Nabla dot B =0 ?

1. Jul 1, 2009

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

Last edited by a moderator: Jul 1, 2009
2. Jul 1, 2009

dx

Re: nabla dot B =0 ??

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

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

3. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

4. Jul 1, 2009

Pinu7

Re: nabla dot B =0 ??

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.

5. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

6. Jul 1, 2009

Staff Emeritus
Re: nabla dot B =0 ??

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.

7. Jul 1, 2009

G01

Re: nabla dot B =0 ??

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:

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

Last edited: Jul 1, 2009
8. Jul 1, 2009

dx

Re: nabla dot B =0 ??

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

9. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

10. Jul 1, 2009

nrqed

Re: nabla dot B =0 ??

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.

11. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

12. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

13. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

14. Jul 1, 2009

diazona

Re: nabla dot B =0 ??

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.

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?
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 in to the region.

15. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

16. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

17. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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

18. Jul 1, 2009

diazona

Re: nabla dot B =0 ??

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.
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.
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.

19. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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 any one 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?

20. Jul 1, 2009

Bob S

Re: nabla dot B =0 ??

For comparison, we should first look at div ε0E = div D = ρe.
The divergence of E represents the existance 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

21. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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?

22. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

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?

23. Jul 1, 2009

Bob S

Re: nabla dot B =0 ??

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 qm (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

24. Jul 1, 2009

rockyshephear

Re: nabla dot B =0 ??

Thanks. So what would it look like?

25. Jul 1, 2009

diazona

Re: nabla dot B =0 ??

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.