Magnetized cylinder magnetas a Magnetic Dipole
Pace / April 9, 2007
a magnetized dipole can be created by operating a current through a loop of line. Most permanent magnets tend to be bar magnets, however, that are generally named creating a dipole area. Inside subject we are going to examine a simple situation when the magnetization of a cylinder magnetis regularly figure out (and show) the resulting magnetic area.
Give consideration to right here a cylinder magnetof length, L, and radius, R, in which the proportion among these values is going to 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 an almost cubic cylinder magnetin which L ≈ R. all these cylinders functions a magnetization, M, Magnetic hooks is distributed by M = α z, where α is a constant. The unit vector z is parallel to L, the axial course for the cylinder. We’re going to quantitatively figure out the magnetized area manufactured in each of these situations.
Figure 1 displays the geometry of this subject. The cylinder magnetis proven to supply viewpoint for different scales becoming considered.
Figure 1: Setup because of this subject showing the cylinder magnetand its magnetization.
If the magnetization of an object is offered, one technique magnetized hooks enable you to determine its magnetized area requires solving for the bound currents. The amount, Jb, and surface, Kb, bound currents are linked to the magnetization by,
\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
where letter represents the vector regular to virtually any area of the cylinder magnet(in other words. each individual surface of cylinder magnethas its very own vector normal and as a consequence its own bound area present).
The certain currents represent all of the current within system∗. The bound volume existing is fixed for as (like the complete cylindrical coordinates curl expression, that 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
while the conceptual solution to understand this zero result is magnetized hooks a consistent area has no curl.
Bound surface currents may occur from the cylindrical surface and on either circular face. For cylindrical area we have,
\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
while the Φ course properly defines the cylindrical surface which means this is a physically reasonable outcome.
Circular faces are located 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 for z = -1/2,
\vecK_b- = \alpha\hatz \times \hat-z = 0
and we now have most of the existing within system.
Truly the only current is directed along +Φ and it is on the cylindrical area. This is certainly equivalent to a ring present, which would be a magnetic dipole. Using the right-hand rule we determine magnetized hooks the resultant magnetic industry must certanly be into the +z direction.
Figure 2 shows the essential outcome with this object. Current flows along the area associated with the cylinder, resulting in a magnetic industry Magnetic hooks is directed along +z regarding cylinder’s axis. This can be like the existing within a solenoid, therefore if the cylinder magnetis very long then your magnetic area is continual inside.
cross-sectional view of cylinder
Figure 2: Cross sectional view of this basic geometry for the magnetic area.
Listed here are explanations for qualitatively describing the industry Magnetic hooks outcomes from each instance of specific cylinder magnetscale.
Case of L << R
dipole magnetized industry
Figure 3: Qualitative view of magnetic industry leading to the situation of L less than roentgen. In this situation the medial side view associated with the cylinder magnet in fact looks like a single range. The magnetic area is exactly the same as magnetized hooks generated by an individual cycle of line. That is really a physically recognized magnetized dipole.
Case of L >> R
dipole area for long cylinder
Figure 4: Qualitative view of this magnetized industry resulting in the way it is of L a lot more than roentgen. This is much like the previous instance at jobs far away through the cylinder. Inside the cylinder magnetit seems as a solenoid and features a continuing magnetized field.
Instance of L ≈ R
dipole magnetic field for square cylinder
Figure 5: Qualitative view associated with magnetic industry causing the truth of L approximately equal to R. Notice Magnetic hooks within the cylinder magnetthe magnetic industry is within the same path whilst the magnetization. If it were possible for inside solid cylinder, then the observed magnetized industry would be like Magnetic hooks of a solenoid.
∗ magnetized areas are often from no-cost currents. There are no free currents within system. In a theoretical treatment such as this, any free up-to-date needs to be put there because of the writer (in other words. you can not solve free-of-charge currents, they may be able only be offered within the topic setup).
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“April 9, 2007 in Physics.
Capacitance of Concentric Cylinders