Magnetic energy ($E_B$)

In vacuum-permeability magnetohydrodynamics, the magnetic energy in a volume is $E_B=\int B^2/(2{\mu }_0)dV$. Its volume density equals the magnetic pressure.

For reversible quasistatic magnetic circuits with flux linkages ${\lambda }_i$, currents $I_i$ and mechanical coordinate $q$, write $d{W}_B={\sum }_i{I}_{i}d{\lambda }_i-F_{q}dq$. The Legendre transform $W_B^{\prime }={\sum }_i{I}_i{\lambda }_i-W_B$ obeys $d{W}_B^{\prime }={\sum }_i{\lambda }_{i}d{I}_i+F_{q}dq$. Thus $F_q=(\partial W_B^{\prime }/\partial q{)}_{I_i}$, while $F_q=-(\partial W_B/\partial q{)}_{{\lambda }_i}$. In a linear reciprocal system $W_B^{\prime }=W_B=\frac{1}{2}{\mathbf{I}}^{\mathsf{T}}L(q)\mathbf{I}$ numerically, but the variables held fixed in their force derivatives differ. The coenergy formulation includes the effect of electrical sources maintaining the currents; it avoids reversing the sign of the magnetic force by differentiating stored energy under the wrong constraint.

The shell-integrated magnetic energy spectrum distributes magnetic energy over wavenumbers. With a solenoidal isotropic spectral tensor normalized to ${\hat{C}}_{ii}=2H(k)$ and the inverse Fourier transform measure $d^{3}k/(2\pi {)}^3$, one has $\mathbb{E}|\mathbf{B}{|}^2={\pi }^{-2}\int k^{2}H(k)dk$. In Gaussian units, $E_B(k)=k^{2}H(k)/(8{\pi }^3)$ integrates to the energy density $\mathbb{E}|\mathbf{B}{|}^2/(8\pi )$. A spectrum per unit mass or normalized to $B^2/2$ has different constant factors. Scale-local fluctuation estimates give $\delta B_l^2$ proportional to $k{E}_B(k)$ at $k\sim l^{-1}$ when the spectrum is normalized consistently.