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Einstein-Laub equation

 Stan Zurek, Einstein-Laub equation , Encyclopedia-Magnetica.com, {accessed 2019-10-23}

Einstein-Laub equation - an equation for calculation of force density acting on an object placed in electromagnetic field.1)2)3)

$$\boldsymbol{F}(\boldsymbol{r},t) = ρ_{free} \boldsymbol{E} + \boldsymbol{J}_{free} × μ_0 \boldsymbol{H} + (\boldsymbol{P} · \boldsymbol{∇})\boldsymbol{E} + (\partial \boldsymbol{P} / \partial t) × μ_0 \boldsymbol{H} + (\boldsymbol{M} · \boldsymbol{∇})\boldsymbol{H} - (\partial \boldsymbol{M} / \partial t) × ε_0 \boldsymbol{E}$$

where: $\boldsymbol{F}$ - force, $\boldsymbol{r}$ - variable of spatial coordinates, $t$ - variable of time, $ρ_{free}$ - density of free charge, $\boldsymbol{E}$ - electric field, $\boldsymbol{J_{free}}$ - density of free current, $μ_0$ - magnetic permeability of free space, $\boldsymbol{H}$ - magnetic field strength, $\boldsymbol{P}$ - electric polarisation, $\boldsymbol{∇}$ - nabla operator, $\partial \ldots / \partial t$ - partial derivative with respect to time, $\boldsymbol{M}$ - magnetisation, $ε_0$ - electric permittivity of free space.

Einstein and Laub did not provide a firm basis for their proposed equation of force density. Their explanation was:

• “Now we have to fit [Maxwell's] equations … to the case where magnetically polarizable bodies are present. Lorentz does this by conceiving of certain electricities as being endowed with cyclical motions; from the standpoint of the pure electron theory this is also the way that is justified. But for the sake of simplicity, we will base ourselves here on the knowledge that, as regards spatio-temporal interrelations, the magnetic polarization is a state wholly analogous to the polarization of dielectrics. Thus, we permit ourselves to conceive of magnetically polarizable bodies as being endowed with bound magnetic volume densities.”

For a similar system, calculation of Lorentz force may lead to introduction of the so-called “hidden momentum”, whereas the Einstein-Laub equation does not require such entities.4) 