endure you for as long

Magnetized cylinder magnetas a magnetized Dipole
Speed / April 9, 2007
a magnetized dipole can be made by operating an ongoing through a loop of line. Most permanent magnets are club magnets, but that are nevertheless commonly described as producing a dipole industry. Within topic we’ll analyze a very simple case when the magnetization of a cylinder magnetis used to determine (and show) the ensuing magnetic industry.

Think about here a cylinder magnetof length, L, and radius, R, where in fact the ratio of the values will undoubtedly be set at three various values. We have a short and fat cylinder magnetin which L << R, a long and skinny cylinder magnetin which L >> R, and a nearly cubic cylinder magnetin which L ≈ R. every one of these cylinders functions a magnetization, M, magnetized hooks is written by M = α z, in which α is a consistent. The machine vector z is parallel to L, the axial direction of cylinder. We are going to quantitatively figure out the magnetized field stated in each one of these instances.

Figure 1 shows the geometry of this topic. The cylinder magnetis shown to offer viewpoint for the different machines being considered.

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

When the magnetization of an object is given, one method magnetized hooks enables you to figure out its magnetized industry involves resolving for its certain currents. The quantity, Jb, and surface, Kb, bound currents are linked to the magnetization by,

\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
where n represents the vector typical to your area of this cylinder magnet(for example. each individual surface of cylinder magnethas its very own vector normal therefore a unique bound area existing).

The bound currents represent the existing within system∗. The bound volume present is solved for as (including the full cylindrical coordinates curl expression, which can be constantly a helpful research),

\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
while the conceptual option to understand why zero result is magnetized hooks a consistent industry has no curl.
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Bound area currents may occur on cylindrical surface as well as on either circular face. For cylindrical area we,

\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
as well as the Φ direction precisely defines the cylindrical area and this is an actually reasonable result.

Circular faces are located at z = ± L/2. The bound surface existing at these faces is, very first for z = +1/2,

\vecK_b+ = \alpha\hatz \times \hatz = 0
after which for z = -1/2,

\vecK_b- = \alpha\hatz \times \hat-z = 0
and then we have all present in this system.

The only current is directed along +Φ and is on the cylindrical surface. That is equivalent to a ring present, which may be a magnetic dipole. With the right-hand rule we determine magnetized hooks the resultant magnetized area must be when you look at the +z way.

Figure 2 reveals the fundamental result with this object. The current flows along the area for the cylinder, resulting in a magnetized field Magnetic hooks is directed along +z regarding cylinder’s axis. It is similar to the current present in a solenoid, so if the cylinder magnetis very long then your magnetic field is constant inside.

cross sectional view of cylinder
Figure 2: Cross sectional view for the fundamental geometry for the magnetic area.

Listed here are explanations for qualitatively explaining the field Magnetic hooks results from each instance of certain cylinder magnetscale.

Case of L << R
dipole magnetic field
Figure 3: Qualitative view of magnetic field causing the outcome of L a lot less than R. In this case the side view for the cylinder magnet in fact looks like one line. The magnetic industry is the same as Magnetic hooks made by just one cycle of line. This really is essentially a physically recognized magnetic dipole.

Instance of L >> Roentgen
dipole field for very long cylinder
Figure 4: Qualitative view associated with magnetized industry resulting in the actual situation of L a great deal more than roentgen. This is nearly the same as the previous instance at jobs distant through the cylinder. In the cylinder magnetit seems as a solenoid and features a continuing magnetic industry.

Instance of L ≈ R
dipole magnetized field for square cylinder
Figure 5: Qualitative view regarding the magnetized industry leading to the case of L approximately equal to R. Notice Magnetic hooks inside cylinder magnetthe magnetic industry is in the exact same course as the magnetization. If it were possible to have within the solid cylinder, then your observed magnetized industry could be like magnetized hooks of a solenoid.

∗ Magnetic industries may also be from no-cost currents. There aren’t any no-cost currents in this system. In a theoretical therapy similar to this, any free up-to-date needs to be placed truth be told there because of the author (for example. you simply can’t resolve for free currents, they may be able simply be provided included in the subject setup).

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“April 9, 2007 in Physics.
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