# Can the electric field have closed field lines?

In electrostatics, we know that $$\vec{\nabla}\times\vec{E} = 0$$ and so, the field lines can't form loops. But when we have time-dependant magnetic fields, there's the Faraday-Lenz law which tells us that $$\vec{\nabla}\times\vec{E} = -\frac{\partial\vec{B}}{\partial{t}}$$. Does that mean that we can have closed field lines for the electric field, let's say for example, when we have moving magnets?

If that's the case, can you have closed field lines in another way that isn't with time-dependant magnetic fields?

• I've found in some places that this seems to be the case, but I just want to make sure. :D May 24 at 19:51
• One of the cooler applications of closed electric field lines is the betatron (en.wikipedia.org/wiki/Betatron). It does, however, require a changing magnetic field. May 24 at 20:14

In short:

Does that mean that we can have closed field lines for the electric field, let's say for example, when we have moving magnets?

Yes.

If that's the case, can you have closed field lines in another way that isn't with time-dependant magnetic fields?

No.

I've found in some places that this seems to be the case

Either you're misreading those situations or there is a time dependent magnetic field, but it's obviously impossible to say without more information.

• With "I've found in some places that this seems to be the case" I was referring to the fact that there's closed electric field lines, not that there is without changing B. I understand that if B does not change in time, curl(E) = 0 and it's the same as in electrostatics. Thanks for the answer! May 24 at 20:30