The Maxwell stress tensor (named after
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
) is a symmetric second-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
in three dimensions that is used in
classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of physics focused on the study of interactions between electric charges and electrical current, currents using an extension of the classical Newtonian model. It is, therefore, a ...
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. 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 nine components of the Maxwell stress tensor appear, negated, as components of the
electromagnetic stress–energy tensor, 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 stress ...
. The latter describes the density and flux of energy and momentum in
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
.
Motivation
As outlined below, the electromagnetic force is written in terms of
and
. Using
vector calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
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, Electrical network, electr ...
, 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 scientific study of matter, its Elementary particle, 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 whi ...
, the Maxwell stress tensor is the stress tensor of an
electromagnetic field
An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
. As derived above, it is given by:
:
,
where
is the
electric constant
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
and
is the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionall ...
,
is the
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
,
is the
magnetic field
A magnetic field (sometimes called B-field) is a physical 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 ...
and
is
Kronecker's delta. In the
Gaussian system, it is given by:
:
,
where
is the
magnetizing field
A magnetic field (sometimes called B-field) is a physical 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 ...
.
An alternative way of expressing this tensor is:
:
where
is the
dyadic product
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
There are numerous ways to multiply two Euclidean vectors. The dot product takes in two ...
, 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.
It has recently been shown that the Maxwell stress tensor is the real part of a more general complex electromagnetic stress tensor whose imaginary part accounts for reactive electrodynamical forces.
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 slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
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 Ricci () is an Italian surname. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), American actress
* Clara Ross Ricci (1858-1954), British ...
*
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 wat ...
*
Electromagnetic stress–energy tensor
*
Magnetic pressure
*
Magnetic tension
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