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## Homework Statement

Imagine a uniform magnetic field, pointing in the ##z## direction and filling all space (##\vec{B} = B_0\hat{z}##). A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?

[There is a footnote for this problem: "This paradox was suggested by Tom Colbert...". The footnote than refers me to a problem about an electric field of the form ##\vec{E} = ax\hat{x}## that in spite of having a uniform charge density by its divergence, is not uniform itself and the problem asks to explain this. The answer is that the question is ill-defined and that the curl and divergence by themselves do not determine the field and appropriate boundary conditions are necessary.]

## Homework Equations

A formula for the induced electric field in terms of the time derivative of the magnetic field:

[tex]\vec{E} = -\frac{1}{4\pi}\frac{\partial}{\partial t}\int\frac{\vec{B}\times\hat{r}}{r^2}d\tau[/tex] where ##d\tau## is the infinitesimal volume element.

## The Attempt at a Solution

Since the magnetic field is uniform and points in the ##z## direction, turning it to (abruptly) change in the ##-z## direction. The change in flux will also be in this direction. Thus, the electric field will have to create flux upwards in the ##z## direction by Lenz's law. In order to do so, it will have to circulate everywhere counterclockwise as viewed from above by the right-hand rule. Since the field was uniform everywhere, the circulations will also be uniform and it seems to me that by symmetry, the force they will exert on the particle will cancel out. Thus, I think that the particle won't move at all. However, this footnote makes me believe that this is a trick question and that I have missed something. Is my reasoning correct or have I missed a catch somewhere?

Any comments will be greatly appreciated!