Maxwell's stress tensor

The Maxwell stress tensor (named after James Clerk Maxwell ) is a symmetric second order tensor used in classical electrodynamics to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as that of a point charge that moves freely in a homogeneous magnetic field , it is easy to calculate the forces on the charge through the Lorentz force .

For more complex problems, the Lorentz force procedure becomes very long. It is therefore useful to collect different quantities of electrodynamics in Maxwell's stress tensor.

In the relativistic formulation of electromagnetism, the Maxwell tensor appears as part of the electromagnetic energy-momentum tensor .

definition

Maxwell's stress tensor is defined in SI units by

${\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {B_{i}B_{j}}{\mu _{0}}}-{\frac {1}{2}}\left(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}}\right)\delta _{ij}}$,

where denote the components of the electric field, the components of the magnetic field and the Kronecker delta. The tensor in cgs units is given by ${\displaystyle E_{i}}$ ${\displaystyle B_{i}}$ ${\displaystyle \delta _{ij}}$

${\displaystyle T_{ij}={\frac {1}{4\pi }}\left(E_{i}E_{j}+H_{i}H_{j}-{\frac {1}{2}}\left(E^{2}+H^{2}\right)\delta _{ij}\right)}$.

Magnetostatics

For purely magnetic fields (which is approximately the case in motors, for example), some of the terms are omitted, which means that the Maxwell tensor becomes

${\displaystyle T_{ij}={\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{ij}}$

simplified

For cylindrical objects - such as B. the rotors of a motor - results

${\displaystyle T_{rt}={\frac {1}{\mu _{0}}}B_{r}B_{t}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{rt}\,,}$

It is the shear in the radial direction (from the cylinder to the outside) and the shear in the tangential direction (around the cylinder). The motor is driven by the tangential force. is the flux density in the radial direction and is the flux density in the tangential direction. ${\displaystyle r}$ ${\displaystyle t}$ ${\displaystyle B_{r}}$ ${\displaystyle B_{t}}$

Electrostatics

In electrostatics, for which the magnetic field disappears ( ), we get the electrostatic Maxwell's voltage tensor. In component notation, this results from: ${\displaystyle {\vec {B}}={\vec {0}}}$

${\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}-{\frac {1}{2}}\varepsilon _{0}E^{2}\delta _{ij}}$

and in symbolic notation

${\displaystyle {\boldsymbol {T}}=\varepsilon _{0}\mathbf {E} \otimes \mathbf {E} -{\frac {1}{2}}\varepsilon _{0}(\mathbf {E} \cdot \mathbf {E} )\mathbf {I} }$

where is the identity tensor. ${\displaystyle \mathbf {I} }$

literature

• David J. Griffiths : Introduction to Electrodynamics. Benjamin Cummings Inc., 2008, pp. 351-352
• John David Jackson : Classical Electrodynamics . 3rd edition, John Wiley & Sons, Inc., 1999.
• Richard Becker: Electromagnetic Fields and Interactions. Dover Publications Inc., 1964.