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Magnetized cylinder magnetas a Magnetic Dipole
Pace / April 9, 2007
a magnetic dipole could be produced by running an ongoing through a cycle of line. Most permanent magnets are club magnets, but that will always be frequently known as producing a dipole field. In this topic we shall examine a simple instance where the magnetization of a cylinder magnetis regularly figure out (and show) the resulting magnetic area.

Think about here a cylinder magnetof length, L, and radius, R, where in actuality the ratio of those values is supposed to be set at three different values. We’ve a quick and fat cylinder magnetin which L << R, a long and skinny cylinder magnetin which L >> R, and an almost cubic cylinder magnetin which L ≈ R. Each of these cylinders functions a magnetization, M, magnetized hooks is provided by M = α z, in which α is a constant. The system vector z is parallel to L, the axial direction for the cylinder. We’re going to quantitatively determine the magnetic industry stated in every one of these situations.

Figure 1 displays the geometry with this subject. The cylinder magnetis proven to provide perspective for different machines becoming considered.

cylindrical geometry
Figure 1: Setup with this topic showing the cylinder magnetand its magnetization.

Once the magnetization of an item is provided, one method magnetized hooks may be used to determine its magnetic industry involves resolving because of its bound currents. The quantity, Jb, and surface, Kb, bound currents are regarding the magnetization by,

\vecJ_b = \vec\nabla \times \vecM \ \ \vecK_b = \vecM \times \hatn
where n signifies the vector typical to virtually any surface for the cylinder magnet(in other words. every individual area of cylinder magnethas a unique vector typical and for that reason unique bound area current).

The certain currents represent every one of the existing in this system∗. The bound amount existing is fixed for as (including the complete cylindrical coordinates curl phrase, which can be always a helpful reference),

\vecJ_b = \vec\nabla \times \alpha\hatz
= \left[ \frac1r\frac\partial M_z\partial \phi – \frac\partial M_\phi\partial z \right] \hatr + \left[\frac\partial M_r\partial z – \frac\partial M_z\partial r\right] \hat\phi + \left[\frac1r\frac\partial\partial r\left(rM_\phi \right) – \frac1r\frac\partial M_r\partial \phi \right]\hatz \ \ \ = \left[ \frac1r\frac\partial\alpha\partial \phi – 0 \right]\hatr + \left[0 – \frac\partial \alpha\partial r \right]\hat\phi + [0-0]\hatz \ \ \ = 0
together with conceptual option to understand this zero outcome is magnetized hooks a consistent area has no curl.

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Bound surface currents may occur on the cylindrical area and also on either circular face. For cylindrical area we now have,

\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
plus the Φ way precisely defines the cylindrical surface so this is a literally reasonable result.

Circular faces are found at z = ± L/2. The bound area current at these faces is, first for z = +1/2,

\vecK_b+ = \alpha\hatz \times \hatz = 0
and then for z = -1/2,

\vecK_b- = \alpha\hatz \times \hat-z = 0
and we also will have all current inside system.

The sole present is directed along +Φ and it is on the cylindrical area. This is certainly equivalent to a ring present, which will be a magnetic dipole. With the right-hand guideline we determine Magnetic hooks the resultant magnetic industry must be into the +z course.

Figure 2 shows the basic outcome because of this item. The existing flows across the surface of this cylinder, resulting in a magnetic field Magnetic hooks is directed along +z in the cylinder’s axis. It is much like the existing found in a solenoid, so if the cylinder magnetis very long then your magnetic industry is continual inside.

cross sectional view of cylinder
Figure 2: cross-sectional view of basic geometry the magnetic area.

The following are explanations for qualitatively explaining the industry Magnetic hooks results from each case of particular cylinder magnetscale.

Instance of L << Roentgen
dipole magnetic area
Figure 3: Qualitative view for the magnetized industry leading to the truth of L significantly less than roentgen. In this case the medial side view associated with the cylinder magnet actually looks like a single line. The magnetic field is exactly the same as magnetized hooks made by an individual cycle of line. This might be really a physically recognized magnetized dipole.

Case of L >> Roentgen
dipole field for long cylinder
Figure 4: Qualitative view of this magnetized field leading to the outcome of L a lot more than roentgen. This really is nearly the same as the prior instance at opportunities far away from cylinder. Inside the cylinder magnetit appears as a solenoid and functions a consistent magnetic area.

Situation of L ≈ Roentgen
dipole magnetic area for square cylinder
Figure 5: Qualitative view of magnetized area causing the case of L about add up to R. Notice Magnetic hooks inside cylinder magnetthe magnetic area is within the exact same course while the magnetization. If it had been possible to get within the solid cylinder, then your observed magnetic industry would-be the same as Magnetic hooks of a solenoid.

∗ magnetized fields may also be from no-cost currents. There aren’t any free currents in this system. In a theoretical treatment like this, any no-cost current has got to be put truth be told there because of the writer (i.e. you can not solve free of charge currents, they could simply be provided as part of the subject setup).

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“April 9, 2007 in Physics.
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