Evolution of Free Magnetic Fields. I. Neglecting Self-Gravitation

Abstract

This paper gives the exact solutions, graphical and analytical, for an idealization of a phenomenon described in every textbook of physics: the decay of a magnetic field after its source current has ceased. The problem is solved for the cylindrically symmetric case where a uniform current in a long straight wire is suddenly brought to zero at an initial time $\ensuremath{\tau}=0$. The probable reason why this apparently simple problem has not been solved before is that the mathematical details are complex. The solutions for the electric and magnetic fields which result from the decay of the initial magnetic field are classified according to the four regions of a space-time diagram in which they appear: Regions $A$ (inside wire) and $O$ (outside wire) are outside the light cone of a point on the boundary of the wire. In accord with causality, the magnetic and electric fields retain their initial values (the latter equal to zero) in region $O$; in region $A$ the magnetic field retains its initial value, but the electric field rises linearly with time, since its time derivative replaces the extinguished constant current density. Region $B$ is between the axis and the wave front from the boundary (which started at $\ensuremath{\tau}=0$) after this wave front has passed through the axis and started outward again. Region $C$ is contained between the initially outgoing wave front which starts from the wire boundary and the initially ingoing wave front after it has passed through the axis and started outward again; it is therefore a region which the initial disturbance from the wire boundary has swept over only once instead of twice, in contrast to region $B$. In both regions $B$ and $C$ the analytical expressions for the electric and magnetic fields $\mathcal{E}$ and $\mathcal{B}$ are very complicated except along certain selected lines in space-time, e.g., at the wire axis ($\mathcal{E}={[\ensuremath{\tau}+{({\ensuremath{\tau}}^{2}\ensuremath{-}1)}^{\frac{1}{2}}]}^{\ensuremath{-}1}$, $\mathcal{B}=0$) or on the light cone of a wire boundary point ($\mathcal{E}=\ensuremath{\tau}[1\ensuremath{-}(\frac{2}{\ensuremath{\pi}})invcos(\frac{1}{{\ensuremath{\tau}}^{\frac{1}{2}}})]\ensuremath{-}(\frac{2}{\ensuremath{\pi}}){(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}$, $\mathcal{B}=\frac{1}{2}(\ensuremath{\tau}\ensuremath{-}1)\ensuremath{-}[\frac{\ensuremath{\tau}}{\ensuremath{\pi}{(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}}]{1\ensuremath{-}[\frac{(2\ensuremath{-}\ensuremath{\tau})}{{(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}}]invsin{[\frac{(\ensuremath{\tau}\ensuremath{-}1)}{\ensuremath{\tau}}]}^{\frac{1}{2}}}$). Quite apart from the specific nature of this problem, the investigations in this and the following papers illustrate a practical method for finding realistic solutions to the Einstein-Maxwell time-dependent equations. These solutions correspond to special cases---worked out explicitly, and in one case independently---of a general family of solutions of the time-dependent electrogravitational equations for cyclindrical symmetry. This general family of solutions has been discussed by Melvin and stachel.