Magnetized cylinder magnetas a Magnetic Dipole
Speed / April 9, 2007
a magnetic dipole are produced by working a current through a cycle of wire. Most permanent magnets are bar magnets, however, that are nevertheless frequently described as making a dipole field. Within subject we’ll examine a simple instance in which the magnetization of a cylinder magnetis regularly determine (and illustrate) the resulting magnetic industry.
Start thinking about right here a cylinder magnetof size, L, and radius, R, where ratio of these values is going to be set at three different values. We’ve a short 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. all these cylinders functions a magnetization, M, magnetized hooks is given by M = α z, in which α is a constant. The system vector z is parallel to L, the axial direction regarding the cylinder. We’re going to quantitatively figure out the magnetic field manufactured in each of these cases.
Figure 1 shows the geometry with this topic. The cylinder magnetis proven to provide viewpoint for the different machines becoming considered.
Figure 1: Setup because of this subject showing the cylinder magnetand its magnetization.
Whenever magnetization of an item is given, one technique Magnetic hooks enable you to determine its magnetic area involves solving for the bound currents. The quantity, Jb, and area, Kb, bound currents tend to be about the magnetization by,
\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
in which n presents the vector normal to any surface of cylinder magnet(for example. every individual area for the cylinder magnethas its vector typical and therefore unique certain area current).
The bound currents represent every one of the existing within system∗. The certain amount existing is solved for as (like the complete cylindrical coordinates curl phrase, that will be always 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
therefore the conceptual method to appreciate this zero outcome is Magnetic hooks a constant field doesn’t have curl.
Bound surface currents may occur regarding the cylindrical area and in addition on either circular face. When it comes to cylindrical area we now have,
\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
while the Φ path correctly defines the cylindrical area so this is a literally reasonable result.
Circular faces are observed at z = ± L/2. The certain 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 now have all current in this system.
The only real up-to-date is directed along +Φ and is on the cylindrical surface. This really is equal to a ring existing, which would be a magnetic dipole. Utilizing the right-hand rule we determine magnetized hooks the resultant magnetic field should be inside +z path.
Figure 2 reveals the essential outcome with this item. The existing flows across the surface for the cylinder, causing a magnetic area Magnetic hooks is directed along +z regarding the cylinder’s axis. It is similar to the present present a solenoid, so if the cylinder magnetis very long then the magnetic field is continual inside.
cross-sectional view of cylinder
Figure 2: cross-sectional view of basic geometry the magnetized area.
Listed below are explanations for qualitatively describing the industry magnetized hooks outcomes from each instance of certain cylinder magnetscale.
Case of L << Roentgen
dipole magnetized field
Figure 3: Qualitative view associated with the magnetic industry resulting in the way it is of L significantly less than R. In this instance the side view of cylinder magnet really looks like one range. The magnetic field is identical to Magnetic hooks produced by one cycle of cable. This might be essentially a physically recognized magnetic dipole.
Case of L >> R
dipole area for long cylinder
Figure 4: Qualitative view of magnetic industry causing the scenario of L a great deal greater than roentgen. This will be much like the last case at jobs distant through the cylinder. Inside the cylinder magnetit seems as a solenoid and features a continuing magnetic area.
Instance of L ≈ R
dipole magnetic field for square cylinder
Figure 5: Qualitative view for the magnetic industry causing the situation of L more or less add up to R. Notice Magnetic hooks inside cylinder magnetthe magnetic field is in the exact same direction since the magnetization. If it had been feasible to obtain in the solid cylinder, then the noticed magnetic area could be just like magnetized hooks of a solenoid.
∗ magnetized fields may also be made by no-cost currents. There aren’t any free currents inside system. In a theoretical treatment such as this, any free current must be put indeed there by the author (in other words. you simply can’t solve free of charge currents, they can simply be offered included in the topic setup).
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“April 9, 2007 in Physics.
Capacitance of Concentric Cylinders