The Maxwell stress tensor (named after
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
) is a symmetric second-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
used in
classical electromagnetism to represent the interaction between electromagnetic forces and
mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the
Lorentz force law
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboa ...
. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.
In the relativistic formulation of electromagnetism, the Maxwell's tensor appears as a part of the
electromagnetic stress–energy tensor
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electrom ...
which is the electromagnetic component of the total
stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
. The latter describes the density and flux of energy and momentum in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
Motivation
As outlined below, the electromagnetic force is written in terms of
and
. Using
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
and
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, symmetry is sought for in the terms containing
and
, and introducing the Maxwell stress tensor simplifies the result.
in the above relation for conservation of momentum,
is the momentum flux density and plays a role similar to
in
Poynting's theorem.
The above derivation assumes complete knowledge of both
and
(both free and bounded charges and currents). For the case of nonlinear materials (such as magnetic iron with a BH-curve), the nonlinear Maxwell stress tensor must be used.
Equation
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the Maxwell stress tensor is the stress tensor of an
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
. As derived above in
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
s, it is given by:
:
,
where
is the
electric constant and
is the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
,
is the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
,
is the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
and
is
Kronecker's delta. In Gaussian
cgs unit, it is given by:
:
,
where
is the
magnetizing field.
An alternative way of expressing this tensor is:
:
where
is the
dyadic product, and the last tensor is the unit dyad:
:
The element
of the Maxwell stress tensor has units of momentum per unit of area per unit time and gives the flux of momentum parallel to the
th axis crossing a surface normal to the
th axis (in the negative direction) per unit of time.
These units can also be seen as units of force per unit of area (negative pressure), and the
element of the tensor can also be interpreted as the force parallel to the
th axis suffered by a surface normal to the
th axis per unit of area. Indeed, the diagonal elements give the
tension (pulling) acting on a differential area element normal to the corresponding axis. Unlike forces due to the pressure of an ideal gas, an area element in the electromagnetic field also feels a force in a direction that is not normal to the element. This shear is given by the off-diagonal elements of the stress tensor.
The Maxwell stress tensor is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
whose real part is the Poynting
momentum flux density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
.
In Magnetostatics
If the field is only magnetic (which is largely true in motors, for instance), some of the terms drop out, and the equation in SI units becomes:
:
For cylindrical objects, such as the rotor of a motor, this is further simplified to:
:
where
is the shear in the radial (outward from the cylinder) direction, and
is the shear in the tangential (around the cylinder) direction. It is the tangential force which spins the motor.
is the flux density in the radial direction, and
is the flux density in the tangential direction.
In electrostatics
In
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
the effects of magnetism are not present. In this case the magnetic field vanishes, i.e.
, and we obtain the ''electrostatic Maxwell stress tensor''. It is given in component form by
:
and in symbolic form by
:
where
is the appropriate identity tensor
usually
.
Eigenvalue
The eigenvalues of the Maxwell stress tensor are given by:
:
These eigenvalues are obtained by iteratively applying the
Matrix Determinant Lemma, in conjunction with the
Sherman–Morrison formula.
Noting that the characteristic equation matrix,
, can be written as
:
where
:
we set
:
Applying the Matrix Determinant Lemma once, this gives us
:
Applying it again yields,
:
From the last multiplicand on the RHS, we immediately see that
is one of the eigenvalues.
To find the inverse of
, we use the Sherman-Morrison formula:
:
Factoring out a
term in the determinant, we are left with finding the zeros of the rational function:
:
Thus, once we solve
:
we obtain the other two eigenvalues.
See also
*
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
*
Energy density of electric and magnetic fields
*
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt p ...
*
Electromagnetic stress–energy tensor
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electrom ...
*
Magnetic pressure
*
Magnetic tension force
References
{{Reflist
*
David J. Griffiths, "Introduction to Electrodynamics" pp. 351–352, Benjamin Cummings Inc., 2008
* John David Jackson, "Classical Electrodynamics, 3rd Ed.", John Wiley & Sons, Inc., 1999.
* Richard Becker, "Electromagnetic Fields and Interactions", Dover Publications Inc., 1964.
Tensor physical quantities
Electromagnetism
James Clerk Maxwell