Editing Momentum (section)

=====Vacuum=====
The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.<ref name="Jackson238" />

In a vacuum, the momentum per unit volume is

<math display="block"> \mathbf{g} = \frac{1}{\mu_0 c^2}\mathbf{E}\times\mathbf{B}\,,</math>

where {{math|{{var|μ}}{{sub|0}}}} is the [[vacuum permeability]] and {{mvar|c}} is the [[speed of light]]. The momentum density is proportional to the [[Poynting vector]] {{math|'''S'''}} which gives the directional rate of energy transfer per unit area:<ref name="Jackson238" /><ref name=FeynmanCh27>[https://feynmanlectures.caltech.edu/II_27.html#Ch27-S6 ''The Feynman Lectures on Physics''] Vol. II Ch. 27-6: Field momentum</ref>

<math display="block"> \mathbf{g} = \frac{\mathbf{S}}{c^2}\,.</math>

If momentum is to be conserved over the volume {{mvar|V}} over a region {{mvar|Q}}, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If {{math|'''P'''{{sub|mech}}}} is the momentum of all the particles in {{mvar|Q}}, and the particles are treated as a continuum, then Newton's second law gives

<math display="block"> \frac{\text{d}\mathbf{P}_\text{mech}}{\text{d}t} = \iiint\limits_{Q} \left(\rho\mathbf{E} + \mathbf{J}\times\mathbf{B}\right) \text{d}V\,.</math>

The electromagnetic momentum is

<math display="block"> \mathbf{P}_\text{field} = \frac{1}{\mu_0c^2} \iiint\limits_{Q} \mathbf{E}\times\mathbf{B}\,dV\,,</math>

and the equation for conservation of each component {{mvar|i}} of the momentum is

<math display="block"> \frac{\text{d}}{\text{d}t}\left(\mathbf{P}_\text{mech}+ \mathbf{P}_\text{field} \right)_i = \iint\limits_{\sigma} \left(\sum\limits_{j} T_{ij} n_j\right)\text{d}\Sigma\,.</math>

The term on the right is an integral over the surface area {{mvar|Σ}} of the surface {{mvar|σ}} representing momentum flow into and out of the volume, and {{math|{{var|n}}{{sub|j}}}} is a component of the surface normal of {{mvar|S}}. The quantity {{math|{{var|T}}{{sub|{{var|i}}{{var|j}}}}}} is called the [[Maxwell stress tensor]], defined as<ref name=Jackson238>{{harvnb|Jackson|1975|pp=238–241}} Expressions, given in [[Gaussian units]] in the text, were converted to SI units using Table 3 in the Appendix.</ref>

<math display="block">T_{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>