Magnetized cylinder magnetas a Magnetic Dipole
Pace / April 9, 2007
a magnetic dipole is produced by working an ongoing through a cycle of cable. Most permanent magnets are bar magnets, however, and these are nevertheless generally known as creating a dipole field. In this subject we’re going to examine a very simple instance where the magnetization of a cylinder magnetis accustomed determine (and illustrate) the ensuing magnetized industry.
Consider right here a cylinder magnetof length, L, and radius, R, in which the proportion of those values may be set at three various values. We 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. every one of these cylinders functions a magnetization, M, magnetized hooks is written by M = α z, where α is a constant. The unit vector z is parallel to L, the axial direction associated with cylinder. We are going to quantitatively figure out the magnetic industry produced in each of these situations.
Figure 1 displays the geometry of this topic. The cylinder magnetis demonstrated to provide viewpoint the various scales becoming considered.
Figure 1: Setup with this topic showing the cylinder magnetand its magnetization.
If the magnetization of an object is provided, one method Magnetic hooks may be used to figure out its magnetized field involves resolving for its certain currents. The quantity, Jb, and surface, Kb, bound currents are pertaining to the magnetization by,
\vecJ_b = \vec\nabla \times \vecM \\ \\ \vecK_b = \vecM \times \hatn
where letter presents the vector normal to any surface regarding the cylinder magnet(in other words. every person area of the cylinder magnethas unique vector normal and for that reason unique certain surface existing).
The bound currents represent every one of the existing inside system∗. The bound volume current is resolved for as (including the complete cylindrical coordinates curl appearance, which can be 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
and the conceptual option to understand why zero result is Magnetic hooks a consistent area doesn’t have curl.
Bound area currents may occur on cylindrical surface as well as on either circular face. The cylindrical surface we,
\vecK_b = \alpha\hatz \times \hatr = \alpha\hat\phi
together with Φ direction correctly defines the cylindrical surface which means this is an actually reasonable result.
Circular faces are observed at z = ± L/2. The certain surface existing at these faces is, very 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 also currently have the existing in this system.
Truly the only up-to-date is directed along +Φ and is located on the cylindrical surface. This is certainly equal to a ring present, which would be a magnetic dipole. Making use of the right-hand guideline we determine Magnetic hooks the resultant magnetic field must certanly be in the +z way.
Figure 2 shows the basic outcome because of this item. Current flows along the surface of this cylinder, leading to a magnetic industry Magnetic hooks is directed along +z on the cylinder’s axis. This is just like the existing found in a solenoid, anytime the cylinder magnetis long then your magnetic area is constant inside.
cross sectional view of cylinder
Figure 2: Cross sectional view of the fundamental geometry the magnetized field.
The following are explanations for qualitatively explaining the industry magnetized hooks outcomes from each instance of particular cylinder magnetscale.
Instance of L << Roentgen
dipole magnetic field
Figure 3: Qualitative view of magnetic area leading to the scenario of L not as than R. In cases like this along side it view regarding the cylinder magnet in fact appears like a single range. The magnetic area is identical to Magnetic hooks produced by an individual loop of wire. That is essentially a physically recognized magnetized dipole.
Case of L >> Roentgen
dipole area for very long cylinder
Figure 4: Qualitative view of magnetized field leading to the scenario of L much higher than R. This might be nearly the same as the earlier instance at positions distant through the cylinder. Inside the cylinder magnetit seems as a solenoid and functions a consistent magnetized industry.
Situation of L ≈ Roentgen
dipole magnetic area for square cylinder
Figure 5: Qualitative view for the magnetic area causing the actual situation of L roughly add up to R. Notice Magnetic hooks inside the cylinder magnetthe magnetized industry is within the same way while the magnetization. If it were feasible to have inside solid cylinder, then noticed magnetic area would be like magnetized hooks of a solenoid.
∗ magnetized areas are often generated by free currents. There are not any free currents in this system. In a theoretical therapy like this, any no-cost up-to-date has to be put there by the writer (for example. you can’t resolve at no cost currents, they could only be provided included in the subject setup).
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“April 9, 2007 in Physics.
Capacitance of Concentric Cylinders