Physics Help Forum Help me understand correct interpretation of Gauss' Magnetic Law

 Aug 16th 2014, 12:06 AM #1 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Help me understand correct interpretation of Gauss' Magnetic Law I have been looking at http://www.maxwells-equations.com and they had a different interpretation to what I expected to Gauss' magnetic law. $\triangledown&space;\cdot&space;\mathbf{B}&space;=&space;0$ I know this is the divergence operator but for the case of a bar magnet inside a sphere (completely) I imagined that it was the number of magnetic force lines leaving one pole and re-entering the other are always equal. However, their explanation is simply that there are no magnetic monopoles. What is the correct interpretation of this law and is that all there is to Gauss' magnetic law, that there exist no monopoles?
 Aug 16th 2014, 06:00 AM #2 Physics Team     Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,344 The origination of Gauss's law for magnetic fields is that there are no magnetic monopoles. The form you cited: $\triangledown&space;\cdot&space;\mathbf{B}&space;=&space;0$ expresses that. The divergence operator tells us the magnitude of "stuff" flowing away from, or into, a point - I like to think of it as the magnitude of flow away from or into a source or sink. For example water flowing out of a tap has positive divergence, while water flowing into a drain has negative divergence. If the divergence of the magnetic field is zero, that means that the flow of magnetic field away from every point must equal the flow of magnetic field back into the point. Thus there can be no source of sink of magnetic flux in isolation. This is what is meant by saying that there are no magnetic monopoles. This is different than the case of electric field lines - if you have an electric charge by itself then it can be the source (or sink) or electric field lines, and the divergence of the electric field is equal to the charge density. Regarding the bar magnet example - the sphere that you described does not need to enclose the entire magnet. If you envision a smaller sphere around just the north pole, it would experience magnetic field line flowing from the north pole out through the surface of the sphere to the south pole looping external to the bar magnet. But there is also an equal number of field lines flowing from the south pole back through the bar magnet itself to the north pole. Thus the total flux through the surface of the sphere is always 0.
 Aug 16th 2014, 10:17 PM #3 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Ok, I thought the purpose of the sphere was that it completely enclosed the bar magnet. Are you thinking that the magnetic field lines continue in a closed loop through the centre of the bar magnet? I am not quite getting why the divergence should be zero in such a case if this is not so. Last edited by kiwiheretic; Aug 16th 2014 at 11:16 PM. Reason: I mean "field lines", not "force lines"
Aug 17th 2014, 06:32 AM   #4
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 Originally Posted by kiwiheretic Are you thinking that the magnetic field lines continue in a closed loop through the centre of the bar magnet? I am not quite getting why the divergence should be zero in such a case if this is not so.
Yes, that's right - the field lines form a complete, continuous loop by returning to the north pole through the interior of the magnet itself. The fact that divergence = 0 means the sphere can be infinitesimally small and would still not be a net source or sink of magnetic field lines.

Aug 18th 2014, 08:21 AM   #5
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 Originally Posted by kiwiheretic ...What is the correct interpretation of this law and is that all there is to Gauss' magnetic law, that there exist no monopoles?
Chip has given the "official" answer, and IMHO expressed it well.

But when you dig around, there are some problems with it. There is no actual source or sink. Nothing is actually flowing out of the magnet's North pole and back into the South pole. In John Jackson’s authoritative textbook Classical Electrodynamics you have to wait until section 11.10 before he says "one should properly speak of the electromagnetic field Fμv rather than E or B separately". The thing is this: an electron doesn't have an electric field, it doesn't have a magnetic field, it has an electromagnetic field. If you are at rest with respect to an electron, it seems to have an electric field. But when you move, it seems to have a magnetic field too. You didn't create the latter just by moving, you just saw a different aspect of the electron's electromagnetic field. And whilst that electron is a charged particle and we talk of electric charge, it's actually electromagnetic charge. So there is no such thing as "magnetic charge".

 Aug 18th 2014, 08:46 AM #6 Physics Team     Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,344 Indeed there is nothing "flowing" in a magnetic (or electromagnetic) field - I use that analogy as it makes the discussion about divergence of the field a bit easier to visualize and understand. Similarly it is common to draw lines closer together to represent a stronger field, which vaguely implies that a stronger field has more field lines, but that's not strictly correct either. Yet it's a useful depiction because it's a bit more intuitive - you can see that the number of field lines entering the sphere in question is equal to the number of field lines exiting, and hence Div B is zero. But I agree that it should be pointed out that these are just convenient short cuts.
 Aug 18th 2014, 09:30 AM #7 Senior Member   Join Date: May 2014 Location: Poole, UK Posts: 132 Sorry Chip, I didn't mean to be critical of you, I thought you gave the stock explanation very well. I guess I just don't like the way electromagnetism is taught.
 Aug 18th 2014, 09:36 PM #8 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Those are some interesting points. So are we saying that the magnetic force is a "fictional" force? I understand that a static electric charge has field lines but no magnetic field. However a uniformly moving charge has both but to which observer is the charge moving? Even more confusing if one has two parallel wires with electric current flowing in opposite direction in the wires. The wires are drawn together by the interaction of the magnetic fields between them. However if we only considers the elctric fields the charges should repel as like charges repel each other. Also when only considering the electric field, say from a single electron, we can place a point charge in the vicinity and note the force on the second electron (presumably by its direction and acceleration away from the first electron). However, what kind of "magnetic charge" can we use to test the strength of a magnetic field? As far as I know there is no magnetic point charge. If there was you could place a magnetic point charge at one pole of a bar magnet and watch it "flow" from one pole to the other along a magnetic field line. Thoughts?
Aug 19th 2014, 05:03 AM   #9
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 Originally Posted by kiwiheretic Those are some interesting points. So are we saying that the magnetic force is a "fictional" force?
No, not at all - magnetic forces are real. But the concept of field lines is a mathematical construct - simply a model that helps us do calculations, but not necessarily real. In the quantum mechanical view of electromagnetism there are no fields at all, just particles interacting.

 Originally Posted by kiwiheretic I understand that a static electric charge has field lines but no magnetic field. However a uniformly moving charge has both but to which observer is the charge moving?
Interesting point. To an observer traveling with the charge it is stationary, so no magnetic field. But to an observer in motion relative to the electric charge there IS a magnetic field.

 Originally Posted by kiwiheretic Even more confusing if one has two parallel wires with electric current flowing in opposite direction in the wires. The wires are drawn together by the interaction of the magnetic fields between them. However if we only considers the elctric fields the charges should repel as like charges repel each other.
Wires don't normally have a charge - they have moving electrons, but the charge on the wire at all points is zero, so there is no electrostatic force at play between the wires.

 Originally Posted by kiwiheretic However, what kind of "magnetic charge" can we use to test the strength of a magnetic field? As far as I know there is no magnetic point charge.
The strength of a magnetic field can be tested either in several ways. A couple of possibilities: (a) seeing how much torque is created on a standard bar magnet, (b) passing a current through a wire and seeing how much force is created in that wire, or (c) shooting an electron through the field and seeing how much it is deflected.

Last edited by ChipB; Aug 19th 2014 at 05:43 AM.

 Aug 19th 2014, 05:11 AM #10 Senior Member     Join Date: Apr 2008 Location: Bedford, England Posts: 668 There is some debate over the existance, or otherwise, of magnetic monopoles (the magnetic equivalent of a point charge). Reading the other posts, I get the impression that magnetism is actually as much product of the electron as electricity. If the electron is moving (relative to the measuring equipment) then a magnetic effect will be observed. This seems to me to indicate that the electron is as much the magnetic point charge as it is the electrical point charge. In my imagination, the magnetic monopole would just be an electron, seen from an alternative perspective. In you question about the parallel wires, the effect of all the protons in the wire will cancel out the electric repulsion effect leaving just the magnetic effect. However, the experiment could be re-stated to use two parallel electron beams fired in oposite directions (in a vacuum) I don't know what the result would be, perhaps someone else does... Last edited by MBW; Aug 19th 2014 at 05:13 AM.

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