Momentum current density

This article deals with the importance of momentum current density in continuum mechanics and electrodynamics; for a different use of the term in the Karlsruhe physics course, see Karlsruhe physics course # content .

The momentum current density is a physical quantity which (analogous to other current densities ) describes the transport of momentum . It is part of the energy-momentum tensor and is particularly important in continuum mechanics and electrodynamics . It is a second order tensor . The dimension of the momentum current density is that of the pressure .

Continuum mechanics

In continuum mechanics, a non-vanishing momentum current density arises in a solid in the course of a deformation or in a flowing fluid . Since the momentum and thus also the momentum density are vectorial quantities, the momentum current density  T is a tensorial quantity.

In a Cartesian coordinate system , the tensor is calculated as:

${\displaystyle T_{ik}=\rho v_{i}v_{k}-\sigma _{ik}}$,

With

• the convective contribution (momentum transport through matter flow) ${\displaystyle \rho v_{i}v_{k}}$
  • the mass density ${\displaystyle \rho }$
  • the components of the velocity vector ${\displaystyle v_{i}}$
• the components of the stress tensor , which describes the contribution caused by surface forces. ${\displaystyle \sigma _{ik}}$

In mechanical equilibrium (see also hydrostatic equilibrium ) is the Tensordivergenz zero pulse current density . This follows from the continuity equation for the momentum density. ${\displaystyle \mathrm {div} \,T}$ ${\displaystyle \mathrm {div} \,T={\vec {0}}}$

Electrodynamics

The momentum current density of an electromagnetic wave or in general of electromagnetic fields ( ) is calculated as: ${\displaystyle \rho =0}$

${\displaystyle \sigma _{ik}=\varepsilon _{0}\left({\frac {1}{2}}E^{2}\delta _{ik}-E_{i}E_{k}\right)+{\frac {1}{\mu }}_{0}\left({\frac {1}{2}}B^{2}\delta _{ik}-B_{i}B_{k}\right)}$(see Maxwell's stress tensor ).

With

• the electric field constant ${\displaystyle \varepsilon _{0}}$
• the magnetic field constant ${\displaystyle \mu _{0}}$
• the electric field strength ${\displaystyle E}$
• the magnetic flux density ${\displaystyle B}$
• the Kronecker symbol . ${\displaystyle \delta _{ik}}$