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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?
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?
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