= Magnetic bending in an incompressible disk
In the <steady state> of the <horizontally invariant magnetized shearing-sheet equations>, let $B_x(\pm H)=\pm B_+$ and $B_y(\pm H)=0$. For $q>0$ and $B_z\ne0$, the symmetric equilibrium has
$$
B_x=B_+\frac{\sin(Kz)}{\sin(KH)},\quad B_y=v_x=0,\quad
v_y=-\frac{B_z}{2\mu_0\rho\Omega}\frac{dB_x}{dz},\qquad
K^2=\frac{2q\mu_0\rho\Omega^2}{B_z^2}.
$$
This follows by combining azimuthal induction with radial <magnetic tension> balance to obtain $B_x''+K^2B_x=0$. Nonzero $B_+$ requires $\sin(KH)\ne0$; even homogeneous additions at $\cos(KH)=0$ are excluded by imposing midplane symmetry.
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