# Quantifying electrical energy stored in a plasma

The energy stored in the electric field in a medium is given by $U = \frac{1}{2} \epsilon E^2$, where $\epsilon$ is the permittivity in the medium and $E$ is the electric field. When considering a plasma, there are no emergent electric fields on a scale larger than the Debye length while far away from any walls/sheath. Thus $E = 0$ and the energy is similarly set to zero.

But when going below the Debye length, there are variations in the electric field on short time and spatial scales, and the electric field can be finite, i.e. $E \neq 0$. There will be both positive and negative electric fields due to both electrons and ions that may be present, and the total $E$ may cancel when averaged over larger scales since it is a vector quantity, but since $E^2$ is positive definite, is not $U$ (i.e. electrical energy density) generally nonzero? When considering $U$ on small scales below the Debye length and integrating it over the volume of the plasma to get a total energy, is this not a significant component that should be accounted for when calculating the internal energy of the plasma and its fields?

As you say, short time-scale and spatial-scale variations in $$E$$ will result in energy stored from energy density in the form $$U = \frac{1}2 \epsilon E^{2}$$. Such is the case during thermal fluctuations in which electrons might vacate a certain region of the plasma leaving ions behind to produce an electric field between them.
Do note, however, that for the region to be effectively vacated, it would have to be a sphere (radius $$r_{v}$$) out of which all electrons exit radially. Then $$E$$ would depend on the charge of ions ($$Q_{i}$$) left behind as $$E = \frac{Q_{i}}{4\pi\epsilon_{0}r^{2}} = \frac{ner}{3\epsilon_{0}}$$ and the energy stored would be equal to the work done by the electrons exiting the sphere,
$$W = \int_0^{r_v} \frac{\epsilon_{0}E^{2}}2 4\pi r^{2} dr = \pi r^{5}_{v} \frac{2n^{2}_{e}e^{2}}{45\epsilon_{0}}$$
But this $$r_{v}$$ turns out to be of a few Debye lengths ($$\lambda_{D}$$) and this is the radius of the largest possible volume that could be spontaneously vacated by electrons in a plasma. And this case is already really unlikely: electrons are moving in and out of space randomly and not radially apart from a specific point in space. The overall $$E$$ will then be $$0$$. In this macroscopic level, the energy stored from the electric field between spontaneous displacement of electrons with respect to ions will also be $$0$$.