Gauss's Law for Magnetism Question

[Only a Mirage]

Hey everyone, I'm new to these forums. Being an electrical engineering major, most of my teachers aren't very concerned with the "physics" side of things. I'm hoping I can gain some insight on Maxwell's equations.

When first stating Gauss's Law for Magnetism, the only reason my electromagnetics text gives for this is that all magnetic field lines close upon themselves. Therefore, the flux due to the B field over a closed surface is zero. This makes perfect sense to me, and I thought that this fact would be true for the H field as well. However, when deriving magnetic boundary conditions, if you assume that the flux due to the B field is always zero, it is impossible that the flux due to the H field is always zero as well. If your Gaussian surface is in free space or in one medium, then both equations can be true, but not if the volume enclosed by your Gaussian surface contains an interface.

My confusion may be a result of not understanding exactly what the difference between B and H is on a fundamental level (I know the constitutive relationships).

What is so special about the B field? Why isn't the flux due to the H field always zero?

 

[Hassan2]

First material reasoning ( You may know all):

One of the Maxwell equations states that

[itex]\nabla . B =0[/itex] eveywhere

From this equation , using divergence theorem ( which is purely mathematical) one concludes that
[itex]\oint B.ds=0[/itex] .

However from

[itex]B= \mu H[/itex]

then we have

[itex]\nabla . B =\nabla \mu.H+\mu \nabla .H=0[/itex]

Since [itex] \mu [/itex] is is discontinuous on the interface, it's divergence is NOT zero everywhere , hence

[itex]\nabla .H≠0[/itex] [itex]\Rightarrow[/itex] [itex]\oint H.ds[/itex] may not be zero.In fact that the normal component of [itex]B[/itex] is continuous everywhere, including on the interfaces and it's the normal component which contribute to the flux. This is the result of [itex]\nabla . B =0[/itex] and is not necessarily true for [itex]H[/itex].

The physical difference between [itex]B[/itex] and [itex]H[/itex] arises from magnetization [itex]M[/itex] of the magnetized media . In fact [itex]H[/itex] has no physical meaning but is defined to make the Maxwell's equations simpler. We just define [itex]H=\frac{1}{\mu_{0}}B-M[/itex]. The discontinuity of the normal component of [itex]H[/itex] is due to discontinuity of the normal component of [itex]M[/itex].

As for the reason given in your textbook, and that you expect the same for H, I don't think we can talk about the lines of H because the lines are the force lines which depends on B. However of I want to do an analogy, I can say the H lines ARE closed on themselves too but they may return " weaker" or "stronger" than when they left! This means the the net flux may not be zero.

 

[Only a Mirage]

Thank you very much for your reply, Hassan! You start your explanation by stating that the divergence of B is zero. My book derives this fact from the assumption that the flux of the B field over a closed surface is zero.

If instead derive the integral form of Gauss's Law for Magnetism from the differential form, where did the differential form come from? I.e., by what reasoning is Div(B) = 0 instead of Div(H) = 0?

You have started to clear up some of my confusion, though. My book never stated that B was the "real" physical field. Also, my book never defined H as you stated- although from what I can tell, there may be some logical gaps in my text when moving into material space. In fact, my book starts off by defining the H field in terms of the Biot-Savart Law, and defines the B field in free space as mu_0 * B.

So B is the only "real" (physically) field? Is the fundamental definition of B in terms of the Biot-Savart Law, or is it defined in some other way?

Thank you again for the response. Neither of my last two electromagnetics teachers (EE dept.) knew the answer to this question.

 

[Hassan2]

Thank you very much for your reply, Hassan! You start your explanation by stating that the divergence of B is zero. My book derives this fact from the assumption that the flux of the B field over a closed surface is zero.

That's why I said mathematical reasoning.

The differential form comes from the assumption ( law) that the filed lines are closed on themselves.

Seems our textbooks have a different approach to electromagnetism. In my textbook, even the Biot-Savart Law is for B. And the physical meaning of B is understood from Lorentz force. F=qvB

For a better understanding of the relation between B and H , read tiny-tim's post in the following thread:
https://www.physicsforums.com/showthread.php?p=3787165 .

 

[Only a Mirage]

Thanks for the link. I'll check it out.