Magnetic Flux through a Closed Surface

[TH93]

Homework Statement

Using the divergence theorem, evaluate the total flux of a magnetic field B(r) across the
surface S enclosing a finite, connected volume of space V, and discuss its possible
dependence on the presence of an electric field E(r).

Homework Equations

∇.B=0

The Attempt at a Solution

The first part was pretty straightforward. Using the Maxwell equation in conjunction with the divergence theorem, it is easy to see that the magnetic flux across a closed surface is 0. The next part somewhat confuses me. My initial thought is that the divergence of B being 0 holds for all cases and therefore the presence of an electric field should have no bearing on the magnetic flux. However, I am not 100% sure though. Any help will be appreciated.

 

[D.W.]

The only thing I could think of for the second part is that a changing electric field produces a magnetic field which would, obviously, affect the magnetic flux across the surface...

 

[TH93]

In what way does this happen? Would that mean a time dependant E field would produce a case in which the divergence of B is not 0. Surely for a closed surface this would not be possible?

In the question itself, the E field mentioned is only dependant on position so a time dependant case is irrelevant.

 

[D.W.]

If the E field is strictly dependent upon position, I'm pretty sure that there is no way that it could affect the flux. But please don't rely on my answer alone. I'm by no means an expert in Electrodynamics

 

[MathiasArendru]

Well the magnetic flux describes the TOTAL flux through the surface. There is still magnetic fields flowing out of the surface, but because no monopoles exist, the flux has to equal 0 because the same amount that exist also enters. Referring to amperes law ∇ x B = $\frac{∂D}{∂t}$ + J That saying that the magnetic flux is equal to the change of the electric field over time. So if you have a changing magnetic field inside the closed surface, you also have magnetic flux. And that's why i think its dependent on the presence of an electric field. And since ∇ . E = $\frac{∑Q}{ε}$ It doesn't matter where it is.