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I am a bit confused by the description of Halbach arrays. It is said that the line of magnets aligned in certain way results in cancellation of magnetic field on one side of the array, and amplification on the other side. And there is a schematic distribution of magnetic flux - The flux diagram of a Halbach array

I don't understand what happens to the poles of magnets, or the direction of magnetic flux? Does it change its direction in each next or third element of array? Or is it possible to make something like a magnetic monopole, so that the direcition of magnetic flux will be uniform on one side of the array?

There are some other images showing different arrays with arrows which, I suppose, represent the direction of the magnetic flux.

Diagram 1 Diagram 1 Diagram 2 Diagram 2

The Diagram 2 looks like a monopole, because there is the same direction of flux all the time? Is it true, or am I misunderstanding something?

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Sorry, a monopole is a pointlike or localized object whose magnetic field is divergent in all directions or convergent in all directions on the surrounding sphere. This is hard and no magnetic monopoles have been seen yet. However, having a magnetic field going in the same direction inside a tubular region isn't a problem - a long magnet or an ordinary solenoid has the property, after all. –  Luboš Motl Dec 19 '12 at 12:12
    
Yes, that is right, but that is solenoid. And I wonder about creating such magnetic field from permanent magnets. –  BartoNaz Dec 19 '12 at 12:16
    
What I have found up to now, is that arrow from Diagram 2 shows not the direction of magnetic flux, but the trajectory of movement of the electron in the free-electron laser. And since it is sinusoidal, the magnetic flux has not uniform direction along the array. –  BartoNaz Dec 20 '12 at 16:22
    
The arrow in diagram 2 is the magnetic flux direction, but there is problem of the electron path in this diagram, see answer below –  hwlau Dec 26 '12 at 5:25
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2 Answers 2

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No, the Halbach array is not related to the monopole at all. It is just an special arrangement of magnet.

Any arrangement of a "dipole" magnet would not result in any monopole. It can be easily understood as follows: If each individual magnet contains no magnetic monopole, i.e., $\nabla\cdot\mathbf{B}_i=0$ true for whole space, then the resulting magnetic field is $\mathbf{B} = \mathbf{B}_1 + \mathbf{B}_2$ when you put them together. Hence the resulting divergence is also $\nabla\cdot\mathbf{B}=0$.

I don't understand what happens to the poles of magnets, or the direction of magnetic flux? Does it change its direction in each next or third element of array?

You should know that magnetic pole is not a very well defined concept as the "boundary" of the magnetic pole is not fixed. The flux line does change the direction across the boundary. But the most important things (in the first figure) is that the magnetic field line is continuous and closed, which means that there is no magnetic monopole.

All the field lines go out have to come back at the next nearest neighbor. The effect of Halbach array is simply to push the closed magnetic field line on one side, hence the resulting magnetic field was enhanced on one side.

The Diagram 2 looks like a monopole, because there is the same direction of flux all the time? Is it true, or am I misunderstanding something?

No, the flux direction is not in the same direction over the whole region, they are alternating instead. You can think of it by putting two array in diagram one (flipped one of them) together. Each of them forming a small field line loop. At the center of two up arrows ($\uparrow$) the magnetic field is pointing up. Also, at the center of two down arrows ($\downarrow$), the magnetic field line is pointing down. Hence, it is clear that the field line is alternating after each next nearest neighbor. The wiggler of the electron path is the result of the alternating magnetic field.

I think there is a problem for the Diagram 2, the oscillating frequency is double of what it is supposed to be. It should be the same as the period of the array.

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Thank you for a detailed explanation. –  BartoNaz Dec 26 '12 at 13:02
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No wonder you are confused, because the diagram ia a cheat ! If you take the picture of the arrangement, with the labels of to and bottom, and turn it by 180 degrees, you get exactly the same arrangement but either displaced by 1 period, or simply with the magnetic lines in opposite direction, but now suddenly the strong field and weak field is reversed. In the picture of the magnetic lines as presented, direction is not shown. The bar is symmetric and there`s no way the field can be unsymmetric. The true arrangement must be something unsymmetric, which is not shown. Regards, Walter Hassenpflug

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Regardless of the orientation, there is still a weak side and a strong side. And if you look close enough, all three images contain the direction is drawn with a single arrow (in the first image, it's a greenish-blue arrow in the center of each block, more clear in the other two). –  Kyle Kanos Feb 4 at 13:47
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