1. May 18, 2006

StatusX

In the course of treating the problem of an infinitely conducting gas accreting to a star in the presence of a magnetic field, I ran across the following problem. If the magnetic field and velocity are confined to a plane (which we'll call the $\theta=\pi/2$ plane), and if they only depend on r, then the only component of E is:

$$E_\theta = \frac{1}{c} (v_r B_\phi - v_\phi B_r)$$

In the steady state, $\nabla \times \mathbf{E} = 0$, and this requires $E_\theta \propto 1/r$. It is reasonable to assume that the density approaches some constant non-zero value at infinity, which by the continuity equation $\rho v_r r^2=$ const implies $v_r \propto 1/r^2$. Also $\nabla \cdot \mathbf{B} =0$ implies $B_r \propto 1/r^2$. Then for some non-zero constant $\kappa$

$$1/r \propto E_\theta \propto (B_\phi - \kappa v_\phi)/r^2$$

But then $B_\phi - \alpha v_\phi \propto r$, so at least one of $v_\phi, B\phi$ must blow up as r goes to infinity, clearly absurd. What is the problem here?

Last edited: May 18, 2006
2. May 25, 2006

StatusX

Anyone have any ideas?