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{{Electromagnetism|cTopic=Electrodynamics}}

The '''Maxwell Stress Tensor''' (named after [[James Clerk Maxwell]]) is a '''second rank [[tensor]]''' used in [[classical electromagnetism]] to represent the interaction between electric/magnetic forces and [[momentum|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 impossibly 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.

==Motivation==
[[File:Lorentz force continuum.svg|200px|thumb|Lorentz force (per unit 3-volume) '''f''' on a continuous [[charge distribution]] ([[charge density]] ρ) in motion. The 3-[[current density]] '''J''' corresponds to the motion of the charge element ''dq'' in [[volume element]] ''dV'' and varies throughout the continuum.]]

As outlined below, the electromagnetic force is written in terms of '''E''' and '''B''', using [[vector calculus]] and [[Maxwell's equations]] symmetry in the terms containing '''E''' and '''B''' are sought for, and introducing the Maxwell stress-tensor simplifies the result.

:{| class="wikitable"
|+ Maxwell's equations in SI units in ''vacuum'' <br/> (for reference)
|-
! Name
! Differential form
|-
| Gauss's law (in vacuum)
| <math>\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}</math>     
|-
| Gauss's law for magnetism
| <math>\nabla \cdot \mathbf{B} = 0</math>    
|-
| Maxwell–Faraday equation<br />(Faraday's law of induction)
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>     
|-
| Ampère's circuital law (in vacuum)<br />(with Maxwell's correction)
| <math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ </math> 
|}

<ol>
<li>Starting with the [[Lorentz force]] law
:<math>\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})</math> 
the force per unit volume for an unknown charge distribution is
:<math>
\mathbf{f} = \rho\mathbf{E} + \mathbf{J}\times\mathbf{B}
</math></li>
<li>Next, ρ and '''J''' can be replaced by the fields '''E''' and '''B''', using [[Gauss's law]] and [[Ampère's circuital law]]:
:<math>
\mathbf{f} = \epsilon_0 \left(\boldsymbol{\nabla}\cdot \mathbf{E} \right)\mathbf{E} + \frac{1}{\mu_0} \left(\boldsymbol{\nabla}\times \mathbf{B} \right) \times \mathbf{B} - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \times \mathbf{B}\,
</math></li>
<li>The time derivative can be rewritten to something that can be interpreted physically, namely the [[Poynting vector]]. Using the [[product rule]] and [[Faraday's law of induction]] gives
:<math>\frac{\partial}{\partial t} (\mathbf{E}\times\mathbf{B}) = \frac{\partial\mathbf{E}}{\partial t}\times \mathbf{B} + \mathbf{E} \times \frac{\partial\mathbf{B}}{\partial t} = \frac{\partial\mathbf{E}}{\partial t}\times \mathbf{B} - \mathbf{E} \times (\boldsymbol{\nabla}\times \mathbf{E})\,</math>
and we can now rewrite '''f''' as
:<math>\mathbf{f} = \epsilon_0 \left(\boldsymbol{\nabla}\cdot \mathbf{E} \right)\mathbf{E} + \frac{1}{\mu_0} \left(\boldsymbol{\nabla}\times \mathbf{B} \right) \times \mathbf{B} - \epsilon_0 \frac{\partial}{\partial t}\left( \mathbf{E}\times \mathbf{B}\right) - \epsilon_0 \mathbf{E} \times (\boldsymbol{\nabla}\times \mathbf{E})\,</math>,
then collecting terms with '''E''' and '''B''' gives
:<math>\mathbf{f} = \epsilon_0\left[  (\boldsymbol{\nabla}\cdot \mathbf{E} )\mathbf{E} - \mathbf{E} \times (\boldsymbol{\nabla}\times \mathbf{E}) \right] + \frac{1}{\mu_0} \left[ -  \mathbf{B}\times\left(\boldsymbol{\nabla}\times \mathbf{B} \right)  \right]
- \epsilon_0\frac{\partial}{\partial t}\left( \mathbf{E}\times \mathbf{B}\right)\,</math>.</li>
<li>A term seems to be "missing" from the symmetry in '''E''' and '''B''', which can be achieved by inserting (∇ • '''B''')'''B''' because of [[Gauss' law for magnetism]]:
:<math>\mathbf{f} = \epsilon_0\left[  (\boldsymbol{\nabla}\cdot \mathbf{E} )\mathbf{E} - \mathbf{E} \times (\boldsymbol{\nabla}\times \mathbf{E}) \right] + \frac{1}{\mu_0} \left[(\boldsymbol{\nabla}\cdot \mathbf{B} )\mathbf{B} -  \mathbf{B}\times\left(\boldsymbol{\nabla}\times \mathbf{B} \right)  \right]
- \epsilon_0\frac{\partial}{\partial t}\left( \mathbf{E}\times \mathbf{B}\right)\,</math>.
Eliminating the curls (which are fairly complicated to calculate), using the [[vector calculus identity]]
:<math>\tfrac{1}{2} \boldsymbol{\nabla} (\mathbf{A}\cdot\mathbf{A}) = \mathbf{A} \times (\boldsymbol{\nabla} \times \mathbf{A}) + (\mathbf{A} \cdot \boldsymbol{\nabla}) \mathbf{A} </math>,
leads to:
:<math>\mathbf{f} = \epsilon_0\left[  (\boldsymbol{\nabla}\cdot \mathbf{E} )\mathbf{E} + (\mathbf{E}\cdot\boldsymbol{\nabla}) \mathbf{E} \right] + \frac{1}{\mu_0} \left[(\boldsymbol{\nabla}\cdot \mathbf{B} )\mathbf{B} + (\mathbf{B}\cdot\boldsymbol{\nabla}) \mathbf{B} \right] - \frac{1}{2} \boldsymbol{\nabla}\left(\epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right)
- \epsilon_0\frac{\partial}{\partial t}\left( \mathbf{E}\times \mathbf{B}\right)\,</math>.</li>
<li>This expression contains every aspect of electromagnetism and momentum and is relatively easy to compute. It can be written more compactly by introducing the '''Maxwell stress tensor''',
:<math>\sigma_{i j} \equiv \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0}  \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)\,</math>,
and notice that all but the last term of the above can be written as the [[divergence]] of this:
:<math>\mathbf{f}  + \epsilon_0 \mu_0 \frac{\partial \mathbf{S}}{\partial t}\, = \nabla \cdot \mathbf{\sigma}</math>,

As in the [[Poynting's theorem]], the second term in the left side of above equation can be interpreted as time derivative of EM field's momentum density and this way, the above equation will be the law of conservation of momentum  in classical electrodynamics.

where we have finally introduced the [[Poynting vector]], 
:<math>\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}</math>.</li>
</ol>

in the above relation for conservation of momentum,  <math>\nabla \cdot \mathbf{\sigma}</math> is the '''momentum flux density''' and plays a role similar to <math>\mathbf{S}</math> in [[Poynting's theorem]].

==Equation==
In [[physics]], the '''Maxwell stress tensor''' is the [[stress tensor]] of an [[electromagnetic field]]. As derived above in [[SI unit]]s, it is given by:
:<math>\sigma_{i j}  = \epsilon_0 E_i  E_j   + \frac{1}
{{\mu _0 }}B_i  B_j   - \frac{1}{2}\bigl( {\epsilon_0 E^2  + \tfrac{1}
{{\mu _0 }}B^2 } \bigr)\delta _{ij} </math>,
where ε<sub>0</sub> is the [[electric constant]] and μ<sub>0</sub> is the [[magnetic constant]], '''E''' is the [[electric field]], '''B''' is the [[magnetic field]] and δ<sub>''ij''</sub> is [[Kronecker's delta]]. In Gaussian [[cgs unit]], it is given by:
:<math>\sigma_{i j}=\frac{1}{4\pi}\left(E_{i}E_{j}+H_{i}H_{j}-
\frac{1}{2}(E^2+H^2)\delta_{ij}\right)</math>,
where '''H''' is the [[magnetizing field]].

An alternative way of expressing this tensor is:
:<math>
\overset{\leftrightarrow  }{ \mathbf{\sigma} } = \frac{1}{4\pi} \left[ \mathbf{E}\otimes\mathbf{E} + \mathbf{H}\otimes\mathbf{H} - \frac{E^2+H^2 }{2} (\mathbf{\hat x}\otimes\mathbf{\hat x} + \mathbf{\hat y}\otimes\mathbf{\hat y} + \mathbf{\hat z}\otimes\mathbf{\hat z}) \right]
</math>,
where ⊗ is the [[dyadic product]].

The element ''ij'' of the Maxwell stress tensor has units of momentum per unit of area times time and gives the flux of momentum parallel to the ''i''th axis crossing a surface normal to the ''j''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 ''ij'' element of the tensor can also be interpreted as the force parallel to the ''i''th axis suffered by a surface normal to the jth axis per unit of area. Indeed the diagonal elements give the [[tension (physics)|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.

==Magnetism only==
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:
:<math>\sigma_{i j} = \frac{1}{\mu_0} B_i B_j - \frac{1}{2 \mu_0} B^2 \delta_{i j} \,.</math>

For cylindrical objects, such as the rotor of a motor, this is further simplified to:
:<math>\sigma_{r t} = \frac{1}{\mu_0} B_r B_t - \frac{1}{2 \mu_0} B^2 \delta_{r t} \,.</math>

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''<sub>''r''</sub> is the flux density in the radial direction, and ''B''<sub>''t''</sub> is the flux density in the tangential direction.

== See also ==
*[[Ricci calculus]]
*[[Energy density#Energy density of electric and magnetic fields|Energy density of electric and magnetic fields]]
*[[Poynting vector]]
*[[Electromagnetic stress-energy tensor]]
*[[Magnetic pressure]]
*[[Magnetic tension force]]

==References==
* David J. Griffiths,"Introduction to Electrodynamics" pp.&nbsp;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.

[[Category:Tensors]]
[[Category:Electromagnetism]]