Magnetized cylinder magnetas a Magnetic Dipole
Speed / April 9, 2007
a magnetic dipole could be made by operating a present through a cycle of wire. Most permanent magnets tend to be club magnets, but that are nevertheless generally described as producing a dipole industry. Inside subject we shall analyze a simple situation when the magnetization of a cylinder magnetis used to determine (and show) the resulting magnetized field.
Consider here a cylinder magnetof size, L, and radius, R, where in actuality the proportion among these values is likely to be set at three different values. We 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. every one of these cylinders features a magnetization, M, magnetized hooks is distributed by M = α z, where α is a constant. The system vector z is parallel to L, the axial way of the cylinder. We are going to quantitatively determine the magnetic industry manufactured in each one of these instances.
Figure 1 displays the geometry with this topic. The cylinder magnetis proven to supply viewpoint when it comes to different machines being considered.
Figure 1: Setup for this topic showing the cylinder magnetand its magnetization.
As soon as the magnetization of an object is offered, one strategy magnetized hooks may be used to determine its magnetic field requires solving for the bound currents. The quantity, Jb, and area, Kb, bound currents are linked to the magnetization by,
\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
where n signifies the vector typical to any area associated with cylinder magnet(for example. each individual surface for the cylinder magnethas unique vector regular and as a consequence its certain area current).
The bound currents represent all of the present within system∗. The certain amount present is solved for as (like the full cylindrical coordinates curl appearance, which is constantly 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
in addition to conceptual way to understand this zero result is magnetized hooks a consistent industry has no curl.
Bound surface currents may occur on cylindrical surface as well as on either circular face. For the cylindrical surface we now have,
\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
together with Φ course properly defines the cylindrical surface so this is a literally reasonable outcome.
Circular faces are found at z = ± L/2. The certain area existing at these faces is, 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 we currently have all of the present in this system.
The only present is directed along +Φ and is on the cylindrical area. This can be equal to a ring present, which may be a magnetic dipole. Using the right-hand rule we determine Magnetic hooks the resultant magnetic industry must be inside +z course.
Figure 2 shows the essential outcome for this item. The current flows across the surface of the cylinder, causing a magnetized field magnetized hooks is directed along +z regarding cylinder’s axis. This can be much like the present within a solenoid, so if the cylinder magnetis very long then the magnetized industry is constant inside.
cross-sectional view of cylinder
Figure 2: cross-sectional view associated with the fundamental geometry the magnetized area.
The following are explanations for qualitatively describing the field magnetized hooks outcomes from each situation of specific cylinder magnetscale.
Situation of L << R
dipole magnetized industry
Figure 3: Qualitative view of the magnetic industry resulting in the scenario of L much less than roentgen. In cases like this along side it view of this cylinder magnet really looks like one line. The magnetized area is identical to Magnetic hooks produced by just one cycle of line. This can be really a physically realized magnetized dipole.
Case of L >> R
dipole area for long cylinder
Figure 4: Qualitative view of this magnetic field resulting in the actual situation of L a lot higher than roentgen. That is much like the prior instance at positions distant through the cylinder. Inside the cylinder magnetit appears as a solenoid and functions a continuing magnetic field.
Situation of L ≈ R
dipole magnetic field for square cylinder
Figure 5: Qualitative view of this magnetic industry causing the truth of L roughly add up to R. Notice Magnetic hooks inside the cylinder magnetthe magnetized industry is within the same direction once the magnetization. If it were possible to get inside solid cylinder, then your observed magnetized field would be similar to magnetized hooks of a solenoid.
∗ magnetized areas can also be generated by free currents. There are not any no-cost currents inside system. In a theoretical treatment such as this, any free present has got to be placed there because of the author (for example. you simply can’t solve free-of-charge currents, they could only be given included in the subject setup).
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“April 9, 2007 in Physics.
Capacitance of Concentric Cylinders