I work as a Teaching Assistant at my university and have an excersise class of elementary electromagnetism. To my surprise, I sometimes come to physical problems which I've never thought before while very simple.
For example, suppose a charged particle confined on a circle. When magnetic field is applied, the particle feels the electric field in the tangential direction (by the induced voltage). The magnitude of the field is . But when the particle is not confined, this problem becomes unsolvable. When the magnetic field changes, there's a electric field
from the Maxwell's third equation. Suppose
, this equation is
. We can't determine what electric field the particle feels because of an uncertainty. For example
and
are both solutions but corresponds to very discrete situations. You'd think this is just like the gauge freedom, but the electric field is a gauge invariant quantity and this interpretation is not valid.
The missing key is the boundary condition. Non-local information is needed to solve some problems.
