction field energy are discussed in full detail. By combining two alternative didactic paths (one
mathematical and the other based on energy conservation principles), it is shown that the interaction field
energy (which is a volume integral of the energy density over the infinite space) is always equal to the potential
energy, regardless of the nature of the electrical charges. For these two characteristic cases of electric charges
(i.e., point-like and uniformly charged spheres), the location and the amount of the major part of the interaction
field energy is discussed; the relevant factoids are documented in the form of two compact theorems in the
Appendix. The case of non-uniformly charged spheres (i.e., spherical conductors) is also discussed to some
extent. In addition to this general presentation, the particular case of the electric dipole is discussed, as a special
case, accompanied with many numerical results.
Key-Words: - Electric dipole, Work from Infinity, Potential energy, Charged spheres, Analytical methods
Received: October 29, 2021. Revised: October 21, 2022. Accepted: November 25, 2022. Published: December 31, 2022.
1 Introduction
Not many years ago, it was proposed that ‘given the
ambiguities and complexities associated with field
energy, a traditional approach focusing on potential
energy is more appropriate for introductory physics
in secondary schools, colleges, and universities’,
[1]. To support his position, the same author refers
to a claim of Feynman that ‘… The concepts of
simple [point-like] charged particles and the
electromagnetic field are in some way inconsistent.’
(see [2], Vol. II, p. 28-1).
Actually, the abovementioned difficulty comes
from the fact that, according to Coulomb’s law, the
electric intensity exactly at a point-like charge
becomes infinite, thus the total electric field energy
cannot be uniquely determined. Although
Coulomb’s law is quite analogous to Newton’s
gravitational law (both of them are inverse square
laws), this difficulty does not appear in the
gravitational theory because (leaving General
Relativity aside for the moment) a point mass
particle (of zero kinetic energy) has zero potential
energy at infinity (i.e., not any additional self field
energy at infinity). In other words, the gravitational
field around two point mass particles separated by a
distance 𝑑, is characterized by only the potential
energy (also called binding energy), which by
definition is the work done to bring one of these
masses from infinity to that point, while keeping the
other mass fixed.
According to Maxwell (1873), [3], the total
energy of the electric field is proportional to the
square of electric intensity (𝑬
𝟐). In 2014, Hilborn,
[1], and Tort, [4], reported their independent studies
on the superposition of the electric fields due to two
point-like charges (𝑞 and 𝑞), i.e., 𝑬
𝑬
𝑬
(see, Fig. 1). Both of the aforementioned authors
have split the field energy in three parts, applying
the identity: 𝑬
𝟐𝑬
∙𝑬
𝑬
𝟐𝑬
𝟐2𝑬
∙
𝑬
. As we shall see in Section 2, the first term
𝑬
𝟐, leads to a self-energy exclusively related to
the electric charge 𝑞, and becomes singular at this
point. Similarly, the second term, 𝑬
𝟐, leads to a
self-energy related to the charge 𝑞, and becomes
singular at this point. The third term, 2𝑬
∙𝑬
, is
twice the dot-product of both constituent point-like
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
212
Volume 19, 2022
charge fields and (after extensive manipulation) has
been proven (see [1] and [4]) to be equal to the
potential energy 𝑈, which –by definition– is the
work done from infinity to bring the two electric
charges together separated by a distance 𝑑.
Regarding the regular integral of the term 2𝑬
∙
𝑬
, Hilborn, [1], has proposed the evaluation of
this integral in three alternative ways, i.e., through
the use of trigonometric substitutions, integral
tables, or symbolic manipulation packages such as
MATHEMATICA®.
To deal with the problem of the abovementioned
infinities at point-like charged particles, following
Corbò, [5], Tort, [4], introduced a regularization and
renormalization scheme, [6]. A more advanced
technique which identifies the infinities as constants
in a perturbative series of Coulomb’s potential, was
proposed by Mundarain, [7]. In any case, a rough
solution is to ignore the field energy related to the
squares of the electric intensity components 𝑬
and 𝑬
, and deal only with the interaction field
energy which is exclusively related to the dot
product 2𝑬
∙𝑬
. An argument in favour of this
approach is that we are mainly interested in energy
changes and not in the absolute value of the field
energy.
My objection to fully ignore the intensity
components 𝑬
and 𝑬
, comes from previous
experience, however related to the field energy of
magnetic dipoles, and particularly with the magnetic
field of the Earth. Clearly, if we calculate the
potential energy which corresponds to the well
known magnetic moment (for the year 2020 is
estimated to 𝑀7.8810 𝐴𝑚𝑚𝑑, 𝑚
pole strength, 𝑑26,371,200 𝑚 Earth’s
diameter thus separation distance of the magnetic
dipole) we will find it to be equal to 𝑈
310 Joule, which is about 2.67 times
less than the total field energy that fills the infinite
exterior space (about 810 Joule), without
considering the interior of the Earth at all (which is
anticipated to be equally large, if not larger than the
exterior one). Therefore, in this particular case, the
influence of the two self-energy terms in the total
field energy becomes imperative.
The counter argument might be that when
considering magnetostatics, there is a question as to
whether a potential energy is defined. In the older
view of Gilbert, Coulomb, Poisson, etc., magnetism
is due to “magnetic charges”/poles, and the
formalism of magnetostatics is the same as that of
electrostatics. A magnetic potential does exist, and
is defined at a point to be the work done to bring
unit poles from infinity to that point, while keeping
all other poles fixed. However, in 1820, Ampere
made the conjecture that magnetism is not due to
“poles”, but to electric currents. In this view, there is
no “magnetic” scalar potential, but rather a magnetic
vector potential 𝐴 (which is not an energy). As far
as we can tell today, Ampere was right, and Gilbert,
Coulomb, Poisson, and others, were wrong (a
comment that of course does not reduce their value
in the process of science). This (experiment) issue is
hard to settle because the two hypotheses as to the
nature of magnetism imply the same results for the
magnetic field outside a magnetic dipole (at
distances large compared to size of the distribution
of poles or currents of the dipole). Therefore, in the
above paragraph I have taken the Gilbertian view of
magnetism, although this view is now disfavoured,
[8].
Within this context, this paper is a contribution
to the estimation of the total electrostatic field
energy, from the abovementioned Maxwellian point
of view, and its aim is threefold. First it discusses
simple remedies for overcoming the shortcoming of
the singularity at the point-like electric charges by
considering two small spheres of charge, which can
be understood by the average student in secondary
schools, colleges and universities. Second, it applies
the law of energy conservation to make clear the
meaning of the total field energy. Third, it discusses
in adequate length the distribution of the interaction
field energy in the surrounding three-dimensional
space, as well as the self-energies of the electric
charges, and presents closed-form analytical
expressions in characteristic zones, where possible.
The study is completed by numerical results
regarding an electric dipole. The first two out of the
five Appendices are in the form of compact
theorems.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
213
Volume 19, 2022
Fig. 1: Electric intensity induced due to two charges of (a) same (here, positive) sign and (b) opposite sign.
2 Basic equations
2.1 Definitions
We recall that the concept of electrostatic potential
energy was first formulated by Poisson (1812), as an
analog of gravitational potential energy, as in both
cases the basic force law has an inverse square
1/𝑟 behavior. By definition, the electric potential
energy 𝑈 of two charges 𝑞 and 𝑞 separated by
distance 𝑑 equals the work needed to bring these
charges together from infinity (where their potential
energy is zero).
The electric scalar potential 𝑉 of a point charge 𝑞
is given by
eq
Vk
r
, (1)
where 𝑘1/4𝜋𝜀
in SI units is the Coulomb’s
constant (8.9875517923(14)×109 kg⋅m3⋅s−4⋅A−2),
with 𝜀 denoting the vacuum permittivity, and 𝑟 is
the distance from the charge to the observer. It is
always useful to remember that 𝜀1/4𝜋𝑘
.
The potential energy of two point charges, 𝑞
and 𝑞 (either opposite-sign or like-sign), separated
by distance 𝑑 is
12
12 21eqq
Uk qV qV
d
. (2)
Clearly, the potential energy 𝑈 equals the work
𝑊 done while bringing 𝑞 from infinity to
distance 𝑑 from charge 𝑞, keeping 𝑞 fixed in
space, and vice-versa. In the first case that 𝑞 is
fixed, 𝑉 is the potential of 𝑞. In the second case
that 𝑞 is fixed, 𝑉 is the potential of 𝑞.
Supposing that electric charge is continuously
distributed with density 𝜌, instead of Eq. (2) we
write 𝑈
∭𝜌𝑉 d𝜐, where 𝑉𝒓
∭𝒓
|𝒓𝒓|
𝑑𝜐 is the scalar electric potential
and d𝜐 is the elementary volume in which the
charges exist. This may be useful when we deal with
distributed charges instead of point charges.
We recall that Isaac Newton showed that the
form (2) holds for the gravitational interaction
energy of spherical shells of mass, as well as for
point masses. By analogy, electric charges
uniformly distributed on the outer surface of a
sphere obey Coulomb’s law with respect to their
centers as well, thus Eq. (2) will be again valid.
So far, we have not mentioned fields. The
previous knowledge was advocated by Coulomb,
Poisson, etc.
It is worthy to mention that, Charles-Augustin de
Coulomb (1785) also gave the force law, 𝑭
on 𝑞
due to 𝑞:
12 12
32
r
ee
qqr qqe
Fk k r
r
, (3)
where vector 𝒓
points from 2 to 1, and 𝒆
is the
corresponding unit vector.
The electric field in electrostatics can be defined
at a point as the force that would be exerted on unit
charge at that point (due to other charges):
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
214
Volume 19, 2022
22
32
11
ii i ri
ee
ii
i
i
qr qe
Ek k
r
r
(4)
where vector 𝒓
𝒊 points from charge 𝑞 to the
observation point 𝑃.
For a continuous charge distribution, we can
write 𝑬𝒓 ∭𝜌𝒓′ 𝒓 𝒓′ 𝑑𝜐/ |𝒓 𝒓′|. This
formula is also useful for distributed electric charges
around a supposed point charge.
This electric field can be related to the electric
potential 𝑉 defined above via
gradEV
(5)
So far, the electric field is just a sort of
computational aid (like the potential 𝑉). From now
on, we pass to the field energy.
2.2 Field energy: A mathematical approach
The concept of magnetic field energy was
introduced by Maxwell (1856), based on Poisson’s
model of magnetic potential energy together with
the magnetic field equations 𝛻
𝑩
4𝜋𝜌
and 𝑩
𝜵
𝑉. James Clerk Maxwell, followed Faraday in
supposing that the electric (and magnetic) field have
a ‘dynamical’ significance in storing energy. Within
this context, in 1873, Maxwell gave the analogous
relation for the electric field 𝑈:
2
0
2
11
dd
22
d
8
f
Volume Volume
e
Volume
UV E
E
k
(6)
which suggests that 22
1
0
2() () (8 )
E
e
uEE k
rr
is the physical density of electric field energy in the
vicinity of point r
.
However, there is a problem with Maxwell’s
formula for the field energy if we consider a point
charge: the integral is infinite! Maxwell worked in a
vision of the “ether” which filled all space. Today
we say that the “ether” is not a ‘mechanical’ entity
(with mass), but just the electric field itself (plus
other fields as well, magnetic, gravitational, Higgs,
and so on).
Henceforth, we focus on Eq. (6) and particularly
on the term 𝐸. As also it was mentioned in the
‘Introduction’ (Section 1), the two components of
the electric intensity (due to two electric charges
only) form the total field 𝑬
𝑬
𝑬
, of which the
main interest is the magnitude 𝐸𝑬
𝟐.
Here, we make use of the well known vector
identity:
𝑬
𝟐𝑬
∙𝑬
𝑬
𝟐𝑬
𝟐2𝑬
∙𝑬
(7)
Substituting Eq. (7) into Eq. (6), we have:
2
21
2
212
dd
88
dd
84
f
ee
Volume Volume
ee
Volume Volume
E
E
Ukk
EEE
kk
(8)
For point-like charges, the first two integrals in Eq.
(8) (with integrands 𝐸
,𝑖1,2), represent the self
energy of the point charges, and are infinite. What to
do? We just ignore these infinite integrals, and say
what matters physically is the third integral, which
we call the “interaction energy”. This is a kind of
"classical renormalization" (see [5]-[7]).
Below we distinguish two cases, the former
being for point-like charges and the latter for
charged spheres of radius 𝑅 and 𝑅.
2.2.1 Point-like Charges
When the electric charges are isolated at infinity
they have their own self-fields 𝑬
and 𝑬
, and
obviously do not interact at all. The electric
potential of point-like charge 𝑖 is 𝑘𝑞/𝑟, where 𝒓
points from the point 𝑖 to the observation point 𝑃.
The electric field 𝑬
equals 𝑘𝑞𝒓
/𝑟, according to
Eq. (4).
Obviously, by virtue of Eq. (4) the volume
integrals of 𝑬
and 𝑬
(self energies in Eq. (8)) are
infinite terms which are successively written as:
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
215
Volume 19, 2022
2
22
2
0
22
2
0
0
d4d
88
1
d,1,2.
22
i
e
i
i
ee
Volume r
ei ei
r
q
k
Er
Urr
kk
kq kq
ri
rr
(9a)
In contrast, the volume integral of 2𝑬
∙𝑬
which
is the third term in the right-hand side of Eq. (8), not
only is bounded but also equal to the potential
energy 𝑈𝑘
, i.e.,
12
12 0 1 2
12d
2e
Volume
qq
UEEk
d
(10)
Thus, the “interaction energy” corresponds to the
“potential energy” of Coulomb and Poisson, while
the total energy includes “something more” that
does not make much sense. This is perhaps the main
reason to be skeptical about field energy (Hilborn,
[1]), and many people remain more comfortable
with the view of Coulomb and Poisson that
emphasizes scalar potentials. But this is a static
view, and Maxwell’s view is much more powerful
in time-dependent problems that constitute the real
world.
The proof of Eq. (10) has been highlighted by [1]
and [4]. A closely related but complete
mathematical proof of Eq. (10), without any gaps of
thought, is given in Appendix A (see, Eq. (A.13)
therein). Another proof, based on physical
principles, is given in Section 3.
2.2.2 Two Charges of Radii 𝑹𝟏 and 𝑹𝟐 Separated
by Distance d
In reality, a point-like charge may be equivalently
substituted by many point-like charges uniformly
distributed over the surface of a small sphere of
radius 𝑅, which logically should not exceed more
than half the distance 𝑑 (no charge penetration, thus
should be 0𝑅
𝑅𝑑). The electric field 𝑬
is zero inside the sphere 𝑖, and equal to 𝑘𝑞𝒓
/𝑟
outside. Taking the limits of the integral in 𝑟
between 𝑅 and infinity (∞), Eq. (9a) changes to:
2()
,1,2.
22
ei ii i i
i
i
kq qV r R
Ui
R
(9b)
In the case of uniformly distributed charges over
two spheres, the interaction energy is difficult to
calculate using the fields 𝑬
. McDonald, [9], has
used the electric potentials of the two equal spheres
in contact to compute, via surface integrals over the
two spheres, and eventually has obtained again Eq.
(10). In the present paper (see Appendix C), we
follow a different technique working in conjunction
with spherical coordinates, inspired by [1], and
generalize the proof for any radius 𝑅 𝑖 1,2,
regardless of whether the two spheres are in contact
or not. In other words, the interaction energy of the
two spheres of radius 𝑅 is the same as if the charge
were concentrated at the centers of the spheres. This
result is also sustained by the findings of Appendix
B (in the form of a theorem), in which it was found
that the action of two point-like charges within any
sphere centered at one of them has a null effect. In
other words, either the spheres of radius 𝑅 are real
(thus with zero electric intensity per se) or they are
fictitious, the volume integral of the dot-product
2𝑬
∙𝑬
over them vanishes.
Therefore, in the case of distributed charges over
the surface of two corresponding spheres of radius
𝑅 and 𝑅, respectively, by virtue of Eq. (10), Eq.
(8) becomes:
22
1212
1212
12
22
fe
qqqq
UUUU k
R
Rd
, (11)
Remark-1: It is worthy to mention that for any
isolated spherical charge of radius 𝑅 and charge 𝑞,
the field energy outside an imaginary sphere of
radius 𝑟 which surrounds it is given in an analogous
way with Eq. (9b), by 𝑈𝑟 𝑘𝑞/2𝑟, with
𝑟𝑅. Therefore, if 𝑈 𝑈
𝑅𝑘
𝑞/2𝑅 is
the maximum possible field energy which
corresponds to the entire exterior of the capacitor (of
radius 𝑅), the previous expression can be also
written as 𝑈𝑟
𝑅𝑟
⁄𝑈
𝑅𝑟
⁄
, and
the corresponding graph is illustrated in Fig. 2. One
may observe that 99% of the total field energy is
trapped into a sphere of radius 𝑟 100𝑅. In other
words, one hundredth of the maximum field energy
value exists in the exterior of a sphere with 𝑟/𝑅
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
216
Volume 19, 2022
100, i.e., at a distance of 100 times the capacitor’s
radius 𝑅.
Remark-2: Regarding the same isolated
spherical charge of radius 𝑅 and uniform charge 𝑞,
it is worth-mentioning that the field energy trapped
between two radii 𝑟 and 𝑟 (with 𝑅𝑟
𝑟
) does
not depend on the particular value of capacitor’s
radius 𝑅. Actually, applying Eq. (9b) twice, one
time for each radius we have: 𝛥𝑈 𝑈𝑟
𝑈𝑟
0.
Obviously, the latter expression does not depend on
𝑅 but only on the radii 𝑟 and 𝑟 as well as the
electric charge 𝑞. This is a consequence of the fact
that the uniform distribution of electric charge on
the surface of the sphere is equivalent to the field
which creates a concentrated (point-like) charge at
its center.
Remark-3: Quite similarly, for a specific sphere
𝑂,𝑟, of radius 𝑟 and centered at the middle 𝑂 of
the two spherical charges, the self-energy outside
the sphere depends only on the value 𝑟 and the
charge 𝑞 (according to 𝑈𝑘
𝑞/2𝑟), while the
energy inside the sphere highly depends on the radii
𝑅 𝑖 1,2 of the two charged spheres as well.
Again, in case of point-like charges the self-energy
inside the sphere 𝑂,𝑟 is infinite, whereas in case
of charged spheres it is bounded given by 𝑘𝑞1/
2𝑅1/2𝑟; obviously the latter occurs because
the field is limited in the interval 𝑅,𝑟.
Fig. 2: Decay of field energy in a charged sphere of
radius 𝑅.
3 Interpretation of the physics
involved
3.1 Basic discussion
In this section we deal again with Eq. (11) but now
using the energy foundations of physics and not the
mathematical physics. To make this point quite
clear, we can assume that the electric charge 𝑞 is
found at the ∞ whereas the charge 𝑞 at ∞, as
shown in Fig. 3(a). Thus, none of them influences
the other one; simply each of them possesses its
self-energy 𝑈 𝑖 1,2 given by Eq. (9b). From the
aforementioned state, we keep 𝑞 fixed while we
slightly push 𝑞 towards 𝑞. Then, due to
Coulomb’s law according to Eq. (3), although the
distance 𝑟 is very large the Coulomb interaction
force will give a small force along the line 12 of the
two point charges. Below we distinguish two cases.
If the sign of the charges is the same (i.e.,
𝑞𝑞0), the interaction force is repulsive thus the
charge 𝑞 will continue staying at ∞. Then we
need to offer a positive amount of work on 𝑞,
which by definition equals to the (here) positive
potential energy, to move 𝑞 at a distance 𝑑 from
the immobile charge 𝑞. At the end of the process, a
positive work 𝑈 𝑞
𝑞/𝑑 will be added to the
initial energy 𝑈𝑈𝑈
0, thus eventually
leading to total field energy equal to 𝑈 𝑈
𝑈𝑈
0.
In contrast, if the sign of the charges is different
(i.e., 𝑞𝑞0), the interaction force is attractive
thus the charge 𝑞 will be accelerated (according to
Newton’s Second Law) until an obstacle
permanently keeps it at a distance d from 𝑞. By
keeping the distance 𝑑 for ever, this kinetic energy
is lost thus the total field energy decreases by the
negative potential energy, leading to 𝑈 𝑈
𝑈|𝑈|0. Alternatively, if we do not like the
scenario of the accelerated charge we can resort to
an external force which cancels the attraction of 𝑞
thus absorbing the negative energy 12e
kqq d from
the initial state of 𝑈𝑈𝑈0, and
therefore eventually leading to total field energy
equal to 𝑈 𝑈
𝑈
|𝑈|0.
Therefore, whatever the sign of the charges is,
we have shown that Eq. (8) is valid, thus we have
𝑈
𝑈
𝑈𝑈
0.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
217
Volume 19, 2022
The above analysis is valid regardless the two
changes are point-like or charged spheres of radii 𝑅
and 𝑅, respectively. The only change for the point-
like charges is that then we may assume that 𝑈
𝑈0.
3.2 Exercise: From infinity to contact and then
separation to the final state
The contents of subsection 3.1 are sufficient for a
student to understand the energy conservation and
the field energy of the final configuration of the two
electric charges separated by distance 𝑑. However,
it is interesting to extend it by changing the virtual
path, starting again from infinity but now
performing a by-pass as follows.
3.2.1 General electric charges
Although the overall idea is very similar, it is
suitable to focus on the particular case of different-
sign charges (i.e., 𝑞𝑞0) where an attractive
Coulomb force dominates. When the charge 𝑞 is on
the right-hand side half-space at ∞ and is
accelerated by attraction to the left charge 𝑞 at
∞, if we wish we may not consider an obstacle
at distance 𝑑, but we may imagine of an artificial
collision between the two charged spheres (where,
due to the attraction, they remain in contact for ever
as shown in Fig. 3(b)). Then we may imagine that a
virtual retouching of them is imposed until their
centers are eventually separated by distance 𝑑
𝑅𝑅) as shown in Fig. 3(c). In this thought-
experiment, the initial energy at infinities is 𝑈
𝑈𝑈
0, while at the collision (where the
distance of centers is 𝑑′ 𝑅𝑅) the interaction
energy is
22
12
12
12
12
22
,
contact ee
total
e
kq kq
URR
kqq dRR
d
(12)
Moreover, since we are concerned with charges
separated by a distance, 𝑑, we can start from the
above virtual contact state described by total field
energy given by Eq. (12), and then perform
additional work to increase the distance between the
centers from 𝑑𝑅
𝑅 to the final value 𝑑, thus
external forces must perform a work equal to:
12 2
12 12
1
111
.
dd
dd e
dd
d
ee
d
WFdrkqqdr
r
kqq kqq
rdd
(13)
Summing up the abovementioned work dd
W [Eq.
(13)] to the initial energy contact
total
U [Eq. (12)], the
terms related to the distance 𝑑𝑅
𝑅 are
cancelled, thus the final field energy of the two
electric charges with opposite charges separated at
distance 𝑑, is given by:
22
1212
12
22
contact
total d d e qqqq
UWk
RRd
(14)
Comparing Eq. (14) with Eq. (11), one may observe
that they coincide. This factoid was anticipated
because the field is conservative and we simply
follow different energy-exchange paths in our
thought experiment. Therefore, the total field energy
has to be preserved.
Fig. 3: Through-experiment: (a) Infinity (b) Contact (c)
After separation by distance 𝑑.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
218
Volume 19, 2022
4 Spatial breakdown of field energy
components
4.1 General
The field energy is divided in two parts, the
interaction energy and the self-energies. For each of
them we distinguish two cases, the former being the
point-like charges and the latter of two charged
spheres of radii (𝑅,𝑅). As has been previously
explained, the type of charges plays a role only
when they belong to the area under consideration,
otherwise not. First we refer to the dashed line
sphere (
𝑂,𝑑/2) (to comply with Hilborn’s, [1],
terminology) as shown in Fig. 4a.
At every point 𝑃 of the electric field made of two
charges, 𝑞 and 𝑞, we can easily determine the
magnitude of electric intensity 𝐸𝐸
as shown in
Fig. 1, thus the energy density 𝑢
⁄
is known. By integrating the quantity 𝑢 in a certain
volume, we can obtain the corresponding electric
field energy.
The aim of this section is to determine the tree
components (𝑈, 𝑈, and 𝑈) [see, Eq. (8)] of the
field energy in characteristic areas of the infinite
field which surrounds the two electric charges.
Fig. 4: Characteristic spheres and positions: (a) Dashed
line sphere 𝑂,𝑑/2, (b) Charge of the same sign
(𝑞𝑞0), (c) Charges of opposite sign (𝑞𝑞0), (d)
Fully surrounding sphere (𝑂,𝑅).
4.2 The Interaction Field Energy
One of these characteristic areas is the dashed line
sphere 𝑂,𝑑/2 which is centered at the middle 𝑂
of the line 𝑄𝑄 connecting either the point-like
charges or the centers of the correspondent spheres
𝑅 and 𝑅. The reason we select this sphere is
because for each point 𝑃 on it the angle 𝑄𝑃𝑄
is 90
right angle (subtended from the diameter 𝑑), thus
the dot-product 𝐸
∙𝐸
vanishes on it. Moreover,
when the point P lies inside this dashed line sphere,
obviously the angle 𝑄𝑃𝑄
is obtuse (> 90o deg),
while when P is outside the same sphere the angle
𝑄𝑃𝑄
is acute (<90o deg). This in turn determines
the sign of the dot-product 𝐸
∙𝐸
, which is
simply depended on the sign of the product 𝑞𝑞,
accordingly.
In more detail, if 𝑞𝑞0
, the dot-product
𝐸
∙𝐸
is negative inside the dashed line sphere
𝑂,𝑑/2 and positive outside it (Fig. 4b). In
contrast, if 𝑞𝑞0
, the dot-product 𝐸
∙𝐸
is
positive inside the dashed line sphere 𝑂,𝑑/2 and
negative outside it (Fig. 4c).
4.2.1 Point-like charges
As shown in Appendix A, the total amount of the
interaction energy 𝑈 is given by:
12
12
e
kqq
Ud
(15)
Moreover, the accurate portion of the interaction
energy inside the dashed line sphere 𝑂,𝑑/2 is
given by:
12
12 (, 2)
12
(2 )
4
0.2854 ,
e
Inside
sphere O d
e
kqq
Ud
kqq
d
(16a)
while outside the dashed line sphere 𝑂,𝑑/2 is:
12
12 (, 2)
12
(2 )
4
1.2854
e
Outside
sphere O d
e
kqq
Ud
kqq
d
. (17a)
In other words, the most amount of the interaction
field energy is outside the small dashed line sphere
𝑂,𝑑/2 which marginally includes both charges.
This may be physically explained by the fact that as
one electric charge proceeds to the other, it is the
far-field potential lines which will interact with each
other.
4.2.2 Charged spheres
The setup is shown in Fig. 5(a). Interestingly, Eq.
(15) holds also for the case that the charges are
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
219
Volume 19, 2022
uniformly distributed along the surfaces of two
spheres of radii 𝑅 and 𝑅, respectively. In contrast,
according to Appendix C, Eq. (16a) and Eq. (17a)
have to be replaced by Eq. (16b) and Eq. (17b),
respectively, while the sum of the interaction energy
remains unaltered despite the differences.
12
12 (, 2)
112
12 1
1
122
12 2
2
2
4
sin 111
2
sin 111
2
e
Inside
sphere O d
e
e
kqq
Ud
R
kqq R
d
dR d
R
kqq R
d
dR d
(16b)
12
12 (,/2)
112
12 1
1
122
12 2
2
2
4
sin 111
2
sin 111
2
e
Outside
sphere O d
e
e
kqq
Ud
R
kqq R
d
dR d
R
kqq R
d
dR d
(17b)
Actually, first of all, we have to mention that the
field outside each charged sphere is identical with
that of a point-like charge. But this notice does not
prove the coincidence between the two fields (point-
like charges versus charged spheres). This is so
because the electric filed inside each charged sphere
of radius 𝑅 vanishes (𝐸
0
) thus the relevant
integral of the previous point-like field is not
encountered in the case of the charged spheres. This
in turn induces a nonzero change 𝛥𝑈 inside
and another nonzero change 𝛥𝑈 outside the
dashed line sphere 𝑂,𝑑/2. The reason that for
point-like charges the sum is preserved invariant is
given in Appendix B, where it is shown that
𝛥𝑈 𝛥𝑈
0. Clearly, it is also shown
that within each such ‘imaginary/virtual’ sphere 𝑅,
of which a certain part is inside the bigger sphere
𝑂,𝑑/2 and the rest outside it, for the particular
case of point-like charges the total volume integral
of 𝐸
∙𝐸
vanishes as well, thus it consists of two
equal but opposite nonzero terms.
Fig. 5: (a) Dashed line sphere 𝑂,𝑑/2 and charged
spheres of radii (𝑅,𝑅), (b) Sphere 𝑂,𝑅 fully
surrounding the two charges.
4.3 The self-energies
Following the methodology used in Section 4.1, and
considering that the point 𝑄 𝑞 is on the right
while 𝑄 𝑞 is on the left, in Appendix D it is
shown that the self-energy of each charged sphere
(𝑅,𝑅) can be analytically found as follows.
For the charge 𝑞 we have:
Inside the sphere 𝑂,𝑑/2:
2
11
1
1
111
ln
4
e
Inside
kq R
Udd dR
(18)
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
220
Volume 19, 2022
Outside the sphere 𝑂,𝑑/2 Right:
2
1
1
1
11
4
e
Outside Right
kq
URd
(19a)
Outside the sphere 𝑂,𝑑/2 Left:
2
11
1
1ln
4
e
Outside Left
kq R
Udd
(19b)
Outside the sphere 𝑂,𝑑/2 Total:
2
11
1
1
11 1
ln .
4
e
Outside Total
kq R
URd d d
(19c)
As a first check, the sum of the above three terms
[i.e., Eq. (18) as well as Eq. (19a) and Eq. (19b)]
equals to the anticipated value 2
11
(2 )
e
kq R :
2
11
1
1
2
1
1
2
11
2
1
1
111
ln
4
11
4
1ln
4
2
e
Total
I
nside
e
Outside Right
e
Outside Left
e
kq R
Udd dR
kq
Rd
kq R
dd
kq
R
(20)
Similarly, for the charge 𝑞 we can write the
analogous field energy expressions:
Inside the sphere 𝑂,𝑑/2:
2
22
2
2
111
ln
4
e
Inside
kq R
Udd dR
(21)
Outside the sphere 𝑂,𝑑/2 Right:
2
22
2
1ln
4
e
Outside Right
kq
R
Udd
(22a)
Outside the sphere 𝑂,𝑑/2 Left:
2
2
2
2
11
4
e
Outside Left
kq
U
R
d
(22b)
Outside the sphere 𝑂,𝑑/2 Total:
2
22
2
2
11 1
ln
4
e
Outside Total
kq R
URd d d
(22c)
As a second check, it can be verified that the sum of
the three portions of 𝑈-self energy is equal to the
anticipated formula 2
22
(2 )
e
kq R :
2
22
2
2
2
22
2
2
2
2
2
2
111
ln
4
1ln
4
11
4
2
e
Total
I
nside
e
Outside Right
e
Outside Left
e
kq R
Udd dR
kq R
dd
kq
Rd
kq
R
(23)
Therefore, the total field energy with respect to the
dashed line sphere 𝑂,𝑑/2 is as follows.
Inside the sphere 𝑂,𝑑/2:
2
(,/2) 11
1
2
22
2
12
112
12 1
1
12
12
111
ln
4
111
ln
4
(2 )
4
sin 111
2
sin
2
Inside O d e
Total Field
e
e
e
e
kq R
Udd dR
kq R
dd dR
kqq
d
R
kqq R
d
dR d
R
kqq d
2
2
2
111
R
dR d
(24)
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
221
Volume 19, 2022
Outside the sphere 𝑂,𝑑/2:
2
(,/2) 11
1
2
22
2
12
112
12 1
1
1
12
111
ln
4
111
ln
4
(2 )
4
sin 111
2
sin
2
Outside O d e
Total Field
e
e
e
e
kq R
Udd dR
kq R
dd dR
kqq
d
R
kqq R
d
dR d
R
kqq
22
2
2
111
R
d
dR d
(25)
5 Field energies in a sphere 𝑶,𝑹𝒑
fully surrounding the two charges
In this section we consider another sphere 𝑂,𝑅,
again centered at the midpoint 𝑂 of the connecting
line 𝑄𝑄, but now with a larger radius so as to
entirely include the two electric charges. The case of
the point-like charges is shown in Fig. 4(d). In case
that the radii of the charged spheres are equal one
another (i.e., 𝑅𝑅
𝑅), it should obviously be
𝑅
𝑅𝑅, (see, Fig. 5(b)).
5.1 Outside the sphere 𝑶,𝑹𝒑, i.e., for 𝒓
𝑹𝒑,𝒎𝒊𝒏
This is shown either in Fig. 4d (point-like) or in Fig.
5b (charged spheres).
According to Appendix E, the self-energy of each
charge in the exterior of the sphere 𝑂,𝑅 is given
by:
2
22
()
42
1ln ,
44 2
(1,2)
p
pp
ei
Outside
irR pp
RRd
kq
URd d Rd
i
, (26)
thus the observer cannot distinguish whether the
charges are point-like of not.
Moreover, regarding the interaction energy
𝑈 in the infinite space outside the sphere
𝑂,𝑅, based on variables
11
,rr
, Eqs.
(E2) to (E.4) of Appendix E show that:
12
12
2
1
1
02
22
2
22 2 2 2
2
sin d.
3cos cos sin
42 4
p
e
rR
pp
kqq
U
dd
Rd d R
(27)
In the context of this paper, it was not possible to
derive a closed–form analytical expression for the
integral in Eq. (27), despite the fact that a
commercial symbol manipulation code was used.
For the moment, it is proposed to calculate it
numerically, for example applying Simpson’s
trapezoidal rule. A short program in MATLAB® is
given at the end of Appendix E.
5.2 Inside the sphere 𝑶,𝑹𝒑, i.e., for 𝒓𝑹𝒑
The field energies in the interior of the sphere
𝑂,𝑅 are easily determined by subtracting Eq.
(26) and Eq. (27) from the corresponding total
energies.
First we recall that in case of charged spheres
the total self-energy is 𝑈𝑈
𝑘
, while it becomes infinity for point-
like charges. Obviously, regarding the self-energies
inside the sphere 𝑂,𝑅 it makes sense to talk
about only for the former case (of charged spheres),
for which we have:
22
22
()
42
1ln ,
244 2
(1,2).
p
pp
ei ei
Inside
irR ip p
RRd
kq kq
URRddRd
i
(28)
Second, regarding the interaction energy
𝑈
, we have:
12
12 12
() p
p
e
Inside rR
rR
kqq
UU
d
, (29)
where
12
p
rR
U is given by Eq. (27).
Note: All the formulas of Section 5 concern the
case that the sphere 𝑂,𝑅 does not intersect the
charged spheres. Therefore, for the particular case
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
222
Volume 19, 2022
that 𝑅𝑅
𝑅, Eqs. (26) to (29) are valid
onlywhen 𝑅
𝑅𝑅,, as shown in Fig.
5(b). Again, the case that 𝑅𝑑/2 is fully covered
in Section 4. Therefore, the very special case
according to which 𝑑/2𝑅𝑑/2𝑅 is not
covered in this paper, and is left as an exercise for
the interested reader.
6 Electric Dipole
6.1 General
The topic of electric dipoles is very interesting in
electrostatics and electrodynamics. The energetics
of a magnetic dipole of moment M in an external
field B, [10], as well as the ‘hidden momentum
forces’, [11], and numerical methods such as the
multipole expansion method, [12], have
been a
matter of intensive research to better understand the
behavior of dipoles. The mathematical complexity
and the involved simplifications sometimes is a
reason for debate, [13], particularly in
electrodynamics. Differences between electrostatics
and magnetostatics regarding the force have been
commented, [14], while in the past the magnetic
force has been discussed in detail, [15]. Not only
theoretical, [16], but also practical topics have
appeared in literature, [17], among many others. For
details the reader is referred to standard textbooks
such as [18] and [19].
Two equal charges, 𝑞, of opposite sign, separated
by a distance 𝑑2𝑙, constitute an electric dipole.
Therefore, the electric dipole is the special case of
the previous sections simply setting (𝑞𝑞
𝑞). The electric moment of a dipole 𝑝 is defined to
have the magnitude 𝑝𝑞𝑑 and points from the
negative charge to the positive charge.
A point 𝑃 is preferably specified by the
coordinates 𝑟 and 𝜃 illustrated in Fig. 1. As
previously, the electric potential at 𝑃 will be given
by the exact relationship:
12
12
eqq
VVV krr
(30a)
Assuming that 𝑟≫𝑑, we can write 𝑟𝑟
≅
𝑑 𝑐𝑜𝑠𝜃 and 𝑟𝑟𝑟
, so Eq. (30a) changes into the
approximation:
22
cos cos
mm
ee
mm
dp
Vkq k
rr
(30b)
Then, based on Eq. (5) [
𝑬
grad 𝑉], we can
determine the electric intensity, which is presented
later in Eq. (35). Alternatively, the electric intensity
may be produced by synthesizing the two vectors,
𝐸
and 𝐸
, due to the separated actions of the two
opposite charges.
The condition 𝑞𝑞
𝑞 induces some
simplifications in the previous formulas which are
outlined below.
Total
interaction
energy:
12 0 1 2
2
12d
2
Dipole All Volume
e
UEE
q
kd
(31)
Total self-
energy:
22
12
12
2
22
e
Dipole
e
qq
UU k
R
R
q
kR
(32)
Total
Field
energy:
2
1212
11
e
Dipole
UUU kqRd
(33)
In addition to Eq. (8) which creates no doubt, Eq.
(33) also shows that in case of charged spheres (of
radii 𝑅𝑅
𝑅), the total field energy is strictly
positive. Actually, the condition of maximum
allowable radius equal to 𝑅 𝑑/2 (the equality
holds for charged spheres in contact) gives the
inequality 𝑅𝑑/2 (no penetration of charges),
which algebraically leads to 1/𝑅 2/𝑑 1/𝑑,
which implies 1/𝑅1/𝑑0
, which proves that
the total field energy of the dipole is positive.
Note: When the two equal spherical charges are
in contact, we have 𝑅𝑑/2 and consequently
𝑈𝑈
𝑈
𝑘
𝑞/𝑑, thus Eq. (33)
becomes:
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
223
Volume 19, 2022
Total Field energy for spherical charges
in contact:
2
1212
2
12
21
Total Field e
Dipole
e
Dipole
UUU kqdd
kq
d
U
(34)
6.2 The semi-analytical approximate model
Let us suppose that we are interested in determining
the radius 𝑅 of a sphere centered at the middle 𝑂 of
a dipole of length 𝑑, with 𝑅𝑑/2, in the exterior
of which the total field energy equals to a certain
given value. This task could be solved analytically
in a single step using Eq. (26) and Eq. (27), but
unfortunately we were not able to derive a closed-
form analytical expression for the portion of the
interaction energy 𝑈 given by Eq. (27).
Below we present a semi-analytical way, where the
first analytical guess can be iteratively improved
easily in about five to six steps.It is worthy to
mention that when established textbooks refer to
electric dipoles, they usually present only far-field
analytical expressions in terms of the dipole
moment 𝒑
(with measure 𝑝|𝒑
|𝑞𝑑) and the
dipole’s length 𝑑2𝑙 (see, [18-20]). In more detail,
at distant points the magnitude of the vector sum
𝑬
𝑬
𝑬
is written in terms of 𝑟,𝜃 as
follows (see, [18], p. 158):
12
2
313cos ,
emm
m
kp
EE r d
r
, (35)
where 𝑟 is the distance with respect to the middle
of the dipole, 𝜃 is the polar angle formed between
the dipole and the position vector as shown in Fig.
1b.
One may observe that in Eq. (35) there is a
fictitious singularity at the middle O of the dipole
(𝑟0), which of course is not true because these
formulas are valid only when 𝑟≫𝑑 (thus not
applicable at the point O). While the field 𝑬
has
been extensively presented in many manuscripts and
textbooks such as, [18-20], the same is not the case
for the field energy of a dipole.
Under the condition 𝑟≫𝑑, the associated field
energy in the infinite space (for 𝑟𝑅
) will be
produced by substituting Eq. (35) into Eq. (6) thus
receiving the following approximate (because it is
valid for 𝑟≫𝑑, practically when 𝑟 10𝑑)
analytical formula:
2
12
22
0
3
()
2
22
4
00
2
3
2
222
33 3
113cos ( sin )
8
13cos sin
8
1
24
83
.
33 3
mp
p
p
approx m m m m m
rR em
V
em
mmm
m
R
e
mR
e
ee
pp p
kp
Urdddr
kr
kdr
pd d r
kpr
kqd
kp kqd
RR R
(36)
In Eq. (36), the quantity 𝑅 represents the radius
of a certain sphere centered at the middle point O of
the dipole (see, Fig. 6) which has to be subtracted
from the integral, otherwise it would be singular.
Obviously, the smaller the radius 𝑅 the greater the
field energy 𝑈. It should become clear that
the field energy 𝑈 is influenced by all the
three intensity components, i.e., 𝑬
, 𝑬
, and
2
𝑬
∙𝑬
which contribute to the two self-energies
plus the interaction energy. Equation (36) includes
all the three terms, i.e., (𝑈,𝑈, and 𝑈 but not the
accurate values which were earlier found (see, Eqs.
(26) to (29)). In other words, it is a rough
approximation when the radius 𝑅 is close to the
dipole whereas it is very accurate for large values of
𝑅. In any case it is sufficient to give us a first rapid
estimation of 𝑅 when we wish to include a certain
amount of field energy outside it (𝑟𝑅
).
As an example, if we temporarily assume that the
total field energy 𝑈 of the dipole outside the
sphere 𝑂,𝑅 (given by Eq. (36)) equals to the
absolute value (|𝑈||𝑈|) of the potential energy
(given by Eq. (2)), i.e., 𝑈𝑘
𝑞/𝑑, then we will
have:
2223
1
(3 )
eapproxe p
UkqdU kqd R , (37a)
whence a first-order estimation 𝑅 of the radius 𝑅
will be (see, Fig. 6):
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
224
Volume 19, 2022
130.6934
3
pd
Rd . (37b)
Therefore, if Eq. (35) were correct everywhere in
the space, then the field energy trapped between the
surface of the sphere of radius 𝑅 given by (37b)
and the infinity would be equal to the potential
energy of the electric dipole with charges separated
at distance 𝑑. In other words, in this particular
example the energy inside the sphere of radius 𝑅
(if were correct) is in excess of the potential energy
(𝑈𝑈
𝑈
|𝑈|).
Moreover, in order to draw definite conclusions
for the accurate boundary of the sphere 𝑂,𝑅 the
interior of which includes the sum (𝑈𝑈
𝑈 |𝑈|) and its exterior includes |𝑈|, we
have to replace the approximate expression (37b),
𝑅, by the accurate one, 𝑅. To this purpose, we
perform the following iterative scheme:
Step-1: Start with the initial guess 𝑅 of Eq.
(37b) and then calculate the accurate field energies
using Eqs. (26) to (29). Then the initial pair of
values (𝑅,𝑈) is known.
Step-2: Increase the initial guess by (say) 10
percent (𝑅
1.1𝑅
) and repeat Step-1 but now
for this new value 𝑅
. Now, the updated pair of
values (𝑅
,𝑈) is known.
Step-3: Perform linear interpolation between the
pairs 𝑅,𝑈 and 𝑅
,𝑈 to obtain the
desired value, here it is 𝑈
|𝑈|, using the simple
formula: 𝑅 𝑅
𝑅
𝑅𝑈
𝑈/𝑈 𝑈
. Based on this new
value 𝑅, repeat Step-1 and then calculate the
accurate field energies using Eqs. (26) to (29).
Step-4: Repeat for a small number of iterations
until the result for the updated value 𝑅,
remains unchanged.
In this specific case in which the prescribed
value of the field energy outside the sphere 𝑂,𝑅
was chosen as 𝑈
|𝑈|, convergence was
achieved in only six steps, and the final result was
found to be:
20.7412
p
Rd (37c)
The fact that the accurate radius 𝑅 (thick solid
line) is greater than the initial guess (dashed line) is
clearly shown in Fig. 6.
Fig. 6: The radius 𝑅 (approximate 𝑅 and accurate 𝑅)
surrounding the dipole.
To make the results of this example neutral, we first
notice that the potential energy 𝑈𝑘
𝑞/𝑑 is
expressed as a unit multiple of the standard ratio
𝜇𝑘
𝑞/𝑑, which could be also equivalently
written in terms of the dipole moment 𝑝 as 𝜇
𝑘𝑝/𝑑. Therefore, instead of numbers it is more
convenient to present the results as factors of the
aforementioned ratio 𝜇.
In this context, the initial guess given by Eq. (37b)
corresponds to 𝑈, 1.3132 𝜇, while the next
iterations are shown in Table 1. One may observe
that while the improvement of the radius from the
initial guess 𝑅 0.6934 𝑑 to the final value
𝑅 0.7412 𝑑 was only +6.9%, the decrease of
the initially erroneous total field energy is about
31%. In other words, for this particular case the
semi-analytical approximate formula which is given
by Eq. (36) overestimates the true total energy by 31
percent. In general, five to six iterations are
sufficient to give us the correct total field energy
which is represented by the unity factor, i.e., in this
example we have 𝑈
1.000𝜇𝑘
𝑞/𝑑
𝑈. Interestingly, the difference between 6.9% and
31% reflects the inaccuracy of the semi-analytical
formula regarding the total field energy.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
225
Volume 19, 2022
Table 1. Iterations for the derivation of the exact total
field energy as a factor of 𝜇𝑘
𝑞
/𝑑
Iteration Factor of 𝜇
1 (initial guess) 1.3132
2 0.8959
3 0.9782
4 1.0020
5 1.0000
6 1.0000
6.3 The accuracy of the semi-analytical
model
Section 6.1 was useful for the reader to understand
that when the distance from the middle of the dipole
is small the inaccuracy of the semi-analytical
formula, given by Eq. (35), is substantial. A bulk of
papers note that the critical point where Eq. (35) is
accurate is about 𝑟 10𝑑 and the overall feeling is
that the field energy vanishes for greater values. In
this section we complete the established knowledge
by drawing definite conclusions.
Having developed in the previous sections all
those analytical formulas that can be easily applied
either near or far away from the electric dipole,
below we shall give some details concerning the
comparison of the accurate field and the
approximate semi-analytical model. For the sake of
completeness, first the comparison is trivial and is
concerned with the electric field itself, a task which
can be easily performed by any student. Fig. 7
shows the comparison of the magnitude of electric
intensity 𝐸𝐸
, where the results are normalized
accordingly (are multiplied by 𝑑/2/𝑝/𝑞), in
two following typical directions:
(i) Along the perpendicular bisector, where
we have 𝜃𝜋/2 thus [according to
Eq. (35)] 𝐸≅𝑘
𝑝𝑟
⁄ (Fig. 7a), and
(ii) Along the dipole axis, where we have
𝜃 0 or 𝜋 thus [according to Eq.
(35)] 𝐸≅2𝑘
𝑝𝑟
⁄ (Fig. 7b).
In a first glance, one may observe that, along the
perpendicular bisector the difference between the
exact and the semi-analytical form becomes small
after a distance 𝑟 of about 𝑑, while along the
dipole axis we need a distance from the midpoint 𝑂
at least 1.5𝑑. The latter is due to the singularity of
the closest charge to the observer, which
(singularity) is at a distance 𝑑/2 from the midpoint
𝑂.
(a)
(b)
Fig. 7: Electric intensity 𝐸: (a) perpendicular bisector, (b)
dipole axis.
In a second glance, a better impression is obtained
when studying the ratio of the exact intensity over
the semi-analytical one, as shown in Fig. 8, where
the difference becomes small for 𝑟8 and 𝑟
4, respectively.
The differences become a little larger when
comparing the field energy densities in Fig. 9, where
it is illustrated that the usually mentioned threshold
of 𝑟 10𝑑 is applicable only along the dipole axis
where it is rather better to consider 𝑟12𝑑 along
the perpendicular bisector.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
226
Volume 19, 2022
(a)
(b)
Fig. 8: Ratio of electric intensity (𝐸
/
𝐸
): (a) perpendicular bisector, (b)
dipole axis.
(a)
(b)
Fig. 9: Ratio of energy densities: (a) perpendicular
bisector, (b) dipole axis.
(a)
(b)
Fig. 10: Total field energy outside a sphere of radius 𝑟,
(a) as factor of 2𝑞
𝑀
/𝑞, (b) as the ratio of actual energy
over the absolute potential energy.
Now, it is interesting to see how the total field
energy reduces outside the sphere 𝑂,𝑟 with
increasing radius 𝑟10𝑑. Actually, Fig. 10 shows
that the total field energy outside a sphere of radius
𝑅
10𝑑 is accurately described by the semi-
analytical formula (35). Also, Fig. 10(b) shows that
although the total field energy is small when 𝑟
10𝑑, it does not entirely vanish even at 𝑟/𝑟
100. In more detail, if we ask to determine the
critical radius 𝑅, which includes the 99% of the
total field energy density, the answer depends on the
chosen radius 𝑅 𝑅𝑅
of the charged
spheres.
6.4 Numerical results for a dipole
Here we present some results regarding uniformly
charged spheres separated by distance 𝑑.
Considering that the infinite space is extended until
𝑟 10𝑑, the results of the field energy density are
shown in Fig. 11, for two cases, i.e., 𝑅𝑅
0.125 𝑑 and also 𝑅𝑅
0.250 𝑑. To avoid any
doubt, the energy density was calculated
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
227
Volume 19, 2022
synthesizing the two vectors 𝐸 and 𝐸 due to the
corresponding charges 𝑞 and q, respectively.
To test the analytical formulas which were
presented in Section 4, first we present Table 2
which is related to the dashed line sphere 𝑂,𝑑/2.
One may observe that as the radius of each
charged sphere increases the amount of interactive
energy inside the sphere 𝑂,𝑑/2 increases as well,
and the same occurs with the absolute value of the
interactive energy outside it. Note that contact
between the two spherical charges occurs when
𝑅𝑅
0.5 𝑑, which corresponds to the last
column of Table 2.
(a)
(b)
Fig. 11: Energy density for charged spheres of radius (a)
𝑅𝑅
0.125 𝑑 (detail) and (b) 𝑅𝑅
0.250𝑑
(full mesh).
When spheres of charge, 𝑞𝑞 and 𝑞𝑞,
and each of radius 𝑅 are considered, the minimum
value of the radius 𝑅 for which Eqs. (26) to (29)
are applicable is 𝑅 𝑑/2𝑅. Based on the
latter constructive element of the dipole, and setting
𝑹𝒅/𝒏 where 𝑛 is a positive integer, after
substitution in the formula
222
1212 (2 ) (2 )
eee
UUU U kq R kq R kqd , we
obtain the following Εq. (38):
2
(1)
e
kq
Un d
. (38)
For 𝑛2,4, and 8, the results are given in Table 3,
Table 4 and Table 5, respectively. In all these three
cases, the results are normalized to the absolute
value of the interaction energy,
0.
Note: Not only the numerical results in Table 2 to
Table 5 are reasonable and show a monotonic
behavior but also a part of them was validated by
performing numerical integration (3×3 Gauss
quadrature) on computational meshes such as that
shown in Fig. 11. Increasing the mesh density the
numerical results were found to coincide with the
analytical ones.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
228
Volume 19, 2022
Table 2. Breakdown of Field Energies for point-like and spherical charges inside and outside the characteristic dashed line
sphere 𝑂,𝑑/2 in the form of factors of the value
0
POINT-
LIKE
CHARGED SPHERES of radius
𝑹𝟏𝑹
𝟐
𝟎.𝟏𝒅 𝟎.𝟐𝒅 𝟎.𝟑𝒅 𝟎.𝟒 𝒅 𝟎.𝟓𝒅
𝑼𝟏𝟐: Inside the sphere 𝑂,𝑑/2 0.2854 0.3855 0.4861 0.5877 0.6910 0.7967
𝑼𝟏𝟐: Outside the sphere 𝑂,𝑑/2 –1.2854 –1.3855 –1.4861 –1.5877 –1.6910 –1.7967
𝑼𝟏: Inside the sphere 𝑂,𝑑/2 ∞ 1.6744 0.5976 0.2823 0.1459 0.0767
𝑼𝟏: Outside the sphere 𝑂,𝑑/2 ∞ 3.3256 1.9024 1.3843 1.1041 0.9233
𝑼𝟐: Inside the sphere 𝑂,𝑑/2 ∞ 1.6744 0.5976 0.2823 0.1459 0.0767
𝑼𝟐: Outside the sphere 𝑂,𝑑/2 ∞ 3.3256 1.9024 1.3843 1.1041 0.9233
SUM: 𝑼𝑼𝑼 ∞ 9.0000 4.0000 2.3333 1.5000 1.0000
Table 3. Breakdown of Field Energies for point-like and spherical charges in contact (𝑛2, i.e., 𝑅𝑅
𝑑/2) (as
factors of the value 𝑘𝑞𝑑
⁄0)
Fully Including the Dipole SPHERE of Radius (𝑹𝒑)
𝒅 𝟐𝒅 𝟓𝒅 𝟏𝟎𝒅 𝟐𝟎 𝒅 𝟏𝟎𝟎𝒅
𝑼𝟏𝟐: Inside the sphere 𝑂,𝑅 -0.1364 -0.5197 -0.8013 -0.9002 -0.9500 -0.9900
𝑼𝟏𝟐: Outside the sphere 𝑂,𝑅 -0.8636 -0.4803 -0.1987 -0.0998 -0.0500 -0.0100
𝑼𝟏: Inside the sphere 𝑂,𝑅 0.3920 0.7390 0.8993 0.9499 0.9750 0.9950
𝑼𝟏: Outside the sphere 𝑂,𝑅 0.6080 0.2610 0.1007 0.0501 0.0250 0.0050
𝑼𝟐: Inside the sphere 𝑂,𝑅 0.3920 0.7390 0.8993 0.9499 0.9750 0.9950
𝑼𝟐: Outside the sphere 𝑂,𝑅 0.6080 0.2610 0.1007 0.0501 0.0250 0.0050
SUM: 𝑼𝑼𝑼 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
229
Volume 19, 2022
Table 4. Breakdown of Field Energies for point-like and spherical charges with 𝑛4, i.e., 𝑅𝑅
𝑑/4 (as factors of
the value
0)
Fully Including the Dipole SPHERE of Radius
(𝑹𝒑)
𝟑𝒅/𝟒 𝒅 𝟐𝒅 𝟓𝒅 𝟏𝟎 𝒅 𝟏𝟎𝟎𝒅
𝑼𝟏𝟐: Inside the sphere 𝑂,𝑅 0.0495 -0.1364 -0.5197 -0.8013 -0.9002 -0.9900
𝑼𝟏𝟐: Outside the sphere 𝑂,𝑅 -1.0495 -0.8636 -0.4803 -0.1987 -0.0998 -0.0100
𝑼𝟏: Inside the sphere 𝑂,𝑅 0.9976 1.3920 1.7390 1.8993 1.9499 1.9950
𝑼𝟏: Outside the sphere 𝑂,𝑅 1.0024 0.6080 0.2610 0.1007 0.0501 0.0050
𝑼𝟐: Inside the sphere 𝑂,𝑅 0.9976 1.3920 1.7390 1.8993 1.9499 1.9950
𝑼𝟐: Outside the sphere 𝑂,𝑅 1.0024 0.6080 0.2610 0.1007 0.0501 0.0050
SUM: 𝑼𝑼𝑼 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000
Table 5. Breakdown of Field Energies for point-like and spherical charges with 𝑛8, i.e., 𝑅𝑅
𝑑/8 (as factors of
the value
0)
Fully Including the Dipole SPHERE of Radius
(𝑹𝒑)
𝟓𝒅/𝟖 𝒅 𝟐𝒅 𝟓𝒅 𝟏𝟎 𝒅 𝟏𝟎𝟎𝒅
𝑼𝟏𝟐: Inside the sphere 𝑂,𝑅 0.1625 -0.1364 -0.5197 -0.8013 -0.9002 -0.9900
𝑼𝟏𝟐: Outside the sphere 𝑂,𝑅 -1.1625 -0.8636 -0.4803 -0.1987 -0.0998 -0.0100
𝑼𝟏: Inside the sphere 𝑂,𝑅 2.3396 3.3920 3.7390 3.8993 3.9499 3.9950
𝑼𝟏: Outside the sphere 𝑂,𝑅 1.6604 0.6080 0.2610 0.1007 0.0501 0.0050
𝑼𝟐: Inside the sphere 𝑂,𝑅 2.3396 3.3920 3.7390 3.8993 3.9499 3.9950
𝑼𝟐: Outside the sphere 𝑂,𝑅 1.6604 0.6080 0.2610 0.1007 0.0501 0.0050
SUM: 𝑼𝑼𝑼 7.0000 7.0000 7.0000 7.0000 7.0000 7.0000
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
230
Volume 19, 2022
7 Electric Charges in the Form of
Conducting Charged Spheres
In the previous Sections, we have considered that
the charged spheres of radii (𝑅, and 𝑅) are not
conductors. This means that in each of them the
electric charge is uniformly distributed either they
are found in the infinity or they are close one
another. In such a case, that is for non-conductive
charged spheres of radius 𝑅 in contact (𝑑2𝑅),
with total charge 𝑄 (each of them carries 𝑞𝑞
𝑄/2), if the charge were uniformly distributed, the
“interaction energy” would take the usual value
(𝑈𝑘
𝑞𝑞/𝑑, with 𝑑2𝑅):
2
(2)(2)
28
ee
QQ Q
Uk k
R
R
(39)
A slightly different problem concerns two
conducting spheres, of charge 𝑞 and 𝑞. Here, the
charge distribution is not uniform, and the
“interaction energy” is not 𝑞𝑞/𝑑. This can be
computed via the so-called “method of images”. A
famous special case is considered in prob. 9 of
Reference [21].
For the particular case that the conducting
spheres of radius 𝑅 are in contact (i.e., are separated
by distance 𝑑2𝑅), it can be shown that their
capacitance is 𝐶2𝑅 𝑙𝑛2, and the “interaction
energy” is
22 2
2
22(2ln2)8ln2
ee e
QQ Q
Uk k k
CR R
(40)
Comparing (40) with (39), one may observe that in
the case of the conductive spheres, the “interaction
energy” is larger by a factor of 2/𝑙𝑛 22.89. In
other words, the conducting capacitor is a better
energy storage device than two separate non-
conducting spheres.
8 Discussion
This study refers to the general case of two electric
charges, 𝑞 and 𝑞 separated by distance 𝑑, either
point-like or of finite size (in the form of charged
spheres of radii 𝑅 and 𝑅, respectively). It was
found that while the “interaction energy” is the same
for finite or point-like spheres, the total field energy
is quite different.
Of particular interest is the case of two spheres
of equal radius 𝑅 that just touch each other (spheres
in contact), such that 𝑑2𝑅. Then:
For 𝑞𝑞
𝑞, the interaction energy is
𝑈 𝑘
𝑞/2𝑅, and 𝑈𝑈
𝑘3𝑞/2𝑅,
While for 𝑞𝑞
𝑞 (electric dipole),
the interaction energy is 𝑈 𝑘
𝑞/2𝑅,
and 𝑈𝑈
𝑈
.
But, clearly, the formulas that were presented in
the previous sections cover all the combinations of
charges and radii, for either the “interaction energy”
or the self-energies.
Regarding the self-energies, the radii (𝑅,𝑅) of
the spherical charges highly influence the field. For
given electric charges (𝑞,𝑞), the smaller the radii
the larger the self-energy. If the radii are given then
there is no problem to estimate the self-energy thus
the total field energy, otherwise we have to guess
them.
Within this context, we recall that for a charged
sphere of radius 𝑅, the uniform surface charge
density is 𝜎𝑞
/4𝜋𝑅
. Therefore, a quite
theoretical lower limit for the radius 𝑅 is probably
Plack length (1.61625502 10 meters), and
another more realistic lower bound comes from an
upper bound set for the surface charge density 𝜎,
i.e., 𝑅
. For example, one could
consider a reasonable radius based on the fact that
the maximum known charge density is probably the
value 1003 μCm−2, which is close to the limit of
dielectric breakdown [22].
The interested reader could extend the same
methodology but now considering a smaller sphere
𝑂,𝑅 than the dashed line sphere 𝑂,𝑑/2, i.e.,
now with 𝑅𝑑/2. Then, he/she has to consider
two cases, the former when this sphere intersects the
charged spheres and the latter when it does not.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
231
Volume 19, 2022
9 Conclusions
It was shown that the field energy due to two
electric charges can be described in full detail using
closed-form analytical expressions. This can be
done either for point-like charges or for charged
spheres of given radii. In the latter case, the charged
spheres may be either in contact (extreme case) or
not. In all these cases, a critical area of the infinite
space is the sphere of which the diameter is the line
segment that connects the centers of the two
charges. It was found that the biggest part of the
“interaction energy” is found in the exterior of the
aforementioned sphere, while the algebraic sum
(inside and outside it) is finite and always equals to
the potential energy (that is, by definition, the work
to bring one charge at a certain distance from the
other). In the exterior space of a big sphere that
surrounds the charges, the total energy does not
depend on whether the electric charges are point-
like or not. In contrast, the total energy in the
interior of this big sphere highly depends on their
type. Clearly, point-like charges are characterized
by infinite self-energy whereas charged spheres are
related to finite self-energies. In the case of an
electric dipole of any type (point-like charges or
charged spheres), the total field energy in the
exterior of a sphere with diameter about one-and-a-
half times the separation distance equals to the
absolute value of the negative “interaction energy”.
Acknowledgement:
The author thanks Professor Emeritus Kirk T.
McDonald, Joseph Henry Laboratories, Princeton
University, for his valuable constructive comments
on a very draft version of this paper, some of which
are contained in [9].
Appendix A
The interactive field energy
Theorem-1: Consider two electric point-like charges, 𝑞
and 𝑞, separated by distance 𝑑, and then a sphere of
diameter 𝑑 centered at the middle 𝑂 of the distance 𝑑.
With regards to the interaction field energy 𝑈 of these
point charges, we have:
(i) Inside the sphere, the field energy
𝑈 has the opposite sign of the
product 𝑞𝑞.
(ii) Outside the sphere (in the infinite space),
the field energy 𝑈 has the same
sign of the product 𝑞𝑞.
(iii) On the surface of the sphere (𝑟𝑑/2) the
dot-product (𝐸
𝐸
) equals to zero, thus it is
a transition state between the
abovementioned (plus to minus) values of
the interaction energy.
(iv) The biggest amount of the interaction field
energy is outside the sphere 𝑂,𝑑/2.
(v) The overall interaction energy,
𝑈 𝑈
𝑈, in the infinite three-
dimensional space is bounded and equals to
the potential energy, which is given by
𝑈𝑘
𝑞𝑞/𝑑, thus is has the same sign of
the product 𝑞𝑞.
Proof
Let us consider the point-like electric charges, 𝑞 at point
𝑄 and 𝑞 at point 𝑄, (of arbitrary values) separated by
distance 𝑑. For the sake of simplicity, it is convenient to
assume that both of them are positive (𝑞0 and 𝑞
0); however the conclusions are of more general
applicability. On an axial (meridian) plane passing
through the line segment 𝑄𝑄, and centered at the
midpoint 𝑂 of this segment, we consider a circle of
diameter 𝑑 thus passing through the two charges, as
shown by the dotted line in Fig. A-1. We also consider an
arbitrary point 𝑃 on the same axial section and a positive
unit charge at it. The electric intensity is produced by the
resultant of two vectors, the former (𝐸
) connecting 𝑃
with 𝑄 and the latter (𝐸
) connecting 𝑃 with 𝑄, as
shown in Fig. A-1.
Fig. A-1: The hypothetical dashed line sphere 𝑂,𝑑 2
⁄.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
232
Volume 19, 2022
The interactive field energy is given by the third term in
Eq. (8), thus it is:
12
12 d
4
All space e
Volume
EE
Uk
(A.1)
In order to calculate the volume integral of Eq. (A.1), it is
convenient to use spherical coordinates, i.e., the
azimuthal (longitudinal) 𝜑, the latitude 𝜃 and the radius 𝑟
measured from the point 𝑄 at which the charge 𝑞 lies.
Therefore, all the other geometrical quantities, such as
distances and angles, can be calculated in terms of the
aforementioned spherical coordinates (𝜑, 𝜃, 𝑟).
Obviously, by definition we have 𝑟𝑟
, while the
distance of 𝑃 from 𝑞 is a secondary variable denoted by
𝑟.
By definition, the dot product 𝐸
∙𝐸
in (A.1) is
written as follows:
𝐸
∙𝐸
𝐸
∙𝐸
∙cos𝜃
, (A.2)
where 𝜃 is the angle formed by the vectors 𝐸
and 𝐸
.
From elementary trigonometry applied to the triangle
(𝑃𝑁𝑄), with right angle at point 𝑁 (see in Fig. A-1), we
have:
12
22
cos
cos PN r d
PQ r
. (A.3)
We distinguish two main parts, the former inside the
dashed line sphere and the latter in all space.
In our proof, we start with last point (v) of the
Theorem.
I. All-space interaction energy
First we proof the identity for the total space, where
the total “interaction energy” is written as:
2
2
12 1 2
000
(,)
1dsind
4
All space erFr
UEErdr
k
(A.4)
Substituting (A.2) and (A.3) in (A.4), and further
considering that 𝐸
𝑘𝑞/𝑟
(with 𝑟𝑟) and 𝐸
𝑘𝑞/𝑟
, which both come from Eq. (4) of the main text,
we obtain:
2
12
0
2
12
22
22
00
1d
4
cos sin d
All space e
ee
r
Uk
qqrd
kk rdr
rr r
(A.5)
After simplifications, including the obvious substitution
2
0d2
, (A.5) simplifies to:
12
12 3
2
00
cos sin
2
e
All space r
rd
kqq
Udrd
r
(A.6)
Then applying the cosine-law in the triangle (𝑄𝑃𝑄),
focusing on the side 𝑃𝑄𝑟
, we receive:
2
2
1
22
2cosdr rrd
(A.7)
Substituting (A.7) into (A.6), we receive:
12
12
00
3
22 2
2
c
2cos
os sin
e
All space
rdr d
kqq
U
rd dr d
r
(A.8)
After elaboration, one can easily validate that the
indefinite integral in 𝑟, of (A.8), is:
3
22 2
2
1
22 2
cos
11
2cos
2co(s)
rd
dr dr
dr r
dr
r
d
(A.9)
and therefore (A.8) becomes:
12
12
0
1
22 2
0
2c
2
1sin
()os
e
All space
dr dr
kqq
U
d
(A.10)
The lower limit (𝑟0) of the definite integral in
(A.10), is clearly the term 1/𝑑, which after the
subtraction becomes 1/𝑑. Moreover, for the upper limit
(𝑟→∞), since 1 cos𝜃 1, we have the inequality:
1
22 2
2c
111
() ()
()os
rd d
dr dr r
(A.11)
Since when 𝑟→∞ both bounds in (A.11) vanish, it
turns out that the upper limit of the bracket in (A.10)
vanishes as well, thus (A.10) becomes:
12
12
0
1sin
2
e
All space d
kqq
Ud
, (A.12)
Since 0sin 2d
, (A.12) eventually gives:
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
233
Volume 19, 2022
12
12
e
All space d
kqq
U, Q.E.D. (A.13)
I. Inside the dashed line sphere
The only difference with the above all-space analysis
is the choice of the limits that have to be imposed for the
integrals in 𝜃 (now 0𝜃𝜋/2) and 𝑟 (now: 0𝑟
𝑑cos𝜃), thus (A.5) is modified to the following
expression:
2
12
0
2cos
2
12
22
22
00
1d
4
cos sin d
Inside e
d
ee
r
Uk
qqrd
kk rdr
rr r
(A.14)
By virtue of (A.9), now we receive:
2
12
12
0
11
sin
2sin
e
inside
kqq
Ud
dd
, (A.15)
and eventually:
12 12
12
20.2854
4
ee
inside
kqq kqq
Udd
(A.16)
Therefore, the part of the interaction field energy inside
the sphere 𝑂,𝑑/2, 𝑈, has the opposite sign of
the product 𝑞𝑞, and this completes the proof of point (i)
of the Theorem.
Regarding the outer part, 𝑈, we can simply
consider the obvious identity,
12 12 12
() () ()
All space Inside Outside
UUU
, (A.17)
which may be solved in 𝑈:
12 12
12
2
( ) 1.2854
4
ee
Outside
kqq kqq
dd
U
. (A.18)
Therefore, the part of the interaction field energy outside
the sphere, 𝑈, has the same sign of the product
𝑞𝑞, and this completes the proof of point (ii) of the
Theorem. Moreover, since 𝑈 is larger than the
𝑈, while 𝑈 is only 28.5 percent of
it (and even with a negative sign), it comes out that the
most part of the interaction energy is outside the dashed
line sphere, and this completes the proof of point (iv) of
the Theorem.
Note: The Theorem of this Appendix-A was highly
inspired by Hilborn (see [1], p. 69), but in the present
paper there is no gap in the proof, which is of major
assistance to the students and teachers. Repeating the
criticism by Hilborn (see [1], p. 69) concerned with
teaching guidelines in USA, “that result tells us that most
of the interaction field energy is found outside the sphere,
not “between the objects” as A Framework asserts in the
quotations cited earlier or as one’s (naive) intuition might
lead one to believe”, we would like to add an important
factoid. In brief, we would like to advise teachers that
when the electric charges are still at infinities (∞ and
∞), as they approach each other to form the final
structure (charges separated by distance 𝑑) in our
thought-experiment, the first area in which they interact
is the far-field first, and this factoid foreshadows that
most interactive energy is anticipated to be in the outer
space. In any case, the pair of equations (A.16) and
(A.18) gives the definite answer and resolves any
misunderstanding.
Appendix B
The interaction field energy within a fictitious
sphere of radius 𝑹 centered at one point-like
charge
Theorem-2: Consider two electric point-like charges, 𝑞
at point 𝑄 and 𝑞 at point 𝑄, separated by distance 𝑑,
and then a dashed line hypothetical sphere of diameter 𝑑
centered at the middle 𝑂 of the distance 𝑑. In addition,
consider a fictitious second sphere 𝑄,𝑅, i.e. centered
at the point 𝑄 where the point charge 𝑞 was
concentrated (as was afore mentioned), but now of radius
𝑅.
For the point-like charges (𝑞,𝑞) show that the
“interaction energy” 𝑈 inside the sphere 𝑄,𝑅
vanishes.
Fig. B-1: Intersection of the two spheres, 𝑂,𝑑 2
⁄ and
𝑄,𝑅.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
234
Volume 19, 2022
Proof
On a median (axial) plane let the two spheres be
intersected at the points 𝐴 and 𝐵. For an arbitrary point 𝑃
inside the sphere 𝑄,𝑅, considering the axial plane
through it, the line segment 𝑄𝑃 intersects the
aforementioned sphere at the points 𝐶 and 𝐷, while the
point 𝑁 is the normal projection of 𝑄 on the straight
segment 𝐶𝐷 (since the angle 𝑄𝑁𝑄
subtends the
diameter 𝑄𝑄 of the dashed line circle), as shown in Fig.
B-1.
In basic lines, we closely follow the procedure of
Appendix A where axis origin of the spherical coordinates
system is again 𝑄 (with 𝑟𝑟
,𝜃𝜃), but now the
most useful relationships are:
12
22
cos
cos
P
Nd r
QP r
(B.1)
1cosQN d
(B.2)
2sinQN d
(B.3)
11
cosPN Q N PQ d r
(B.4)
In an analogous way with (A.14), the interactive field
energy under consideration will be given as:
max
2
12
12
0
3
2
0
d
4
cos sin d
D
C
e
rR
r
rr
kqq
U
dr
dr
r
, (Β.5)
where 𝜃 𝜃
, is the maximum latitudinal angle
(𝑄𝑄𝐴
), and 𝑟𝑄
𝐶,𝑟𝑄
𝐷 the minimum and
maximum values of the variable 𝑟 (position vector),
which using elementary trigonometry are given by:
222
min 1
222
max 1
cos sin
cos sin
C
D
rrQNNCd Rd
rrQNNDd Rd
(B.6)
Since (A.7) is still valid, the integral in 𝑟 of (Β.5) is
described again by the opposite of (A.9)), thus is given
by:
1
22 2
2
2, 2, 2 2
11
2cos
()()
()
11 1 1 11
0
S
D
C
R
rr rr
rr
rr
DC
r
dr dr
rr QDQCRR
, (Β.7)
and since 𝑟, 𝑟
, 𝑅 (see Fig. B-1), it vanishes.
Therefore, the entire integral 𝑈 in (Β.5) vanishes
as well, and this completes the proof of Theorem-2.
Appendix C
Interaction field energy within and outside the
sphere 𝑶,𝒅/𝟐 for uniformly charged spheres of
radii 𝑹𝟏,𝑹𝟐
Find the active part of the interaction field energy
𝑈,
within the dashed line sphere 𝑂,𝑑/2,
and then outside it, now by excluding the corresponding
dead areas of the charges, of radii 𝑅 and 𝑅 (see, Fig. C-
1).
Fig. C-1: Interaction between two charged spheres
separated by distance 𝑑, within and outside the dashed
line sphere 𝑂,𝑑/2.
Solution: Considering the entire dashed line sphere
𝑂,𝑑/2, we have previously derived (in Appendix A)
Eq. (A.16), which is valid inside the sphere (𝑂,𝑑/2) for
point-like charges. But now that we have electric charges
with a radius (𝑅 and 𝑅), we need to remove that part of
the spherical charges which is inside the dashed line
sphere (because in the volume 𝑄𝐵𝐶𝐴𝑁𝑄 for 𝑞, and
similar for 𝑞, the electric intensity vanishes therein).
Following the previous Appendices, when considering
the charge (of radius 𝑅) at 𝑄 the axis origin is taken at
𝑄 as shown in Fig. C-1, the latitudinal angle is 𝜃𝜃
and the radius is 𝑟𝑟
. In this case the limits in Eq.
(Β.7) now become
,
,
,
,
, with sin𝜃, sin𝜃
𝑄
𝐴𝑄
𝑄
⁄𝑅
/𝑑 (see,
Fig. C-1) thus cos𝜃, 1𝑅/𝑑 (note that
𝜃, 𝑄
𝑄𝐴 in Fig. B-1). Therefore, the final integral
in 𝜃𝜃
will be:
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
235
Volume 19, 2022
max
2
max
12
12
2
0
12
2
0
12
max max
2
122
12 2
2
11
2sind
4sin
1sin d
2
11
cos 1
2
sin 111
2
e
Inside
rR
e
e
e
kqq
UdR
kqq
dR
kqq
dR
R
kqq R
d
dR d
(C.1)
A similar expression may be derived for
1
12
I
nside
rR
U
(inside the charged sphere of radius 𝑅 centered at 𝑄),
which is produced from (C.1) by simply replacing 𝑅
with 𝑅.
Then the clear portion of interaction energy within the
dashed line sphere 𝑂,𝑑/2 will be the algebraic sum of
Eq. (A.16) and the term
12
12 12
Inside Inside
rR rR
UU
,
which eventually is:
12
12 (,/2)
112
12 1
1
122
12 2
2
2
4
sin 111
2
sin 111
2
e
Inside
sphere O d
e
e
kqq
Ud
R
kqq R
d
dR d
R
kqq R
d
dR d
(C.2)
Regarding the net amount of the interaction energy
outside the dashed line sphere,
12 (,/2)
outside
sphere O d
U, the
opposite of the lost energy inside it (due to the fact that
one of (𝐸 or 𝐸) vanishes within the charges sphere) is
added to the external term, thus it becomes:
12
12 (,/2)
112
12 1
1
122
12 2
2
2
4
sin 111
2
sin 111
2
e
Outside
sphere O d
e
e
kqq
Ud
R
kqq R
d
dR d
R
kqq R
d
dR d
(C.3)
Adding (C.2) and (C.3) we derive the same amount,
𝑈 𝑘
𝑞𝑞/𝑑, as it were for point-like
charges.
Appendix D
Self-field-energy with respect to the dashed line
sphere 𝑶,𝒅/𝟐 for a uniformly charged sphere
of radius 𝑹𝑹
𝟐
We distinguish two areas, the former being inside the
dashed line sphere 𝑂,𝑑/2 while the latter outside it. In
both cases the charged sphere of radius 𝑅 is not included
(because within it the electric intensity vanishes, i.e.,
𝐸0).
Here, it is convenient to take the axis origin at point
𝑄, so it will be 𝑟 𝑟. For the exterior of the sphere
𝑄,𝑅 but within the sphere 𝑂,𝑑/2, for the three
spherical variables we have the following lower and
upper limits: 0𝜑2𝜋, 𝑅𝑟𝑟
𝑑cos𝜃
, and
0𝜃𝜃
𝜃
,, with cos𝜃, 𝑅/𝑑 (see Fig. D-
1).
A. Inside the dashed line sphere 𝑶,𝒅/𝟐, not
including the inner part of 𝑸𝟐,𝑹
We have 𝑅𝑟𝑑𝑐𝑜𝑠𝜃, and 0𝜃𝜃
,, with
cos𝜃, 𝑅/𝑟.
Therefore, the field self-energy 𝑈 within this sphere
becomes:
2,max
2,max
2,max
2,max
2,max
2cos
22
2
24
00
cos
2
2
2
0
cos
2
2
0
2
2
0
2
2
0
2
2
sin
8
1
2sin
8
1
sin
4
11
sin
4cos
1sin
tan
4
4
d
e
Inside rR
d
e
rR
d
e
rR
e
e
e
kq r
Ud dr
r
kq dr
r
kq
r
kq
dR
kq d
dR
kq
2,max
0
2
2
1sin
tan
111
ln
4
e
d
dR
kq R
dd dR
(D.1)
B. Outside the dashed line sphere 𝑶,𝒅/𝟐 , not
including the outer part of 𝑸𝟐,𝑹
Outside the sphere 𝑂,𝑑/2 we distinguish two areas,
one at the left and one at the right, depending on the
angle 𝜃𝜃
, as follows.
B1. Outer space on the right of charge 𝑞: 0𝜃
𝜃,.
In this case we have:
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
236
Volume 19, 2022
2,max
2,max
2,max
2,max
2,max
2
22
2
24
00 cos
2
2
2
0cos
2
2
cos
0
2
2
0
2
2
0
2
2
sin
8
1
2sin
8
1
sin
4
1
sin
4cos
1tan
4
1ln
4
e
Outside Right d
e
d
e
d
e
e
e
kq r
Uddr
r
kq dr
r
kq
r
kq d
d
kq d
d
kq R
dd
(D.2)
B2. Outer space on the left of charge 𝑞: 𝜃, 𝜃𝜋.
In this case we have:
2,max
2,max
2,max
2
22
2
24
0
2
2
2
0
2
2
2
2
2
2
sin
8
1
2sin
8
1
cos
4
1
1
4
11
4
e
Outside Left rR
e
rR
e
rR
e
e
kq r
Uddr
r
kq dr
r
kq
r
kq R
dR
kq
Rd
(D.3)
Fig. D-1: Overlapping between the spheres 𝑂,𝑑/2 and
𝑄,𝑅.
Appendix E
Energy breakdown outside a sphere 𝐎,𝐑𝐩 with
𝐑𝐩𝒅/𝟐
The case of a sphere 𝑂,𝑅 with 𝑅𝑑/2 (see, Fig. E-
1) is of great interest when we wish to determine the
particular value of 𝑅 so as the exterior of this sphere
(𝑟𝑅
) contains a prescribed amount of total field
energy.
Fig. E-1: Field energy with respect to the surface of the
sphere 𝑂,𝑅.
First we start with the electric charge 𝑞 located at the
point 𝑄, so it is convenient to select the main spherical
variables 𝑟𝑟
,𝜃𝜃 with respect to axis origin at
𝑄, as shown in Fig. E-1. For each point 𝑃𝑟,𝜃 in the
exterior of the sphere 𝑂,𝑅, we bring the line 𝑄𝑃
which intersects the sphere 𝑂,𝑅 at a point 𝑊, as
shown in Fig. E-1. Then, the self-energy 𝑈 in the
exterior of the sphere 𝑂,𝑅 becomes:
2
2
2
2
2
2
2
2
2
2222
2
2
00
2
()
2
2
0
2
2
2
0
2
2
0
8
1(sin)
8
() 1
2sin
8
sin
4
1
4
p
W
W
W
W
rR e
V
e
err
rr e
err
e
rr
e
r
E
Ud
k
kq
drdrd
kr
kq dr d
kr
kq rdr d
kq
r
2
2
0
sin
sin
4
e
w
d
kq d
r
(E.1)
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
237
Volume 19, 2022
If we bring the normal projection 𝑁 of the midpoint 𝑂
onto the line 𝑄𝑊𝑃 (see, Fig. E-1), we have:
22
22
22
22
22
cos ( ) ( )
2
cos ( sin )
22
cos cos
244
W
p
p
rQWQNNW
dOW ON
dd
R
ddd
R
(E.2)
Since 𝑅𝑑/2, the integrand in Eq. (E.1) will be always
positive. The latter is symbolically ensured by setting the
following auxiliary variable:
2
22
2
pp
d
RR
(E.3)
Substituting (E.3) into (E.2) and then into (E.1), the latter
becomes:
2
2
22
022
sin
4cos cos
24
p
e
rR
p
kq
Ud
dd
R
(E.4)
After analytical integration and imposition of the limits,
Eq. (E.4) progressively becomes:
12
22 2
22
22 21/2
2
222
0
12
22
2
221/2
2
2
12
22
2
221/2
2
2
0
1
22
2
2
cos cos 4
cos 1 ln ( cos 4 ) cos
44 4
41ln ( 4 )
44
41ln ( 4 )
44
4
2
4
p
p
ep
rR pp
p
ep
p
p
ep
p
p
e
dR
kq d
UdRd
dRR
dR
kq dR d
dR
dR
kq dR d
dR
dR
kq
2
221/2 221/2
2
12
22 221/2
2
2
2221/2
11
ln ( 4 ) ln ( 4 )
4
4(4)
1ln
42 (4)
pp
p
pp
e
pp
dR d dR d
ddR
dR dR d
kq
dRdRd
(E.5a)
Then, substituting 𝑅
by its definition, i.e. Eq. (E.3), Eq.
(E.5a) eventually becomes:
2
2
222
42
1ln
42
4
p
pp
e
rR pp
RRd
kq
UdRd
Rd
(Ε.5b)
A similar expression is immediately derived for
𝑈, simply by replacing in (Ε.5b), 𝑞 with 𝑞.
Finally, using now different than previously variables
𝑟𝑟
,𝜃𝜃 to comply with Appendix A (see, for
example the integrand of Eq. (A.5)), the “interaction field
energy” in the infinite space outside the sphere 𝑂,𝑅,
i.e., with 𝑟𝑅
, is given by:
2
12
0
2
2
121 1
11 11
22
12 2
0
2
2
12
1
22
12 2
0
1d
4
cos sin d
1cos
2sind
4
p
T
T
rR e
ee
rr
ee
err
Uk
qqrd
kk rdr
rr r
qqrd
kk rdr
krrr
1
2
12 11
3
2
0
22
(.9)
12 12 1
11 1
1
00
cos sin d
2
sin1sin d d ,
22
W
W
e
rr
Aee
W
r
I
kqq rd dr
r
kqq kqq
rr
(Ε.6)
with
1
()
W
rQW, (E.7)
as shown in Fig. E-1.
Now we are seeking to express the edge 𝑟𝑄
𝑊,
which appears in the denominator of the integrand of Eq.
(Ε.6), as a function of the angle 𝜃. Actually, applying the
cosine-rule to the triangle 𝑂𝑄𝑊
for the edge 𝑂𝑊
𝑅, after the solution of a quadratic polynomial in
𝑄𝑊𝑟
,, we obtain:
1
22
2
1,min 1 1 1
()cos sin
22
p
dd
rQW R
(E.8)
Then applying for a second time the cosine-rule to the
outer triangle 𝑄𝑄𝑊
for the desired edge 𝑄𝑊, after
the elimination of the common edge 𝑄𝑊, and using Eq.
(E.8) again (it ensures a positive inner radicand in Eq.
(E.8) and Eq. (Ε.9)) below, we eventually derive the
lengthy expression:
1
1
12
2
22
22 2 2
11 1
()
cos cos cos
22
W
pp
rQW
dd
Rd d R
(Ε.9)
Unfortunately, the substitution of Eq. (Ε.9) into the
denominator of the integrand in Eq. (Ε.6) does not lead to
a well known closed-formed analytical expression (this
claim was checked using the software MATLAB® and
MATHEMATICA® as well). Therefore, it is proposed to
calculate it numerically, for example applying Simpson’s
trapezoidal rule. A short program in MATLAB® which
calculates the integral 𝐼 in (Ε.6), with
1
2
1
1
0
sin d
W
Ir
, is given below.
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
238
Volume 19, 2022
Obviously, when working in the above numerical
way, one may wish to skip the introduction of the
auxiliary variable 𝑅
in (Ε.9) which was used to ensure a
positive base, thus instead it is proposed to directly apply
Eq. (27) of the main text.
%% CALCULATE THE INTEGRAL USEFUL FOR INTERACTION ENERGY
U12
%------------------------------------------------------
clear all
clc
%------------------------------------------------------
L=1; %[m] Dipole's half-length
d=2*L; %[m] dipole's length (separation distance)
Rcritical=1.5*d;%Rp[m]=critical distance (manually)
Rp=Rcritical;
%------------------------------------------------------
Rpbar=(Rp^2-(d/2)^2)^(1/2); %auxiliary variable
%------------------------------------------------------
%---We apply Simpson's integration in the interval
[0,pi]:
%------------------------------------------------------
nseg=10*180; %number of segments
h=pi/nseg; %step
Ith=0; %initialize integral
for i=1:nseg+1
th1=(i-1)*h;
nominator = sin(th1);
denominator=(Rpbar^2+d^2-d^2/2* ...
cos(th1)^2-d*cos(th1)* ...
(Rpbar^2+d^2/4*cos(th1)^2)^(1/2))^(1/2);
integrand = nominator / denominator;
if( i==1 || i==(nseg+1) )
Ith = Ith + integrand *(h/2);
else
Ith = Ith + integrand *h;
end
end
fprintf('Ith=%20.15f\n',Ith);
References:
[1] Hilborn R.C., What should be the role of field
energy in introductory physics courses?, American
Journal of Physics 82 (1), 66-71 (2014)
[2] Feynman R.P., Leighton R.B., and Sands M., The
Feynman Lectures, Addison-Wesley (1965).
[3] Maxwell J.C., A Treatise on Electricity and
Magnetism, Vol. 2, Clarendon Press (1873)
[4] Tort A.C., On the electrostatic energy of two point
charges, Revista Brasileira de Ensino de Física 36
(3), 3301 (2014)
[5] Corbò G, Renormalization in classical field theory,
Eur. J. Phys. 31 L5–8 (2010).
[6] Tort A.C., Another example of classical
renormalization in Electrostatics, Eur. J. Phys. 31,
L49-L50 (2010).
[7] Mundarain D., On the infinities in the electric
potential of unbounded charge distributions, Eur. J.
Phys. 41 (2020) 035205 (8pp)
[8] McDonald K.T., Private Communication (July
2022)
[9] McDonald K.T., Field Energy of Two Spheres in
Contact (2022),
http://kirkmcd.princeton.edu/examples/2spheres.pdf
[10] Goedecke G.H., Wood R.C., and Nachman P.,
Magnetic dipole orientation energetics, American
Journal of Physics 67, 45 (1999);
[11] Griffiths D.J., and Hnizdo V., The torque on a
dipole in uniform motion, American Journal of
Physics 82, 251 (2014)
[12] Kort-Kamp W.J.M., and Farina C., On the exact
electric and magnetic fields of an electric dipole,
American Journal of Physics 79, 111 (2011)
[13] McDonald K.T., Radiated energy and momentum
for time-dependent dipoles, American Journal of
Physics 90, 247 (2022)
[14] Boyer T.H., The force on a magnetic dipole,
American Journal of Physics 56, 688 (1988)
[15] Greene J.B., and Karioris F.G., Force on a Magnetic
Dipole, American Journal of Physics 39, 172-175
(1971)
[16] Lattes C.M., Schönberg M., and Schützer W.,
Classical Theory of Charged Point-Particles with
Dipole Moments, Brazilian Journal of Physics 52,
49 (2022)
[17] Heilmann A., Damasceno Ferreira L.D., and Dartora
C.A., The Effects of an Induced Electric Dipole
Moment due to Earth’s Electric Field on the
Artificial Satellites Orbit, Brazilian Journal of
Physics 42, 55–58 (2012)
[18] Griffiths D.J., Introduction to Electrodynamics, 4
th
ed., Pearson (2017)
[19] Pollack G.L., and Stump D.R., Electromagnetism,
Addison Wesley (2002)
[20] Halliday D., and Resnick R., Physics (Parts I and
II), John Wiley & Sons (1966)
[21] McDonald K.T., Ph501: Electrodynamics Problems
Set 2 (1998),
http://kirkmcd.princeton.edu/examples/ph501set2.pdf
[22] Wang J., Wu C., Dai Y., et al., Nat Commun 8, 88
(2017)
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.24
Christopher G. Provatidis
E-ISSN: 2224-3410
239
Volume 19, 2022
