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The gauge invariant formulation of Maxwell's Laws (7.13):

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Indicates that the transverse electric field is the time derivative of the transverse vector potential.

This gauge invariant vector potential increases without bound as long as there exists a static electric field. Indeed, even when the electric field is removed, there appears to be no mechanism by which the gauge invariant vector potential disappears.

In what way am I misinterpreting (7.13)?

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  • $\begingroup$ A transverse electric field requires a changing current, so I think we need not worry about unbounded growth in the vector potential, because that requires an unbounded increase in current density. $\endgroup$ – user27118 Mar 23 '15 at 19:58
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This gauge invariant vector potential increases without bound as long as there exists a static electric field. Indeed, even when the electric field is removed, there appears to be no mechanism by which the gauge invariant vector potential disappears.

Static electric field has zero transversal component; entire field is longitudinal.

The unbounded increase of vector potential would occur if you used the gauge where the static electric field is given as time derivative of vector potential.

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