Understand Magnetic Field Divergence: Nabla dot B =0 Explained

[rockyshephear]

I prefer to discuss it in terms of a bar magnet. Can we then say that the number of flux lines leaving the north is equal to the number of flux lines entering the south end? If so, can you rearrange the equation such that you have something on the left hand side of the equation equaling something on the right hand side of the equation. Then I can follow your word statement in terms of the actual equation.

 

[diazona]

Thanks to all trying to help me get a grasp on this.

When I see an equation like below, I like to read it in terms of words since I'm much more a visual thinker than an abstract thinker.

LaTeX Code: \\oint\\limits_S \\vec{B} \\cdot d \\vec{A} = 0

So the above equation is saying...

The surface integral of a magnetic field, dot producted with the rate of change of the vector A = zero.

There is no rate of change of the vector A. (If something like \frac{\mathrm{d}\vec{A}}{\mathrm{d}t} were in this equation, that would be the rate of change of the vector A.) That A is just an "area vector". \mathrm{d}\vec{A} just represents an infinitesimally small patch of area; the vector itself points perpendicular to the patch and has a magnitude equal to the (infinitesimal) area of the patch.

Here's how I might put it: the sum (\oint) of the magnetic flux (\vec{B}) perpendicular to (\cdot) each infinitesimal patch of area (\mathrm{d}\vec{A}) on the surface is equal to zero (=0).

I prefer to discuss it in terms of a bar magnet. Can we then say that the number of flux lines leaving the north is equal to the number of flux lines entering the south end?

We can, if and only if we assume that all flux lines that leave the magnet do so through the north end, and that all flux lines that enter the magnet do so through the south end.

If so, can you rearrange the equation such that you have something on the left hand side of the equation equaling something on the right hand side of the equation. Then I can follow your word statement in terms of the actual equation.

How about this:
\oint_\text{north} \vec{B} \cdot \mathrm{d}\vec{A} = -\oint_\text{south} \vec{B} \cdot \mathrm{d}\vec{A}
This is saying that the amount of flux leaving the north end of the magnet is equal to the amount of flux entering the south end.

 

[Phrak]

Thanks. So magnetic monopole flux lines look just like the electric field around a single point charge. So Would there be any difference whatsoever. Maybe a single point charge IS the elusive magnetic monopole! :)

I'm going to back up because you where pretty close to it, here.

A magnetic monopole would look like the picture of the electron, but the fields would be the magnetic fields, rather than the electric fields.

The amount of electric field coming out of a spherical surface tells you how much charge is inside. The same goes for magnetic charge. The amount of magnetic field coming out of the spherical surface tells you how much magnetic charge is contained inside.

For this to work we'd have to change Gauss' Law for Magnetism.

\oint\limits_S \vec{B} \cdot d \vec{A} = 0

would become

\oint\limits_S \vec{B} \cdot d \vec{A} = Q_{m} \; ,

where Q_{m} is magnetic charge.

As no one has found any magnetic charge, or magnetic monopoles, one should have to say why magnetic charge has remained hidden.

 

[rockyshephear]

So magnetic flux always has a divergence of zero because the flux lines are continuous, meaning if x lines leave a magnet, x lines come back into it.
Can you think of any other real world system whereby the divergence is zero? Is magnetism special because of this? Does the current in a simpel (battery, resistor circuit) have a zero divergence because the current the leaves a part of the wire, comes back due to the circuitous nature of the circuit? Do horse on a track have zero divergence since they leave one part of the track and come back around do it?
I think I'm getting confused with the surface that you measure the sink and source from. Does it matter if its a 2D surface shaped like a peanut or a 3D cylinder? So many questions.

 

[rockyshephear]

No one really know what differentiates a magnetic field from an electric field, right? An positive point charge will attract a negative point charge. The south end of a magnet attracts the north end of a magnet. What is the essential difference? I think it's making some sense.

 

[rockyshephear]

You know what makes the study of this so difficult? It never seems to have any practical examples whereby you can gain a deeper appreciation for the math based on observing the real world system, and a problem that is solved by the use of these equations.

As far as Gauss' Law goes, all I have to go on is a visual of a bar magnet and the flux coming out one end and going back in the other. Is that sufficient proof of the law? lol

Yet, I don't know what goes on in the length of the magnet. Do the lines continue?

 

[rockyshephear]

If you take a magnet and cut it down all the way to an atom, would you have a singular magnetic charge?

 

[Born2bwire]

You get two answers from me.

No one really know what differentiates a magnetic field from an electric field, right? An positive point charge will attract a negative point charge. The south end of a magnet attracts the north end of a magnet. What is the essential difference? I think it's making some sense.

We do know the difference between magentic and electric fields. The behaviors and properties are very distinct. They are produced by different phenomenon (despite being coupled) and react differently with charges and materials (polarization vs. magnetization for example). The most elementary source for an electric field is a monopole, for a magnetic field it is a dipole. Current theory does not allow for monopole magnetic sources, although the use of ficticious magnetic currents are sometimes used for calculation and simulation purposes (plus it makes things nice by balncing Maxwell's Equations but again this is not meant to be physically correct).

If you take a magnet and cut it down all the way to an atom, would you have a singular magnetic charge?

No. There is no magnetic charge, the basic source of a magnetic field is a dipole. No matter what you will just get smaller and smaller dipoles.

 

[rockyshephear]

So how would you make the smallest, most reductivist magnet since you need a dipole. Do you need 2 quarks, 2 atoms, side by side? Does it depend on which element of the periodic table or can you make a manet from things smaller than the atom. I understand that magnetizing something aligns atoms somehow? How many atoms aligned to you need minimally? What is the smallest magnet possible?

 

[Phrak]

No one really know what differentiates a magnetic field from an electric field, right? An positive point charge will attract a negative point charge. The south end of a magnet attracts the north end of a magnet. What is the essential difference? I think it's making some sense.

Without getting into details, the electric and magnetic fields can be described by a more primative vector field. Still roughly speaking, the electric field is the divergence of this field, and the magnetic field is the curl. Because of this the divergence of B is zero--no magnetic charge.

There's at least way to both describe electromagnetism using this primative field, and allow magnetic charge, leaving Maxwell's equations relatively intact, but additional assumptions on the structure of spacetime are required.

 

[rockyshephear]

Hi Phrak,
You're comments are very interesting. This fundamental vector field to which you state that both the electric field and magnetic field arise (the former based on divergence and the latter, curl)...please do get into detail. I've never heard of this before.

primative field with zero divergence = no charge
primative field with curl = magnetism
(what if the field has both curl and divergence?)

Thx

 

[Bob S]

...Current theory does not allow for monopole magnetic sources...

Current theory does allow for monopoles. There is no theoretical proof that monopoles cannot exist.

 

[rockyshephear]

I think it's fair to say that magnetic flux is an infinite sphere weakening with distance from the center of radiation, but traveling infinitely thru the universe. Sound right?

I can't imagine why anyone would conceive of a magnetic monopole since it's antithetical to the definition of a dipole radiation. That's like conceiving of a dog with wings. Science has yet to find one.

 

[Phrak]

Current theory does allow for monopoles. There is no theoretical proof that monopoles cannot exist.

True. I was speaking of classical theory.

 

[Phrak]

Hi Phrak,
You're comments are very interesting. This fundamental vector field to which you state that both the electric field and magnetic field arise (the former based on divergence and the latter, curl)...please do get into detail. I've never heard of this before.

primative field with zero divergence = no charge
primative field with curl = magnetism
(what if the field has both curl and divergence?)

Thx

Look up "vector potential". You should find references all over the internet.

 

[rockyshephear]

Drawing A shows the geometric orientation of the lines of flux in the vicinity of an electrically charged object. The intensity of the field is inversely proportional to the separation between the lines of flux. The flux density , and hence the electrostatic field strength, decreases as the distance from the charged object increases. Electrostatic flux density is inversely proportional to the distance from the charge center.

This seems to say that flux density relates to distance between lines of flux. But I understood vectors to be at every point in a magnetic field. Which is it? Lines of flux like strings forming an apple like geometry with distance between each line of longitude or the skin of the apple which is discontinuous?
Thanks for all the help.
Rock

 

[Count Iblis]

Actually, we can be almost sure that monopoles do exist:

http://en.wikipedia.org/wiki/Magnetic_monopole

Within theoretical physics, modern approaches agree that monopoles exist. In particular, Grand Unified Theories and string theory both require them. Joseph Polchinski, a prominent string-theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".

One of the successes of inflation theory is that it solves the monopole problem of standard Big Bang theory, i.e. that so many monopoles should have been produced that the universe should have collapsed soon after the Big Bang.

The existence of magnetic monopoles also limits how strong the magnetic field strength can be, see page 9 of this article:

http://arxiv.org/abs/astro-ph/0002442