# and even though these cost a lot

Magnetized cylinder magnetas a Magnetic Dipole
Pace / April 9, 2007
a magnetic dipole could be made by working a current through a cycle of wire. Many permanent magnets are bar magnets, but and these continue to be commonly named producing a dipole industry. Inside topic we shall examine a simple situation where the magnetization of a cylinder magnetis regularly figure out (and illustrate) the resulting magnetized industry.

Give consideration to here a cylinder magnetof length, L, and radius, R, where the ratio among these values should be set at three different values. We now have a quick 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. Each of these cylinders functions a magnetization, M, Magnetic hooks is distributed by M = α z, in which α is a consistent. The machine vector z is parallel to L, the axial course regarding the cylinder. We shall quantitatively determine the magnetic field stated in every one of these situations.

Figure 1 shows the geometry with this topic. The cylinder magnetis proven to provide perspective for the various machines being considered.

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

Once the magnetization of an item is given, one method magnetized hooks enables you to figure out its magnetic industry requires resolving for the certain currents. The amount, Jb, and surface, Kb, bound currents tend to be related to the magnetization by,

\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
in which letter presents the vector normal to your area of cylinder magnet(i.e. every person area of this cylinder magnethas a unique vector normal and for that reason unique bound area present).

The certain currents represent every one of the current in this system∗. The certain amount existing is resolved for as (such as the complete cylindrical coordinates curl expression, that is always a helpful guide),

\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
therefore the conceptual option to appreciate this zero outcome is Magnetic hooks a continuing industry does not have any curl.

Bound surface currents may exist on cylindrical surface also on either circular face. When it comes to cylindrical area we,

\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
therefore the Φ path properly defines the cylindrical area and this is a physically reasonable result.

Circular faces are found at z = ± L/2. The certain area existing at these faces is, initially for z = +1/2,

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

\vecK_b- = \alpha\hatz \times \hat-z = 0
and then we now have most of the present within system.

The only real current is directed along +Φ and it is on the cylindrical surface. This might be comparable to a ring current, which will be a magnetic dipole. Using the right-hand rule we determine magnetized hooks the resultant magnetized area must certanly be in the +z course.

Figure 2 shows the fundamental result with this item. The present flows across the surface of this cylinder, leading to a magnetized field magnetized hooks is directed along +z from the cylinder’s axis. This can be like the present present a solenoid, anytime the cylinder magnetis very long then your magnetized industry is continual inside.

cross-sectional view of cylinder
Figure 2: Cross sectional view associated with the fundamental geometry the magnetic field.

The following are explanations for qualitatively explaining the industry Magnetic hooks outcomes from each situation of certain cylinder magnetscale.

Instance of L << R
dipole magnetic field
Figure 3: Qualitative view of magnetic area leading to the way it is of L not as than R. In this instance the medial side view of this cylinder magnet actually looks like an individual range. The magnetized area is identical to magnetized hooks generated by just one cycle of cable. This can be basically a physically realized magnetic dipole.

Situation of L >> R
dipole area for very long cylinder
Figure 4: Qualitative view of magnetic industry leading to the scenario of L a lot higher than R. This is nearly the same as the previous case at jobs distant through the cylinder. Inside cylinder magnetit appears as a solenoid and functions a constant magnetized field.

Case of L ≈ Roentgen
dipole magnetic field for square cylinder
Figure 5: Qualitative view associated with the magnetic area causing the case of L approximately corresponding to R. Notice Magnetic hooks inside the cylinder magnetthe magnetic industry is within the same course while the magnetization. If it were possible to obtain inside solid cylinder, then your noticed magnetic area would-be just like Magnetic hooks of a solenoid.

∗ Magnetic industries can also be from free currents. There are not any no-cost currents within system. In a theoretical therapy like this, any free current must be put here by the writer (in other words. you simply can’t solve at no cost currents, they could simply be given as part of the topic setup).

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“April 9, 2007 in Physics.
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