This article shows how to get the flux of momentum in the electromagnetic field from the Maxwell stress tensor in the scope of classical electromagnetism.

1. Momentum flux in the electromagnetic field
Sergio A. Prats López, September 2015
SergioPL81@gmail.com
Abstract
The purpose of this paper is to give an expression for the flux of momentum in the
electromagnetic field in the scope of the classical electromagnetic theory.
The momentum flux is obtained in a straightforward way from the Maxwell stress tensor .
However, I state that the Maxwell stress tensor is not the flux itself, to get the flux you have to
derive the stresses on each of the three coordinate planes through its perpendicular direction.
After a brief explanation about the Poynting theorem and the Maxwell stress tensor, I expose
the arguments to defend my statement, and the expression for the momentum flux.
Introduction
The Poynting vector = × is the quantity that satisfies the continuity equation of the
electromagnetic field energy. It is used in the Poynting theorem to achieve the local
conservation of energy in the EM field.
Let it be the density of electromagnetic energy:
= + .
The continuity equation for the energy in the electromagnetic field is:
= −∇ · − ·
Where S is the energy flux and J·E is the energy that the field exchanges with the charges due
to the Lorentz Force.
Since S is the flux of energy, the space density of momentum is / since momentum is the
flux of mass and = .
Since the density of momentum must be also a local conserved quantity, it has to satisfy its
own set of continuity equations, which consists in three equations, one for each dimension. In
this equations the Maxwell stress tensor plays the same role than the Poynting vector in the

2. continuity equation for energy, however the Maxwell stress tensor does not contains flow of
momentum but the stress that every point creates in every direction.
The continuity equation for the momentum is:
+ × = ∇ · −
1
Where is the Maxwell stress tensor, S the Poynting vector and the term in the first part of
the equation the Lorentz force.
In this equation “∇ · ” is the net pressure or normal stress done on a point, taking the surface
that it is perpendicular to that point. Therefore, the evolution of the momentum in a point is
the net pressure transferred from the contiguous field minus the Lorentz force.
It’s important to remark that the continuity equations for energy and momentum work only in
classical EM with “independent” spatial densities of charge (and current). If we use particles
with a discrete charge and a fixed shape, or punctual particles with no infinitesimal charge,
some issues about conservation of energy arise. These issues can be treated with the
Abraham-Lorentz force but it is out of the scope of this document to deal with these issues.
The Maxwell stress tensor
The Maxwell stress tensor, , is a 3x3 tensor part of the electromagnetic energy-stress tensor
The Maxwell stress tensor is defined as:
σ =
1
4π
E ⊗ E +
1
H ⊗ H − ∗
1
2
E +
1
H
Where is the identity matrix and ⊗ is the dyadic product.
This tensor, whose units are force per unit of surface, satisfies the continuity equation given in
the previous section:
+ × = ∇ · −
1

3. The stress done in some direction ̂ it’s simply = ̂ σ, it’s remarkable that the stress done in
a direction and its opposite will be always the same except because it will point to the opposite
direction.
If we use a coordinate system in which the vector E + H is aligned with the X axis we can see
that in the X the shear stresses disappear and the normal stresses take the same value except
for the sign:
σ =
1
2
E +
1
H 0 0
0 −
1
2
E +
1
H 0
0 0 −
1
2
E +
1
H
This can be interpreted in the following way: the electromagnetic field creates a stress that is
inwards in the E + H axis and outwards in the plane parallel to that axis. Notice that a 180º
rotation over any of these axes will produce the same stress tensor.
Momentum flux in the electromagnetic field
First of all, the momentum flux is a field that shows how much spatial density of momentum is
travelling in each direction. Since the momentum is a vector quantity and it can travel in any
direction the momentum flux is a tensor field expressed with a 3x3 matrix.
I define the momentum flux as the net momentum transferred from a point to its neighbors in
each of the three spatial axes. This transferred is caused by the difference of stresses which is
expected to be infinitesimal.
The reasons because I have rejected the Maxwell stress tensor as the flux of momentum are
two: first, the Maxwell stress tensor depends on orientation, which means that it takes the
opposite value when looking at the opposite direction and that does not fit with a flux
definition.
The second reason is that the stress is force per unit of surface, so it transfers a surface density
of force whereas the EM’s Poyting vector contains spatial density of momentum so it does not
make sense to have a flux of “surface density of momentum” over all the space… It would lead
to a renormalization in classical theory!
Let it be the vector quantity _ = , , the stress done by the field in
direction X and _ and _ are the stresses done on directions Y and Z respectively. The
momentum flux in a point can be expressed as show in the next picture:

4. Figure 1. Diagram of how momentum flows due to differences of stress. The
brown dot represents the point where we are evaluating the momentum flux
and the green dots represent its neighbor points on the X, Y and Z axes.
The difference in stress, ∂ _, between the point P and its neighbor + means that there
is an infinitesimal net stress among these two points, so it will be an infinitesimal transmission
of “momentum per unit of surface” per unit of time. Since the field has spatial - not surface -
density of momentum, this infinitesimal on surface density transforms into a no infinitesimal
measured over spatial density of momentum, letting the momentum to evolve with time. If
this was not so, the divergence term ∇ · at the continuity equation would be zero.
Telling it with other words, ∇ · reflects how the momentum changes in a point while the
momentum flux tells also where this moment comes from.
To refer momentum flux, I will use the symbol . The subscript ‘S’ comes from the letter used
to refer the Poynting Vector and the letter ‘J’ evokes a current although the momentum flux
technically is not a current because the transported “substance” – the momentum – cannot be
considered a charge.
Therefore, the momentum flux can be written as:
J = ∇~ =
∂ 0 0
0 ∂ 0
0 0 ∂
=
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
Where I have defined the operator ∇~ as
∇~ =
∂ 0 0
0 ∂ 0
0 0 ∂
In this tensor, each column contains the flux of each a component of the momentum, the first
column contains the “current” of momentum for the X coordinate - ∂ _ - that means the
momentum that flows in X direction, the second column would apply for Y coordinate and the
third one for Z coordinate.

5. A corollary is that the flux of momentum does not depend directly on the flux of energy. For
example in a locally constant field with both Electric and magnetic fields there is momentum
and therefore flux of energy but there is no flux of momentum.
A digression about the ~ operator
To finish, I would like to describe the key concept in the momentum flux, the operator "∇~"
which I have defined as:
Since I have not been able to find in literature a definition for the previously defined operator,
I have defined the symbol "∇~" to identify it. For vectors, this operator can be defined from
the Del / Nabla operator in this following way:
Let it be = + + ̂ any vector, I define ∇~ as:
∇~ = + + ̂ ̂ · = + + ̂
The adaptation to matrixes can be easily done if we regard that matrixes have not one but two
coordinates on each element and we take into account the rules of matrix multiplication: the
second coordinate of the first matrix contracts with the first coordinate of the second matrix
standing only the first coordinate of the first matrix and the second coordinate of the second
matrix.
The Maxwell stress tensor with coordinates would be seen like this:
=
̂
̂
̂ ̂ ̂ ̂
For example, to get the Y component of the momentum flux in X direction (row 1, column 2),
we will have:
· = · =
As the remaining components are it’s the component that goes on row 1 (x) and column 2
(y). Obviously neither nor ̂ ̂ can put any contribution on this parcel.

6. References
Electromagnetism courses:
Classical electrodynamics Part II
Robert G Brown, Duke University
http://www.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/
Classical electromagnetism
Richard Fitzpatrick, University of Texas
http://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/
Remarkable articles from Wikipedia:
https://en.wikipedia.org/wiki/Poynting%27s_theorem
https://en.wikipedia.org/wiki/Poynting_vector
https://en.wikipedia.org/wiki/Maxwell_stress_tensor
https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor
https://en.wikipedia.org/wiki/Cauchy_stress_tensor
Special thanks to the members of Physics Forums that had helped me resolve my doubts and
specially to DaleSpam.
PS: If you have any suggestion or comment related to this article please send me it to my mail
put under the title, I will be happy to receive it.