Understand Magnetic Field Divergence: Nabla dot B =0 Explained

[rockyshephear]

nabla dot B =0 ??

I've read the physical explanation for this eq is that magnetic monopoles do not exist. A poor explanation in my opinion. :) So, I would like it explained along these lines. (Obviously I don't unuderstand this but am giving an example of how I would like it explained). Thanks!

nabla dot B =0 means The divergence of a magnetic field is zero because at any given point in space inside a magnetic field the tangent line on the curve of the magnetic field line is always perpendicular to the direction of the current causing the magnetic field.

I would say that magnetic fields have divergence because they weakening with distance. So the geometric structure of the direction and magnitude of each vector is different than it's neighbor. Isn't that the definition of divergence?

What am I missing here. and please...in Sesame Street terms. lol
Thx

 

[dx]

I would say that magnetic fields have divergence because they weakening with distance. So the geometric structure of the direction and magnitude of each vector is different than it's neighbor. Isn't that the definition of divergence?

That's not the definition of divergence. The divergence of a vector field A(x,y,z) is defined as

\nabla \cdot A = \partial_{x} A_{x} + \partial_{y} A_{y} + \partial_{y} A_{y}

 

[rockyshephear]

Excuse me but that's exactly what "I" said.
Thx

 

[Pinu7]

If the divergence of a scaler function is the zero vector, the function is constant since the partial derivatives with respect to x,y and z are all zero.

Answer your question?

 

[rockyshephear]

But B is not a scalar function. It's a magnetic field which is a vector function. I need an explanation in 'words' as to how a magnetic field's divergence is zero. Thx

 

[Vanadium 50]

Rocky, you seem to want an explanation in words as to why your mental vision of what a divergence is doesn't match the mathematical definition. I suspect that it will be difficult to change the mathematical definition to agree with your mental vision, so it's probably best to do the reverse. Have you read the book Div, Grad, Curl and All That? It's by Schey.

 

[G01]

Excuse me but that's exactly what "I" said.
Thx

No it wasn't.

The divergence of a vector field does not measure the change in magnitude of the field as you move farther away from the origin. So, even though the B field gets weaker the farther away you go from the origin, it doesn't mean that the field has non zero divergence. That is just not what divergence measures.

In simple, graphical, non-quantitative terms the divergence at a point measures the tendency of the vectors in a vector field to point towards or away from that point That is, if the field vectors tend to point towards or away from one point (the sink or source) then the field has a divergence at that point. If the divergence is positive, then the vectors tend to point away from that point. If the divergence in negative, they tend to point towards it. If the divergence is zero there is no tendency to point towards or away. Either the vectors point in directions that circle around the point or there are equal numbers pointing towards and away.

You will notice that all of the magnetic systems you've encountered in classes fall into the third category. They have no tendency to point towards or away from any given point. Thus, the divergence of a magnetic field is 0:

\nabla \cdot \vec{B} =\vec{0}

 

[dx]

If the divergence of a scaler function is the zero vector, the function is constant since the partial derivatives with respect to x,y and z are all zero.

Answer your question?

Divergence is not defined for scalar functions. Maybe you're thinking of gradient.

 

[rockyshephear]

No. Actually I'm thinking of Divergence. The tendency for fluid particles to leave a source or enter a source, for example. Fluid and Magnetic flux are vector fields. The degree to which they spread out or converge, to me, is divergence.
How does magnetic force decrease with distance if there is no divergence. 2nd question: Does magnetic flux decrease with distance because of the charges becoming smaller with distance or because lines of flux have greater space between them?
Thx

 

[nrqed]

No. Actually I'm thinking of Divergence. The tendency for fluid particles to leave a source or enter a source, for example. Fluid and Magnetic flux are vector fields. The degree to which they spread out or converge, to me, is divergence.

This is where your interpretation is invalid. You can have a vector field which spreads out or converges and still have the divergence of that vector field equal to zero. To picture the divergence, build a small volume element. If the number of field lines entering is equal to the number of field lines leaving, the divergence is zero. You can clearly have a divergence equal to zero even if the fields lines spread out or converge toward one another.

 

[rockyshephear]

Now you've confused me. So you're saying a vat of water with a hole at the bottom and water coming in the top can be described with the concept of divergence. If 100 gals per min enters and 100 gals per min exits, then that is divergence? What about the direction of the infinite elements as they enter and exit?
Maybe divergence in that case is misnamed. Maybe it should be called zero sum movement or something else.
Thx

 

[rockyshephear]

It seems to be that this Wikipedia definition is contrary to your statement.

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Thx

 

[rockyshephear]

But regardless, how does all this relate to magnetic flux? Is it saying that for every line of flux exiting the end of a dipole, you have one coming back in the other end?
Thx

 

[diazona]

Now you've confused me. So you're saying a vat of water with a hole at the bottom and water coming in the top can be described with the concept of divergence. If 100 gals per min enters and 100 gals per min exits, then that is divergence? What about the direction of the infinite elements as they enter and exit?
Maybe divergence in that case is misnamed. Maybe it should be called zero sum movement or something else.
Thx

uh... it's sort of like that. More precisely, if water enters the tank at the same rate it leaves, then the integral of the divergence of the flow over the volume of the tank is zero. Technically divergence is defined at each particular point, so you could have negative divergence at one point (if water is building up at that point) and positive divergence at another point (if water is draining away from that point), but over the whole tank it would all cancel out.

It seems to be that this Wikipedia definition is contrary to your statement.

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Thx

I don't understand the contradiction you're seeing. The Wikipedia article is consistent with what everyone else has been telling you. Maybe you can explain in more detail why you think there is a contradiction?

But regardless, how does all this relate to magnetic flux? Is it saying that for every line of flux exiting the end of a dipole, you have one coming back in the other end?
Thx

Yeah, but it's not just true for dipoles - it's true for any region of space (no matter whether there is something in that space or not). For every line of flux that exits the region, you have one coming into the region.

 

[rockyshephear]

I'm thinking of a magnet and lines of flux drawn around it like you see in all the illustrations.
If I pick a point in space on a flux line, of course the line enters that point and leaves that point. But if nabla dot B did not equal zero what would the physical description look like? A flux line entering a point on a flux line and the line stopping dead...right at the point?
I'm not sure my question is being understood.
Thx

 

[rockyshephear]

Plus the Wikipedia definition does not deliniate anything to do with DIRECTION of the vectors in the concept that is being integrated as far as sink and source is concerned. I think I need a very accurate physical definition of DIVERGENCE. I'm not sure engineers can think in terms of images vs mathematical statements. Maybe I need an engineer who is also an 3D artist to answer my question. lol
Thx

 

[rockyshephear]

One final thing. I always thought of INTEGRAL as area under a curve and DERIVATIVE as rate of change. So why isn't DIVERGENCE a function of the derivative and not an integral since we are talking about the rate of change of some fluid exiting vs entering a system? Or is it INTEGRAL because we are discuss the volume the enters vs exits?
Thx

 

[diazona]

I'm thinking of a magnet and lines of flux drawn around it like you see in all the illustrations.
If I pick a point in space on a flux line, of course the line enters that point and leaves that point. But if nabla dot B did not equal zero what would the physical description look like? A flux line entering a point on a flux line and the line stopping dead...right at the point?
I'm not sure my question is being understood.
Thx

Yes, that's right: if \vec{\nabla}\cdot\vec{B}\neq 0, a flux line would enter the point and stop right at that point. There would probably also be another flux line entering from the other side and also stopping right at the point.

Plus the Wikipedia definition does not deliniate anything to do with DIRECTION of the vectors in the concept that is being integrated as far as sink and source is concerned. I think I need a very accurate physical definition of DIVERGENCE. I'm not sure engineers can think in terms of images vs mathematical statements. Maybe I need an engineer who is also an 3D artist to answer my question. lol
Thx

Vectors always point away from a source and toward a sink. Is that what you're confused about?

Here's a website that may be of interest of you: http://www.math.umn.edu/~nykamp/m2374/readings/divcurl/
It shows examples of positive and negative divergence.

One final thing. I always thought of INTEGRAL as area under a curve and DERIVATIVE as rate of change. So why isn't DIVERGENCE a function of the derivative and not an integral since we are talking about the rate of change of some fluid exiting vs entering a system? Or is it INTEGRAL because we are discuss the volume the enters vs exits?
Thx

The divergence is a derivative.
\vec{\nabla}\cdot\vec{B} = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}
But divergence is something that depends on position. It might be positive at one point, negative at another point, zero at another point, etc. In the fluid flow example (with the water in the tank), what I was saying is that if you add up the divergences at every point in the tank, the total would be zero (as long as water is entering at the same rate it's leaving, of course). That's what an integral is: adding up of a bunch of infinitesimal points. (Getting the area under a curve is just one use of an integral - it's basically adding up the areas of all the points under the curve)

But with water flow, the divergence could be positive at some points and negative at some others, as long as the total is zero. With magnetic flux, however, \vec{\nabla}\cdot\vec{B} = 0 means that the divergence is zero at every point.

 

[rockyshephear]

Ah. So the meaing of
LaTeX Code: \\vec{\\nabla}\\cdot\\vec{B} = 0
is that at any given point in a sphere of magnetic flux, if you pick a point, the integral of the divergences equals zero. So, if we equate this to forces that move an object, the point object would have zero force in anyone direction and as such the point object would not move in any direction.

How about a sphere in the river analogy. Water passing by it tends to move it downstream. If
LaTeX Code: \\vec{\\nabla}\\cdot\\vec{B} = 0
B were the flow of water, then there is an equal amount of water flowing uphill, flowing in from the right and left and any of infinite angles, all the same magnitude or at least the total sum is such that the ball stays in a stable position?

 

[Bob S]

For comparison, we should first look at div ε₀E = div D = ρ_e.
The divergence of E represents the existence of the electric monopole, which is -e. So why then is div B = 0? (why is there no magnetic monopole?)
Maxwell's equations would be symmetric if there were a magnetic monopole. I know Luis Alvarez in the early 1960s was grinding up meteorites found in Antartica and spinning them in toroids (using compressed air), looking for a dc output voltage. So he thought they may exist. He never found any.
~ ≈ ≠ ≡ ≤ ≥ « »
α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[rockyshephear]

One thing that confuses me are illustrations of vector fields. They show maybe 59 vectors in a space. Are these infinite such that the space between these vectors has more vectors but just are not shown for purposes of clarity? Or do you finally get a point at the size of an atom or below where there are no more vectors in the inbetween spaces?

 

[rockyshephear]

What would a magnetic monopole look like. A bar which emits flux in sort of a hemisphere around the bar but not being attracted to the other end of the bar?? Maybe like an umbrella? It almost seems logical that there are no magnetic monopoles if you think of flux as like current which needs a complete path to maintain a current. You need a north and a south to complete flux. Is this correct?

 

[Bob S]

What would a magnetic monopole look like. A bar which emits flux in sort of a hemisphere around the bar but not being attracted to the other end of the bar?? Maybe like an umbrella? It almost seems logical that there are no magnetic monopoles if you think of flux as like current which needs a complete path to maintain a current. You need a north and a south to complete flux. Is this correct?

If you integrate the electric field E normal to the surface of a sphere surrounding an electron, you get -e/ε₀. Because the electron is a point electric monopole, the sphere can be any size, as long as it is greater than about an electron Compton wavelength, you will get the same answer. So if the magnetic monopole were a point monopole also, and you integrated the normal component of B on the surface of a sphere surrounding it, you would get q_m (the monopole "charge"). The sphere might have to be greater than a (monopole) magnetic Compton wavelength, in order to avoid renormalization effects.

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[rockyshephear]

Thanks. So what would it look like?

 

[diazona]

If you drew the magnetic field around a magnetic monopole, it would look like a bunch of lines of flux emanating from a point in all directions. (Not just a hemisphere.) It would look like this (the picture is of an electric field around an electric monopole, but the magnetic field around a magnetic monopole would look the same).

Also, in the illustrations of vector fields: they are "infinite" as you say. That is, in reality there are vectors at all points, not just the 59 or however many that are drawn. Obviously, you could not draw all the vectors at every single point; the picture would be entirely black. So when you're drawing a vector field, you just pick a few points and draw the vectors at those points.

 

[rockyshephear]

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! :)

 

[Phrak]

But regardless, how does all this relate to magnetic flux? Is it saying that for every line of flux exiting the end of a dipole, you have one coming back in the other end?
Thx

I believe I understand the source of your confusion from your original question.

First, though, I hope you realize that there are not physical lines of force, separated in space from one another, but a continuous field. Just making sure.

As you follow along the direction of the field, it might spread-out and become weaker with distance as you say. But the area it passes through is now greater.

So, say we have this tube that is bigger on one end than the other. It's surface lies tangent to the field. The ends are caped with surfaces perpendicular to the field The product of the field strength times the area on each end of the tube will be equal.

What I am hinting at is the integral form of the equation. It's equivalent to the differential equation.

Look for Gauss' law of magnetism
http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/maxeq.html"

 

[arunma]

Rocky, I think others have alluded to this, but it may help to consider the integral form of Gauss' Law for magnetic fields, which often makes more conceptual sense if you're not an expert on vector differential calculus,

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

This states that the number of field lines which enter any closed surface S will always equal the number of field lines leaving that surface. This is because the magnetic field has no sources (i.e. places from which magnetic fields originate). Does this make sense?

BTW: honestly the only way I understand the divergence is in terms of surface integrals. You might want to check out the top part of the http://mathworld.wolfram.com/Divergence.html" on divergence. In essence, the divergence is defined as the surface integral per unit volume around a point.

 

[rockyshephear]

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.

Or
The degree to which- the sum of some infinite calculation related to a surface- points in the same direction as the rate of change of some vector A. Since this degree is zero, that means the angle between these two poorly described vectors is 90 degrees.
If I am not right on with this, could you please refine the sentence so as to make it a more correct wording of the equation.

Thanks

 

[diazona]

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! :)

No it's not. A point charge is an electric monopole, not a magnetic monopole.

 

[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