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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 [itex]\theta=\pi/2[/itex] plane), and if they only depend on r, then the only component of E is:
[tex]E_\theta = \frac{1}{c} (v_r B_\phi - v_\phi B_r)[/tex]
In the steady state, [itex]\nabla \times \mathbf{E} = 0[/itex], and this requires [itex]E_\theta \propto 1/r[/itex]. It is reasonable to assume that the density approaches some constant non-zero value at infinity, which by the continuity equation [itex]\rho v_r r^2=[/itex] const implies [itex]v_r \propto 1/r^2[/itex]. Also [itex]\nabla \cdot \mathbf{B} =0[/itex] implies [itex]B_r \propto 1/r^2[/itex]. Then for some non-zero constant [itex]\kappa[/itex]
[tex]1/r \propto E_\theta \propto (B_\phi - \kappa v_\phi)/r^2[/tex]
But then [itex]B_\phi - \alpha v_\phi \propto r[/itex], so at least one of [itex]v_\phi, B\phi[/itex] must blow up as r goes to infinity, clearly absurd. What is the problem here?
[tex]E_\theta = \frac{1}{c} (v_r B_\phi - v_\phi B_r)[/tex]
In the steady state, [itex]\nabla \times \mathbf{E} = 0[/itex], and this requires [itex]E_\theta \propto 1/r[/itex]. It is reasonable to assume that the density approaches some constant non-zero value at infinity, which by the continuity equation [itex]\rho v_r r^2=[/itex] const implies [itex]v_r \propto 1/r^2[/itex]. Also [itex]\nabla \cdot \mathbf{B} =0[/itex] implies [itex]B_r \propto 1/r^2[/itex]. Then for some non-zero constant [itex]\kappa[/itex]
[tex]1/r \propto E_\theta \propto (B_\phi - \kappa v_\phi)/r^2[/tex]
But then [itex]B_\phi - \alpha v_\phi \propto r[/itex], so at least one of [itex]v_\phi, B\phi[/itex] must blow up as r goes to infinity, clearly absurd. What is the problem here?
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