Editing Virial theorem (section)

== Inclusion of electromagnetic fields ==

The virial theorem can be extended to include electric and magnetic fields. The result is<ref>{{cite book |first=George |last=Schmidt |title=Physics of High Temperature Plasmas |edition=Second |publisher=Academic Press |year=1979 |pages=72}}</ref>

<math display="block">
\frac12\frac{d^2I}{dt^2}
+ \int_Vx_k\frac{\partial G_k}{\partial t} \, d^3r
= 2(T+U) + W^\mathrm{E} + W^\mathrm{M} - \int x_k(p_{ik}+T_{ik}) \, dS_i,
</math>

where {{mvar|I}} is the [[moment of inertia]], {{mvar|G}} is the [[Poynting vector|momentum density of the electromagnetic field]], {{mvar|T}} is the [[kinetic energy]] of the "fluid", {{mvar|U}} is the random "thermal" energy of the particles, {{math|''W''{{isup|E}}}} and {{math|''W''{{isup|M}}}} are the electric and magnetic energy content of the volume considered. Finally, {{math|''p<sub>ik</sub>''}} is the fluid-pressure tensor expressed in the local moving coordinate system

<math display="block">
p_{ik}
= \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma
- V_iV_k\Sigma m^\sigma n^\sigma,
</math>

and {{math|''T<sub>ik</sub>''}} is the [[Maxwell stress tensor|electromagnetic stress tensor]],

<math display="block">
T_{ik}
= \left( \frac{\varepsilon_0E^2}{2} + \frac{B^2}{2\mu_0} \right) \delta_{ik}
- \left( \varepsilon_0E_iE_k + \frac{B_iB_k}{\mu_0} \right).
</math>

A [[plasmoid]] is a finite configuration of magnetic fields and plasma. With the virial theorem it is easy to see that any such configuration will expand if not contained by external forces. In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish. Since all the other terms on the right hand side are positive, the acceleration of the moment of inertia will also be positive. It is also easy to estimate the expansion time {{mvar|τ}}. If a total mass {{mvar|M}} is confined within a radius {{mvar|R}}, then the moment of inertia is roughly {{math|''MR''<sup>2</sup>}}, and the left hand side of the virial theorem is {{math|{{sfrac|''MR''<sup>2</sup>|''τ''<sup>2</sup>}}}}. The terms on the right hand side add up to about {{math|''pR''<sup>3</sup>}}, where {{mvar|p}} is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for {{mvar|τ}}, we find

<math display="block">\tau\,\sim \frac{R}{c_\mathrm{s}},</math>

where {{math|''c''<sub>s</sub>}} is the speed of the [[ion acoustic wave]] (or the [[Alfvén wave]], if the magnetic pressure is higher than the plasma pressure). Thus the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time.