Magnetized cylinder magnetas a magnetized Dipole
Pace / April 9, 2007
a magnetized dipole can be produced by running a current through a cycle of cable. Most permanent magnets are club magnets, but that continue to be generally named making a dipole field. Inside subject we’re going to analyze a simple situation where the magnetization of a cylinder magnetis always figure out (and show) the resulting magnetic industry.
Consider right here a cylinder magnetof length, L, and radius, R, in which the ratio of the values are 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 a nearly cubic cylinder magnetin which L ≈ R. Each of these cylinders functions a magnetization, M, Magnetic hooks is provided by M = α z, in which α is a consistent. The unit vector z is parallel to L, the axial course for the cylinder. We shall quantitatively determine the magnetized industry manufactured in each of these cases.
Figure 1 shows the geometry of the subject. The cylinder magnetis shown to provide viewpoint the various scales becoming considered.
Figure 1: Setup for this topic showing the cylinder magnetand its magnetization.
Whenever magnetization of an object is given, one technique Magnetic hooks may be used to determine its magnetic field involves solving for the certain currents. The volume, Jb, and surface, Kb, bound currents are associated with the magnetization by,
\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
where n signifies the vector typical to any surface regarding the cylinder magnet(i.e. every person surface associated with the cylinder magnethas its vector regular and therefore its bound area existing).
The certain currents represent the current within system∗. The bound volume present is solved for as (such as the full cylindrical coordinates curl appearance, that will 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
additionally the conceptual method to understand why zero outcome is Magnetic hooks a continuing area has no curl.
Bound surface currents may occur on cylindrical surface and also on either circular face. For the cylindrical surface we’ve,
\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
and Φ path correctly defines the cylindrical surface which means this is a physically reasonable result.
Circular faces are located at z = ± L/2. The certain surface current at these faces is, very first 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 we also currently have all existing in this system.
Truly the only up-to-date is directed along +Φ and it is found on the cylindrical area. This is certainly comparable to a ring existing, which would be a magnetic dipole. Using the right-hand rule we determine magnetized hooks the resultant magnetic field must certanly be in the +z course.
Figure 2 shows the essential result because of this object. Current flows across the surface of cylinder, resulting in a magnetized area Magnetic hooks is directed along +z regarding the cylinder’s axis. This really is similar to the present present a solenoid, therefore if the cylinder magnetis long then magnetized area is continual inside.
cross sectional view of cylinder
Figure 2: cross-sectional view of the standard geometry the magnetized area.
The following are explanations for qualitatively describing the field Magnetic hooks results from each situation of specific cylinder magnetscale.
Situation of L << Roentgen
dipole magnetized field
Figure 3: Qualitative view associated with magnetized field resulting in the scenario of L much less than roentgen. In this instance the side view for the cylinder magnet in fact appears like one range. The magnetized field is just like Magnetic hooks generated by an individual cycle of line. This might be really a physically recognized magnetic dipole.
Case of L >> R
dipole field for long cylinder
Figure 4: Qualitative view associated with the magnetized area causing the way it is of L a lot higher than roentgen. That is very similar to the previous situation at positions far-away from cylinder. Within the cylinder magnetit seems as a solenoid and functions a consistent magnetized field.
Case of L ≈ Roentgen
dipole magnetized field for square cylinder
Figure 5: Qualitative view regarding the magnetic field leading to the scenario of L roughly corresponding to R. Notice Magnetic hooks within the cylinder magnetthe magnetic field is within the same direction as the magnetization. If it had been possible for within the solid cylinder, then your observed magnetized area is like magnetized hooks of a solenoid.
∗ magnetized fields can also be produced by free currents. There are not any no-cost currents inside system. In a theoretical therapy like this, any no-cost present has got to be placed indeed there by the author (i.e. you can not resolve 100% free currents, they are able to simply be provided included in the subject setup).
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“April 9, 2007 in Physics.
Capacitance of Concentric Cylinders