Solution (source code)

= Solution

For an isotropic <weak electron-magnetohydrodynamic cascade>, set $l_\parallel\sim l_\perp\sim l$. Part (b) gives $\delta B_l\propto l^{1/2}$. With the one-dimensional shell convention for the <magnetic energy spectrum>, fluctuations on scale $l\sim k^{-1}$ satisfy $\delta B_l^2\sim kE_B(k)$ up to normalization constants. Therefore
$$
\boxed{E_B(k)\propto k^{-2}.}
$$
This is the shell-integrated spectrum. The three-dimensional <spectral tensor> coefficient has two additional powers of $k^{-1}$ and scales as $H(k)\propto k^{-4}$. These exponents apply within the assumed local, weak cascade range.