#### Momentum
emd.ce.mom

From Newton's second law, \[ {\boldsymbol F} = \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt} \] we have \[ \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt} = -\varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau + \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} \] in which the first integral can be interpreted as the momentum stored in the EM fields, and the second is the momentum per unit time flowing in through the surface.

This is thus simply a conservation law for momentum, with

Momentum density in the EM fields

\[ {\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B} \tag{gExB}\label{gExB} \]

In a region in which the mechanical momentum is not changing due to external influences, we then have the

Continuity equation for EM momentum

\[ \frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0 \tag{contg}\label{contg} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42