Why does my textbook only show magnetostatics when curl of H = 0?

[pivoxa15]

In my textbooks it shows curl of H = 0 is a situtaion of magnetostatics but in here http://en.wikipedia.org/wiki/Magnetostatics it shows otherwise assuming J can be anything. Which is correct?

Magnetostatics is defined to be when the magnetic field is constant so H should be a vector field with scalar components suggesting that curl of H=0.

 

[quasar987]

curl of H will be 0 outside regions with currents.

 

[pivoxa15]

I also should have mentioned that magnetic fields can only arise when a current is present. So a constant current must be present. But it means H will be a non constant field so can't be magnetostatics?

Outside regions with current, magnetic fields don't even exist - which can't be magnetostatics can it? i.e. no charges is not electrostatics.

We seem to have a problem either way. What am I missing?

 

[quasar987]

I also should have mentioned that magnetic fields can only arise when a current is present. So a constant current must be present. But it means H will be a non constant field so can't be magnetostatics?

I wonder how you make that conclusion. It is readily seen from say Biot savard law, or Ampere law over a circle centered on the axis of a wire in which a steady current flows that the megnetic field is constant in time.

Outside regions with current, magnetic fields don't even exist

wahoo! Then how can a magnet work?! No, a steady current creates a steady magnetic field in all space.

 

[quasar987]

The relation between the magnetic field and its div and curl is the following:

https://www.physicsforums.com/showthread.php?t=127264&highlight=helmholtz

Open the doc file called "Question.doc" and replace F by H in that equation. The integrals are over the whole universe. The first integral is always 0 because the divergence of H is always 0. But you see that as soon as there is a current somewhere in space, the the integral is non- vanishing.

The main lesson here is that $\nabla\cdot \vec{H}=0$ and $\nabla\times \vec{H}=0$ at some points $\vec{r}$ does not imply $\vec{H}(\vec{r})=0$. The value of H at some point depends on the value of J everywhere in space.

 

[pivoxa15]

I wonder how you make that conclusion. It is readily seen from say Biot savard law, or Ampere law over a circle centered on the axis of a wire in which a steady current flows that the megnetic field is constant in time.

I am thinking about the curl of H = 0 => constant vector field H => magnetostatics.

But on the website curl of H = nonzero constant current density => H is a nonconstant vector field => B is non constant vector field hence not megnetostatics.

The problem is the current density should not be 0 for anything to do with magnetics but that leads to nonconstant B field so no magnetostatics.

wahoo! Then how can a magnet work?! No, a steady current creates a steady magnetic field in all space.

In classical physics, we are taught to think about magnets as having mini current loops inside hence with many magnetic dipoles meaning magnetised material.

 

[quasar987]

I think I've hit it now:

The curl operator has nothing to do with how B behaves in time! It says something about how H changes in space. Magnetostatic is by definition a case where B is constant in time, not where the magnitude of B is the same at all points in space.

In classical physics, we are taught to think about magnets as having mini current loops inside hence with many magnetic dipoles meaning magnetised material.

Well exactly. A magnet's magnetic field is cause by its surface current. But the point is that it is a current that causes the field, but there is a non zero B field all around the magnet, not only where the current itself it.

 

[pivoxa15]

I think I've hit it now:

The curl operator has nothing to do with how B behaves in time! It says something about how H changes in space. Magnetostatic is by definition a case where B is constant in time, not where the magnitude of B is the same at all points in space.

Okay that makes more sense now. But there is still the question why my textbook shows magnetostatics only when curl of H = 0. The book later addressed with the fact that there are no free current in magnetostatics, which means the B fields in magnetostatics are never generated by electrical coils but rather by the magnets themselves (bound currents). But you could set up a constant B field with electrical coils without a magnet in sight couldn't you?

 

[Meir Achuz]

Magnetostatics is defined to be when the magnetic field is constant so H should be a vector field with scalar components suggesting that curl of H=0.

Your problem is that this statement is wrong.
Magnetostatics means that all partial time derivatives are zero, so steady currents are allowed. The key equations are Curl H=4 pi j/c and
div j=0.

 

[quasar987]

Okay that makes more sense now. But there is still the question why my textbook shows magnetostatics only when curl of H = 0. The book later addressed with the fact that there are no free current in magnetostatics, which means the B fields in magnetostatics are never generated by electrical coils but rather by the magnets themselves (bound currents). But you could set up a constant B field with electrical coils without a magnet in sight couldn't you?

Certainly. Talk to your instructor about it.