# Energy Densities of Electric and Magnetic Fields

That electric (${\displaystyle E}$) and magnetic (${\displaystyle B}$) fields hold energy shouldn’t be surprising; the electromagnetic fields transporting energy from the Sun to Earth warm us up (and give us sunburns) when we stand outside. But the fact that macroscopic kinetic energy could disappear into an invisible field and be recovered later did indeed come as a surprise to 18th and 19th century physicists.

In astrophysical applications ranging from radiative transport to magnetohydrodynamics (the behavior of conducting fluids such as plasmas), our understanding of the energy dynamics would be fundamentally lacking if we didn’t include the energy of the electric and magnetic fields in our tally. The short answer is that, in CGS units, the energy density ${\displaystyle U_{tot}}$ of the ${\displaystyle E}$ and ${\displaystyle B}$ fields is given by

${\displaystyle U_{tot,CGS}={\frac {1}{8\pi }}E^{2}+{\frac {1}{8\pi }}B^{2}\,\!}$

It intuitively makes sense that energy should be proportional to the square of the amplitude of the ${\displaystyle E}$ and ${\displaystyle B}$ fields, just as power is proportional to the square of voltage in electrical circuits. What may be less intuitive is that the scalar amplitudes of the electric and magnetic fields are intrinsically densities, in the sense that they are per-volume quantities. This is because the ${\displaystyle E}$- and ${\displaystyle B}$-field amplitudes are local quantities intrinsic to a point in space, and must be integrated over volume to become energy.

As an aside, the symmetry in expression for ${\displaystyle U_{tot}}$ with respect to ${\displaystyle E}$ and ${\displaystyle B}$ is much more apparent in the CGS expression than in MKS, where the presence of ${\displaystyle \epsilon _{0}}$ and ${\displaystyle \mu _{0}}$ are distracting:

${\displaystyle U_{tot,MKS}={\frac {\epsilon _{0}}{2}}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\,\!}$

to me, this symmetry is one of the more compelling reasons why I prefer CGS to MKS; ${\displaystyle E}$ and ${\displaystyle B}$ fields are already in the same units!