9.1.Magnetization
Each atomic current is a tiny closed circuit of atomic dimensions, so it seems reasonable that the “distant” magnetic field of the atom can be appropriately represented by a magnetic dipole field. In fact, a wide body of experimental studies as well as the formulation of quantum mechanics, which is our most accurate method for calculating atomic phenomena, tell us that the dominant part of the distant magnetic field due to a single atom is determined by specifying its magnetic dipole moment, m.
We sum up vectorially all of the dipole moments in a small volume element v and then divide the result by v. The resulting quantity.
Is called the magnetic dipole moment per unit volume, or simply the magnetization.
9.2.The Magnetic Field Produced by Magnetized Material
Instead of calculating B directly, we find it expedient to work with the vector potential A, and to obtain B subsequenly by means of the curl operation.
A(x,y,z) =
=
9.3.Magnetic Scalar Potential and Magnetic Pole Density
Called the magnetic pole density, and
M(r’) = M(r’) n
The surface density of magnetic pole strenght
These quantities are quite useful even through rather artificial. They play the same role in the theory of magnetism that P and P play in dielectric theory. The units of and M are A/m2 and A/m, respectively.
9.4.Sources of The Magnetic Field: Magnetic Intensity
B(r) =
Where
the volume V extends over all current-carrying regions and over all matter.
The surfaces S includes all surfaces and interfaces between different media.
The current density J includes only conventional currents of the charge transport variety, whereas the effect of atomic currents is found in the magnetization vector M (and potential
H =
H(r) =
9.5.The Field Equation
, J
We should now like to see how these equations are modified when the magnetic field B includes a contribution from magnetized material.
The curl equation is the differential form of Ampere’s circuital law. Here we must be careful to include all types of currents that can produce a magnetic field. Hence, in the general case, the equation is properly written as
J + JM)
9.6.Magnetic Susceptibility, Permeability, and Hysteresis
In order to solve problems in magnetic theory, it is essential to have a relationship between B and H or, equivalently, a relationship between M and one of the magnetic field vectors. These relationships depend on the nature of the magnetic material and are usually obtained from experiment.
In larges class of materials, there exist an approximately linear relationship between M and H. If the material is isotropic as well as linear.
M = xmH
Where the dimensionless scalar quantity xm is called the magnetic susceptibility.
9.7.Boundary Conditions on The Field Vectors
B2 n2 S + B1 n1 S = 0
Where n2 and n1 are the outward-directed normals to the upper and lower surfaces of the pilbox. Since n2 = n1 and since either of these normals may serve as normal to the interface,
(B2 - B1) n2 = 0
Thus, the normal component of B is continuous across an interface.
9.8.Boundary-Value Problems Involving Magnetic Materials
Which is Laplace’s equation. Thus, the magnetic problem reduces to finding a solution to Laplace’s equation that satisfies the boundary conditions. H may then be calculated as minus the gradient of the magnetic potential, and B obtained from
or
B = (H + M)
Whichever is appropriate.
9.9.Current Circuits Containing Magnetic Media
Htl = NI
Here the subscript stands for the component tangent to the path, and l = 2 r is the total path lenght.
Bl = l
Thus, the magnetic field differs from that in the vacuum case by the additive terms l
9.10.Magnetic Circuits
The magnetic flux lines, as we have seen, form closed loops. If all the magnetic flux (or substantially all cf it) assosiated with a particular distribution of currents is confined to a rather well-defined path, then we may speak of a magnetic circuit.
Let us consider a more general series circuit of several materials surrounded by a toroidal winding of N turns carrying current I.
9.11.Magnetic Circuits Containing Permanent Magnets
The magnetic circuit concept is useful also when applied to permanent magnet circuit-that is, to flux circuits in which has its origin in permanently magnetized material. We shall find it convenient to use the abbreviation P-M for permanent magnet. Because of the complicated B-H relationship in P-M material, the procedure outlined in the preceding section is not well suited to the problem at hand. Instead we start again with Ampere’s circuital law, applied now to the flux path of the P-M circuit:
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