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Can someone explain Gauss' Law for Magnetism

  1. Jul 5, 2009 #1
    I need an explanation that relates the equation to what I learned about the dot product of two vectors being the magnitude of the parallelogram formed by the two vectors in 2D or similarly by the paralleliped formed by three vectors in 3D.

    I want to know for instance if the resultant is normal to the 2 vectors in 2D, how can a resultant be normal to three vectors. Wouldn't you require a fourth dimension?

    So basically I want to look at Gauss' Law and see how Nabla relates to the first vector, dot is the operator and B relates to the second vector. Then I want to see a new resultant vector that is normal to both Nabla and B...and see how that is zero.

    I believe the resultant is zero when the two vectors are pointing in the same direction.

    Can someone clear this up for me?

    Thanks
     
  2. jcsd
  3. Jul 5, 2009 #2

    Doc Al

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    Sounds like you are confusing the "dot product" (a scalar product) with the cross product (a vector product). The dot product of two vectors is a scalar, not a vector.
     
  4. Jul 5, 2009 #3
    I think nabla is an operator and in some way behaves like a vector, however nabla is not a real vector,so you don't need a fourth dimension to contain it and I don't think "normal" is defined for nabla, instead it's called "divergence free" or "curl free"
     
  5. Jul 5, 2009 #4

    Doc Al

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    Yes, nabla is a vector operator. The notation del dot B, defines the divergence of the vector field B; you can think of it as the scalar product of the vector operator and the vector B.
     
  6. Jul 5, 2009 #5
    Oh poop. You are right. Let me please rephrase.

    I need an explanation that relates the equation to what I learned about the dot product of two vectors being the degree to which two vectors point in the same direction.

    So nabla, dot producted with B is the sum of the rates of change of B to the x, y, and z unit vectors. Is this right? What in laymens term does this mean? I can't get my head around the meaning of this.

    Thanks
     
  7. Jul 5, 2009 #6
    Just regard nabla as :[tex]\nabla[/tex]=[tex]\frac{\partial}{\partial x}[/tex][tex]\hat{i}[/tex]+[tex]\frac{\partial}{\partial y}[/tex] [tex]\hat{j}[/tex]+[tex]\frac{\partial}{\partial z}[/tex][tex]\hat{k}[/tex]
     
  8. Jul 5, 2009 #7
    So what does nable dot B mean in words???
     
  9. Jul 5, 2009 #8
    Intuitively that means "how many" magnetic lines are "flowing out" from a small closed surface.
     
  10. Jul 5, 2009 #9
  11. Jul 5, 2009 #10
    How many magnetic lines flowing out. According to what I've read here its infinite. Unless flux lines have distance separation. Are they infinite or finite? And if finite, what equation defines how far apart the lines of flux are apart????????
     
  12. Jul 5, 2009 #11
  13. Jul 5, 2009 #12
    Obviously it's way to difficult a concept even for the best of Physics Forum to put into words. Imagine the planet described by longitude and latitude. These have finite separation. Is magnetism the same way or are there infinite lines of magnetism? Simple question. Finite or Infinite. Thx
     
  14. Jul 5, 2009 #13
    Well,the answer should be infinite. But magnetic line is not an essential description for magnetic fields when you get down to Maxwell's equation, so I only use it for an intuitive description purpose. To understand what divergence is, you should read the wikipedia article with patience
     
  15. Jul 5, 2009 #14

    dx

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    The divergence of a vector field is the flux density. This means that if you have a small volume dV, then the flux of the vector field out of that volume is (divergence)dV.
     
  16. Jul 5, 2009 #15
    small volume of what??? 2D area or 3D volume? What shape is the volume?? Like a box or a cone or a cylinder? What is small? A foot? A nanometer? How do flux lines interact with volume? Do they leave the volume perpendicular to the surface or at any angles? How do they come into a volume? Does each dipole emit it's own shape of flux which interferes with it's neighboring dipole? I need these kind of questions answered if anyone can do that.
    Thx
     
  17. Jul 5, 2009 #16

    dx

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    By small, I mean 'infinitesimal' in the calculus sense. So the flux out of any volume V will be

    [tex] \int_{V} (\nabla \cdot B)dV [/tex]

    The rest of your questions are best answered by a textbook.
     
    Last edited: Jul 5, 2009
  18. Jul 5, 2009 #17

    jambaugh

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    It is easiest to visualize the various vector derivative operations in terms of fluid flow.

    Imagine we are considering air in dynamic motion within a room. At an instant in time we imagine a vector field expressing the flux of air, F. You may think of F(x) as the density times the air velocity at a given point.

    Now the divergence of F, [itex] \nabla \bullet F[/itex] expresses the expansion of air at a given point. To maintain a steady-state positive divergence at a point we would have to have a source of air. Imagine say a speck of frozen air evaporating at that point. (And of course we can have the reverse, a negative divergence with a spec of frozen air growing) We can however allow a transient divergence at a point which must with conservation of the flowing quantity correspond to a change in concentration. We have the classic conservation equation:

    [tex] \nabla \bullet F = -\frac{d}{dt}\rho[/tex]
    Divergence equals the decrease in density over time if the vector field expresses the current or flux of the quantity.

    Similarly one can consider the vorticity of a fluid flow via the curl. Imagine a hurricane. Draw a circle around the hurricane and you'll note that as you trace around this circle the wind is always flowing in the direction you are tracing. Thus there is a net curl inside this circle (crossing any surface bound by the circle. But be careful here, the curl will not be distributed through out the hurricane but rather is concentrated along the eye-wall. Draw a small circle in the storm but off of the eye-wall and the curl inside will be near zero. As you trace along the circle you'll see for a larger part of the circle farther from the eye you get the wind traveling one way but for the smaller part nearer the eye where the wind is reversed you also have stronger winds. These cancel out. You'll see that the source of the curl is actually the dynamic part of the storm where energy is being converted from heat of the ocean to wind.

    Finally the easiest to visualize is the gradient of say a scalar density.
    [tex] \nabla \rho[/tex]
    expresses a vector in the direction of maximum increase of the density with length equal to the rate of increase in that direction. For a smoothly changing density you'll also see that the rate of increase in an arbitrary direction (unit vector u) will be:
    [tex] u\bullet \nabla \rho[/tex]

    With these expositions you should then take a look at Stokes and Gauss' laws. I in fact was invoking Stokes formula implicitly when describing the curl in terms of paths in the hurricane.

    You may find it instructive, with these fluid analogues of the vector operations to then look at Maxwell's mechanical model. He imagined space filled with little ball bearings (two sizes) which rotated against each other as well as moving about each other. The smaller bearings expressed flowing charges (with relative density expressing charge density) and the rotation of the other expressed magnetic fields. This is of course a model and not to be taken seriously as a physical description of reality. But it gives a good scaffolding to understand the interplay of electric and magnetic fields.
     
  19. Jul 5, 2009 #18

    nrqed

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    Just to add to this explanation. The divergence is the NET flux flowing out of an infinitesimal volume. So it is the flux leaving minus the flux entering.
     
  20. Jul 5, 2009 #19
    I'm going to persist with my inquiry.

    If you have the smallest dipole magnet possible that is perfectly stationary with respect to the earth but it's in a vaccum or in sometype of container that no type of magnetism or radiation of any kind may penetrate-and magnet is 1 inch wide, 2 inches tall and .5 inches thick.......describe in the most detail possible to science the magnetic flux in and around the magnet.

    My assumption is that the magnetic flux would be shaped the the letter C with a backwards C beside it. Or like a pair of ears if you will, meeting along the centerline of the magnet. I assume it radiates to infinity and it's force covers all parts of the xyz coordinate system (ie no space between flux lines) but is only experimentially appreciable with a few inches of the magnet. What the flux is made up of, I have not idea. Is it like gravity? Or the exchange of gluons like in strong force? Or is it just a manifestation of the electric force?

    Can anyone attempt to answer this without ref to equations. This kind of information does not appear to be available anywhere.
     
  21. Jul 5, 2009 #20

    diazona

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    Seems like you have a contradiction there.

    You've got another contradiction there. Describing anything in "the most detail possible to science" means using math - in fact, math was invented to describe nature, since no other method of description was sufficiently detailed.
     
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