This document brings a solution for the "4/3 electromagnetic problem" that shows a discrepancy between the overall momentum for the EM field created by a charged sphere shell and its energy. The solution comes by including a term caused by the charge-field interaction over the sphere (j·E) multiplied by the distance to the center of mass (the center of the charged sphere). The idea of center of mass displacement on interactions can be applied to other electromagnetic problems, as long as there are particles or systems with some extension, and to other fields of physics.

1. The Center of Mass Displacement Caused by Field-Charge
Interac on Can Solve the Electromagne c 4/3 Problem
Sergio Prats López, August 2023
Abstract
The 4/3 problem happens when calcula ng the electromagne c (EM) momentum and energy
(ME) for a spherical shell uniformly charged that can be considered a "par cle". It turns out
that for low veloci es the rela onship between the momentum p and the energy U is
𝒗
instead of being just 𝑈 as one would expect.
I assert in this paper that, for a system interac ng with its environment in a way where it is
ge ng energy gains in some regions and energy losses in other regions, there may be a net
contribu on to the system's momentum propor onal to the amount of energy gained at each
point mul plied by the space vector from that point to the system's center of mass. This can
add an extra term to the par cle's momentum even for a system whose overall energy is kept
constant but where the gains and losses of energy happen at different places.
For a charged shell moving with velocity v, the region where 𝑱 · 𝑬 > 0, "the front", is the one
where the EM field is losing energy while the opposite region "the back" is the one where the
field gains energy, I will show how this effect adds a term of momentum whose value is −
1
3
𝒗
𝑐2
.

2. Rela on between iner al momentum and center of mass evolu on
For a classical, non-rela vis c point par cle with mass m and at posi on x, the rela onship
between the iner al momentum and the mass is straigh orward:
𝒑 = 𝑚𝒙̇ [1]
This rela onship is barely modified by the special rela vity, it only adds the "rest mass", which
is constant while the actual mass depends on the speed as 𝑚 = = 𝛾𝑚 , 𝛾 =
𝒑 = 𝛾𝑚 ∗ 𝒙̇ [2]
This defini on can be extended to a non-point par cle whose density at every point is
expressed as 𝐽 = [𝑢, 𝒑], by using the center of mass:
𝑿𝟎 = ∫ 𝑢 ∗ 𝒙𝑑𝒙 [3a]
𝑈 = ∫ 𝑢𝑑𝒙 = 𝑚𝑐 [3b]
The par cle momentum can be defined as the way the center of mass changes per unit of
me, mul plied by the iner al mass:
𝑷 = 𝑿
̇ 𝟎 = 𝑚𝑿
̇ 𝟎 [4]
According to [4], a moving charged sphere whose EM field has energy U, should have a
momentum of 𝒗𝑈, not 𝒗𝑈, which is in fact the issue.
Classically, the only source of movement considered is the flow of mass-energy, which can be
obtained by integra ng the local momentum p, I will call 𝑿
̇ 𝟎 to this quan ty, so that 𝑿
̇ 𝟎 will
include the term introduced in this document.
𝑷′ = ∫ 𝒑𝑑𝒙 [5a]
𝑿
̇ 𝟎 = 𝑷′ [5]
Effect of interac ons in the par cle's center of mass
For any par cle that can be represented as a field 𝐽 , the con nuity equa ons will grant that,
in absence of external sources, energy and momentum will be conserved locally, therefore
expression [1] should not only hold globally, that is, like expression [4], but also locally at every
point of space: the energy at any point flows with velocity 𝒗 =
𝒑
, this is for example the case
for the electromagne c field in open space.
On the other hand, when a system is interac ng with an external source, it is no longer granted
that its momentum-energy (ME) will be conserved, either locally or globally, since the
conserva on rules apply to the sum of the interac ng system, not to each of them individually.
Now I will consider two different scenarios: the first one in which the global ME is not
conserved:

3. [𝑈, 𝑷] = 𝑈̇ , 𝑷
̇ ≠ [0, 𝟎] [6]
The second scenario is the one in which ME is conserved globally but not locally (no ce that if
ME was conserved locally everywhere, there would be no interac on at all).
𝑈̇ , 𝑷
̇ = [0, 𝟎], [𝑢̇, 𝒑] ≠ [0, 𝟎] [7]
In the second scenario the interac on is not affec ng globally the basic physical proper es of
energy and momentum, however, it could be affec ng other proper es such as the angular
momentum or the center of mass, and by affec ng the center of mass, according to [4] the
interac on would be affec ng the global momentum for the par cle. In the second scenario,
the effect of interac on in the center of mass can be defined as:
𝑿
̇ 𝟎 = 𝑿
̇ 𝟎 + ∫(𝒙 − 𝑿𝟎)𝑃 𝑑𝒙 [8]
In the previous expression 𝑿
̇ 𝟎 is the quan ty defined at [5], that is, the classical way to
calculate the iner al momentum.
I will call "Interac on Term" to the second term from [8], it contains the "displacement" caused
by the source or sink of energy (the interac on) in the center of mass. 𝑃 is the amount of
power, transferred from the par cle to the interac ng system. For example, if the EM field
interacts with a charge, the amount of power the EM field would be transferring to the charge
would be 𝑱 · 𝑬, therefore for the charge 𝑃 = 𝑱 · 𝑬, while for the field it would be the opposite,
𝑃 = −𝑱 · 𝑬.
Expression [8] shows us that the existence of sources and sinks (places where our system
interacts with the environment) can affect the way the center of mass of a par cle evolves with
me, if we want [4] s ll to be valid when an interac on is happening then we need to add the
interac on term to the defini on of global momentum:
𝑷 = ∫ 𝒑𝑑𝒙 + ∫(𝒙 − 𝑿𝟎)𝑃 𝑑𝒙 [9]
This is the expression for the iner al momentum of a par cle that I propose. The second term
should be included any me that the par cle is interac ng with its environment, not only in
the interac ons of "second type" defined by [7] but also in the "first type" ones, defined by [6].
It is important to remark that because of the interac on term, the total momentum is no
longer the integra on of local momentums.
One could be tempted to modify the local momentum by doing something like this: 𝒑 = 𝒑 +
(𝒙 − 𝑿𝟎) . However, the idea of adding the interac on term to the local momentum must
be discarded because we want the local momentum to tell us how much is moving the energy
at some point, we expect that, for a point with current [u, p], the energy at this point is moving
with velocity 𝒗 =
𝒑
, with no dependency on where the center of mass is. Besides, the
con nuity equa on 𝜕 𝑢 + 𝑐 ∇ · 𝒑 = 𝑃 is altered in an undesirable way by an unwanted (𝒙 −
𝑿𝟎) · ∇𝑃 + 3𝑃 term.
From the previous considera ons I assert that the interac on term is needed to calculate the
par cle's center of mass me evolu on and therefore, it should be included in the total
momentum, but this term should not be added to the local momentum, which means that the

4. interac on makes the physical proper es for the par cle no to be only the sum/integra on of
its local proper es.
Calcula ng the Effect of the Interac on Term in the 4/3 Problem
Let we have a par cle whose charge is uniformly spread over a spherical surface of radius R,
the overall charge is Q and it is moving with velocity 𝑣 𝑧̂, being v much smaller than the speed
of light (𝑣 ≪ 𝑐) and suffering no accelera on. The EM field is zero inside the sphere and
outside it is the same that would be created by a charge of value q located in the center.
For rela vis c low veloci es we can approximate the EM field energy as the one for a sta c
charge:
𝑈 = ∫ 4𝜋𝑟 𝑑𝑟 = ∗ ≡ [10]
The momentum in the Z direc on can be obtained to first order of v by applying the cross product
to the radial component of E against B.
𝑝 = (𝜀 (𝑬 × 𝑩) · 𝑧̂)𝑧̂ ≅ E (𝑟̂ × (𝑣𝑧̂ × 𝑟̂)) = E sin (𝜃) [11]
When integra ng the momentum over the sphere we get:
𝑃 = ∫ ∫ E sin (𝜃) ∗ 2𝜋𝑟 ∗ sin(𝜃) 𝑑𝜃𝑑𝑟 = = 𝑈 [12]
For the previous result I have used ∫ sin (𝜃)𝑑𝜃 = − cos(𝜃) +
( )
To calculate the "interac on term", since we are studying the EM field created by an isolated
par cle, we only need to evaluate the interac on between the charged surface and the field. If
we center the par cle and use spherical coordinates, the density of charge can be described as:
𝑞 = 𝛿(𝑟 − 𝑅) [13]
Since the charge is moving with velocity v, the current can be defined as:
𝒋 = 𝒗𝑞 = 𝑣𝑞𝑧̂ [14]
Next, we need to calculate the interac on between field and charge, that interac on for the
field is −𝑬 · 𝒋, thus, we need to calculate the electric field on the spherical surface. This is in
fact, troublesome, since the electric field for a spherical surface is defined for values greater or
smaller than the shell radius:
𝑬 = 0 𝑓𝑜𝑟 𝑟 < 𝑅 [15a]
𝑬 = 𝑟̂ 𝑓𝑜𝑟 𝑟 > 𝑅 [15b]
To solve this, we can turn this surface into a very thin sphere, whose thickness is 𝑑𝑙 ≪ 𝑅. We
can now assert that each of the infinitesimal shells creates an electric field that affects only to
the shells outside it, this way, if the charge is between radius R-dl and R, the electric field in the
region where the charge exists will be:

5. 𝑬 = 𝑟̂ = 1 − 𝑟̂, 𝑅 − 𝑑𝑙 ≤ 𝑟 < 𝑅 [16]
𝑄 = 𝑄 1 − is the charge from the shells that are closer to the center than r. This is the
field that a charged shell surface would see at distance r, if we integrate to average it, the result
we get is that the effec ve EM field in the surface is:
𝑬𝒔 = 𝑟̂ [17]
We can define the interac on term on each point of the sphere as:
𝑃 = −𝑬𝒔 · 𝒋 = − ( )
𝑐𝑜𝑠(𝜃) [18]
Now, if we integrate on the surface replacing 𝒙 with 𝑅𝑐𝑜𝑠(𝜃) (since we are only interested in
the Z component), we get the interac on term contribu on to the momentum:
𝑷𝒊𝒕 =
1
𝑐
2𝜋𝑅 ∗ 𝑅𝑐𝑜𝑠(𝜃) ∗ −
1
2
𝑄2
(4𝜋)2𝜀0𝑅4 𝑐𝑜𝑠(𝜃) ∗ sin(𝜃) 𝑑𝜃 𝑧
𝑷𝒊𝒕 = −
𝒗
𝜀0
= −
𝒗
𝜀0
= −
𝒗
𝑈 [19]
For the previous result we use ∫ 𝑐𝑜𝑠 (𝜃) sin(𝜃) 𝑑𝜃 = − cos (𝜃).
Now, when we add the interac on term to the momentum, we get the quan ty in accordance
with the EM's energy:
𝑷 = 𝑷 + 𝑷𝒊𝒕 = 𝑈 − 𝑈 = 𝑈 = 𝑣 ∗ 𝑚 [20]

6. Discussion about the interac on between a par cle's charge and field
In the interac on between the charge and the field from the same par cle several ques ons
may arise. The first one is where the 𝑬𝒔 · 𝒋 energy term is transferred. The integral of this term
around the sphere is zero so there is no net energy transfer between the charge and the field
as one would expect for a charge with no accelera on, however it is also relevant what is
happening locally with the energy and momentum being transferred.
For this ques on I am not going to write an answer in this paper but only to discuss some
possibili es, one might be to consider that the par cle has a "material" mass associated to the
charged surface, that is, where there is charge (and current) there is also mass (and
momentum) in a way that we can define a constant 𝑘 = 𝑚 /𝑞 to define the rela onship
between the quan ty of mass and charge at any point, and this constant will also hold for the
rela onship between the momentum and current.
With this approach, we can assume that the momentum-energy for the field is transformed
into mechanical ME locally, then some forces such as the Poincaré stresses would balance
locally the forces caused by the EM field on the charge, making possible for the charged sphere
to remain in the same shape.
While the previous approach seems consistent on the EM field side, it is not a well-defined
solu on from the "material" or charge side since some ques ons remain: How are the
Pointcaré stresses created? What is causing them? Is there some poten al holding these
stresses?
I would like to suggest another possibility to avoid this problem, it would be to add an internal
EM field that is transparent to everybody except to one charged par cle, which creates it. This
par cle is the only one that can interact with this field which I call "correc ve field".
We can split the EM tensor 𝐹 for the overall EM field into two parts:
𝐹 = 𝐹 ( ) + 𝐹 ( ) [21]
The first part is the field caused by the single par cle's charge distribu on, the second one is the
field created by the rest of the world. The effect of 𝐹 ( ) on the par cle (the self-force), can be
cancelled by crea ng the correc ve field as follows:
The correc ve field is created by an imaginary charge distribu on which is exactly equal to the
par cle's (real) charge distribu on mul plied by the imaginary number i.
𝑞 = 𝑞 ∗ 𝑖 [22]
There will be a correc ve EM field space created by this imaginary charge and nothing more,
since the rest of the charges in the world are not contribu ng to this EM field, they are not going
to interact with it, this field is invisible for them. The field will take this value:
𝐹 ( ) = 𝑖 ∗ 𝐹 ( ) [23]
The field is purely imaginary, but its effects are real since it interacts with 𝑞 , which is also
imaginary ge ng a minus sign, since 𝑖 = −1. The force caused by 𝐹 ( ) on 𝑞 is real and it
cancels the force caused by 𝐹 ( ) on q, since the par cle is experiencing both forces, the result
is that the field created by the charge is not ac ng over it. The correc ve field also has a

7. nega ve density of energy and a momentum that cancels exactly the energy and momentum
that the internal field has.
By using this correc ve field, the Poincaré stresses are no longer needed to keep the charged
shell stable, however it goes further and removes the whole energy from the par cles field,
and causing that, let this approach be valid, the energy in the EM field would be only caused by
the interac ons between different par cles.
The correc ve field approach is not free of issues since it would cancel the Larmor radia on
term when the par cle is accelera ng, this term, connected to the Abraham-Lorentz-Dirac
force, seems to be real, therefore this approach should be enhanced and that goes beyond the
aim of this discussion.
Conclusions
When working with extended par cles, the interac on with external systems can cause a flow
of energy at different points that can cause the center of mass to move, this movement should
be computed as part of the par cle's iner al momentum, although it is not of a local nature
and should not be considered to compute how fast the mass at some point is moving.
By evalua ng the interac on between a par cle's charge and the EM field created by it, we
find that the displacement of the center of mass adds a contribu on to the par cle's field
momentum that makes changes it from 𝑈 to 𝑈 solving the issue.
One last insight can be obtained that can be brought to another areas of physics:
Some mes global magnitudes for a par cle are not equal to the integra on of its local
counterparts but they may also include other terms of a global nature.
References
- The Feynman Lectures on Physics, Volume II – Richard P Feynman:
h ps://www.feynmanlectures.caltech.edu/II_28.html
- Classical theory of radia ng electrons - P. A. M Dirac, 1938:
h ps://royalsocietypublishing.org/doi/10.1098/rspa.1938.0124
- Inclusion of a Field Shi Term for the 4/3 Classical Electrodynamics Problem:
h ps://www.slideshare.net/SergioPL81/adding-a-shi -term-to-solve-the-43-problem-
in-classical-electrodinamics
- An Apology to Dirac's Reac on Force Theory -
h ps://www.slideshare.net/SergioPL81/an-apologytodiracsreac onforcetheory
- Electromagne c mass: h ps://en.wikipedia.org/wiki/Electromagne c_mass