Topological - ttrr's blog

Topological

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 $1/2\pi R (d\Phi /dt)$ . But when the particle is not confined, this problem becomes unsolvable. When the magnetic field changes, there's a electric field $\nabla \times \vec{E} = -\frac{\partial \vec{B}(t)}{\partial t}$ from the Maxwell's third equation. Suppose $\vec{B}(t)=(0,0,-Bt)$ , this equation is $\partial_{y}E_{x}-\partial_{x}E_{y} = B$ . We can't determine what electric field the particle feels because of an uncertainty. For example $E_{x} = By, E_{y}=0$ and $E_{x} = 0,E_{y} = -Bx$ 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.