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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 \mathbf and \mathbf. 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 \mathbf and \mathbf, and introducing the Maxwell stress tensor simplifies the result. in the above relation for conservation of momentum, \boldsymbol \cdot \boldsymbol is the momentum flux density and plays a role similar to \mathbf in Poynting's theorem. The above derivation assumes complete knowledge of both \rho and \mathbf (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: :\sigma_ = \epsilon_0 E_i E_j + \fracB_i B_j - \frac\left(\epsilon_0 E^2 + \fracB^2\right)\delta_ , where \epsilon_0 is the electric constant and \mu_0 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 ...
, \mathbf 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 ...
, \mathbf 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 \delta_ is Kronecker's delta. In Gaussian cgs unit, it is given by: :\sigma_ = \frac\left(E_i E_j + H_i H_j - \frac\left(E^2 + H^2\right)\delta_\right), where \mathbf is the magnetizing field. An alternative way of expressing this tensor is: : \overset = \frac \left \mathbf \otimes \mathbf + \mathbf \otimes \mathbf - \frac\mathbb \right where \otimes is the dyadic product, and the last tensor is the unit dyad: :\mathbb \equiv \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end = \left(\mathbf \otimes \mathbf + \mathbf \otimes \mathbf + \mathbf \otimes \mathbf\right) The element ij 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 ith axis crossing a surface normal to the jth 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 ij element of the tensor can also be interpreted as the force parallel to the ith axis suffered by a surface normal to the jth 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: :\sigma_ = \frac B_i B_j - \frac B^2 \delta_ \,. For cylindrical objects, such as the rotor of a motor, this is further simplified to: :\sigma_ = \frac B_r B_t - \frac B^2 \delta_ \,. where r is the shear in the radial (outward from the cylinder) direction, and t is the shear in the tangential (around the cylinder) direction. It is the tangential force which spins the motor. B_r is the flux density in the radial direction, and B_t 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. \mathbf = \mathbf, and we obtain the ''electrostatic Maxwell stress tensor''. It is given in component form by : \sigma_ = \varepsilon_0 E_i E_j - \frac\varepsilon_0 E^2\delta_ and in symbolic form by : \boldsymbol = \varepsilon_0\mathbf \otimes \mathbf - \frac\varepsilon_0(\mathbf \cdot \mathbf)\mathbf where \mathbf is the appropriate identity tensor \big(usually 3\times3\big).


Eigenvalue

The eigenvalues of the Maxwell stress tensor are given by: :\ = \left\ These eigenvalues are obtained by iteratively applying the Matrix Determinant Lemma, in conjunction with the Sherman–Morrison formula. Noting that the characteristic equation matrix, \overleftrightarrow - \lambda\mathbf, can be written as : \overleftrightarrow - \lambda\mathbf = -\left(\lambda + V\right)\mathbf + \epsilon_0\mathbf\mathbf^\textsf + \frac\mathbf\mathbf^\textsf where : V = \frac\left(\epsilon_0 E^2 + \fracB^2\right) we set : \mathbf = -\left(\lambda + V\right)\mathbf + \epsilon_0\mathbf\mathbf^\textsf Applying the Matrix Determinant Lemma once, this gives us : \det = \left(1 + \frac\mathbf^\textsf\mathbf^\mathbf\right)\det Applying it again yields, : \det = \left(1 + \frac\mathbf^\textsf\mathbf^\mathbf\right) \left(1 - \frac\right) \left(-\lambda - V\right)^3 From the last multiplicand on the RHS, we immediately see that \lambda = -V is one of the eigenvalues. To find the inverse of \mathbf, we use the Sherman-Morrison formula: : \mathbf^ = -\left(\lambda + V\right)^ - \frac Factoring out a \left(-\lambda - V \right) term in the determinant, we are left with finding the zeros of the rational function: : \left(-\left(\lambda + V\right) - \frac\right) \left(-\left(\lambda + V\right) + \epsilon_0\mathbf^\textsf \mathbf\right) Thus, once we solve : -\left(\lambda + V\right) \left(-\left(\lambda + V\right) + \epsilon_0 E^2\right) - \frac\left(\mathbf \cdot \mathbf\right)^2 = 0 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