Static Electromagnetic Fields in Matter
7.2 Magnetic Induction Field Due to a Magnetized Material
it is in air. The diminution may be attributed to shielding produced by the aligned dipoles at the air-dielectric interface. The thin air layer, on the other hand, has a significantly increased electric field due to the dielectric, and is a major contributor to the voltage difference between the plates.
=µ0
$J(r) +∇×M(r)
%
δ(r−r)d3r
=µ0
$J(r) +∇ × M(r)
%
(7–27) Again we find that, according to equation (7–27), the curl of the magnetization behaves like a current density. Just as the polarization contributes an effective surface charge at discontinuities, a discontinuity in magnetization, as, for example, at the boundaries of magnetic materials, contributes an effective surfacecurrent.
In arriving at (7–26), we eliminated the surface integral by forcing the volume of integrationτto extend beyond the region containing magnetized materials. The penalty for this procedure is that the discontinuity of M at the boundary of the material makes∇ × M undefined. If we had instead taken the region of integration to be the “open” region containing all the magnetization, but not the boundary, we could exclude the discontinuity from the region where the curl must be computed.
We would then obtain, instead of (7-26), A( r) = µ0
4π
S∈τ
J(r) +∇×M(r)
|r−r| d3r+µ0 4π S
M(r)×dS
|r−r| (7–28) The added term is just that which would have been produced by a surface current j=M ×nˆ. This same conclusion can of course be reached from (7–27), as is shown below. Consider a uniformly magnetized bar with magnetizationM =Mzˆk. Inside the magnet, ∇ × M = 0, implying that∇ × B =µ0J; the uniform magnetization makes no contribution to the induction field! At the boundary, since ∇ × M is undefined, we resort to a different stratagem. We draw a thin rectangular loop with long sides parallel to the magnetization M straddling the boundary (Figure 7.4) and integrate the curl equation over the area included in the loop:
(∇ × B) ·dS=µ0
(∇ × M)·dS (7–29) For simplicity we have takenJ= 0 and takeM to be directed alongz.
By means of Stokes’ theorem (18) both surface integrals may be recast as line integrals
B ·d=µ0 M ·d (7–30)
Figure 7.4:The Amp`erian loop lies in thex-zplane straddling the boundary.
Figure 7. 5: The short side of the loop, inside the bar magnet, is taken parallel to the magnetization.
which, after letting the width shrink to zero, givesBintz −Bzext =µ0M. A surface currentj=Mˆwould produce exactly this kind of discontinuity inB. Thus, at the boundary, the effect of a discontinuity in the magnetization is exactly the same as that of a surface currentj =M ×n.ˆ
Example 7.3: Find the magnetic induction field inside a long (assume infinite) bar magnet with uniform magnetizationM.
Solution: We integrate the equation
∇ × B(r) =µ0∇ × M(r) (Ex 7.3.1) over the area of a rectangular loop with one short side inside the magnet and its other short side sufficiently far removed thatB vanishes, as shown in Figure 7.5:
(∇ × B) ·dS=µ0
(∇ × M)·dS (Ex 7.3.2) Applying Stokes’ theorem to both integrals we obtain
B ·d=µ0 M ·d (Ex 7.3.3)
The translational invariance of the problem makes the integral ofB along the long (perpendicular to the bar) side cancel (they would each vanish when they’re per- pendicular toB in any case). The integral along the distant short side also vanishes sinceB vanishes, reducing the integral forB to BzL. Similarly, the integral of the magnetization reduces toMzL. We conclude therefore that the magnetic induction field inside the magnet isB =µ0M. Having foundB inside the magnet, we can now repeat the argument with the distant end of the loop brought close to the magnet to obtainB = 0 outside the magnet.
Example 7.4: Find the magnetic induction field along the center line of a uniformly magnetized cylindrical bar magnet of lengthL.
Solution: Recognizing that we may replace the magnetization by a surface current, we note that this problem is identical to that of a solenoid of lengthLwith azimuthal
current I = ML/N. The equivalent current density along the periphery of the magnet isj =M, giving a currentdI =jdz =Mdzfor a current “loop” of width dz centered atz. According to example 1.8, the field due to a circular plane loop of radiusR, carrying currentdI centered atz, is
d B(0,0, z) = µ0R2dIˆk
2 [R2+ (z−z)2]3/2 = µ0R2Mdz
2 [R2+ (z−z)2]3/2 (Ex 7.4.1) Integrating this expression from −12L to 12L to obtain the contribution from all parts of the magnet, we find that
B(0, 0, z) = µ0 2
12L
−12L
M R 2dz
[R2+ (z−z)2]3/2 = µ0R2M
2 · −(z−z) R2
R2+ (z−z)2
1 2L
−12L
= µ0M 2
12L+z
(12L+z)2+R2 +
1 2L−z
(12L−z)2+R2
(Ex 7.4.2)
Note the resemblance of this result to the equivalent result (Ex 7.1.1) for the polar- ized dielectric cylinder if this result is restated forD =ε0E+P and the appropriate translation of the origin is made.
As was true in the case of electric polarization, it is frequently preferable to ascribe the magnetization to the medium as an attribute rather than having it act as a source of the appropriate field. We write equation (7-16),∇ × B =µ0(J+∇ × M), in more convenient form. Gathering all the curl terms on the left, we obtain
∇ × B
µ0−M
=J (7–31)
The quantity H ≡B/µ 0−M is called themagnetic field intensity. H satisfies
∇ × H =J (7–32)
Because H is directly proportional to the controllable variable J, the relation between B and H is conventionally regarded as B being a function ofH with H the independent variable. This perspective leads us to write B =µ0[H +M(H)].
In the linear isotropic approximation, M(H) = χmH, whereχm is known as the magnetic susceptibility. Under this approximation,
B ≡µ0(H +M) =µ0(1 +χm)H ≡µ H (7–33) where µ is known as themagnetic permeability. Materials withχm>1 are para- magnetic whereas those with χm < 0 are diamagnetic. For most materials, both 7.2.1 Magnetic Field Intensity
paramagnetic and diamagnetic,|χm| 1. Whenχm1, the material is ferromag- netic. In this latter case the relationB =µ0(H +M) still holds, butM is usually a very complicated, nonlinear, multivalued function ofH.
It is worth pointing out that to this pointH andD appear to be nothing but mathematical constructs derivable from the fields B and E. The Lorentz force on a charged particle is F = q(E +v×B). This means the fields E and B are detectable through forces they exercise on charges. The magnetic flux through a loop remains Φ = B ·dS. However, as we are gradually converting to the view that fields, possessing momentum and energy have an existence independent of interacting particles, there is no reason to suggest that D and H should have less reality thanE andB. As we recast Maxwell’s equations into their eventual form, H and D will assume an equal footing with E and B. It is worthwhile to shift our perspective on the fields, regardingE andB as the fields responsible for forces on charged particles, whereas H and D are the fields generated by the sources.
In gravitational theory, by contrast, the gravitational mass and inertial mass are identical meaning that the source and force fields are the same.
In the following sections we will briefly discuss the microscopic behavior of materials responsible for polarization and magnetization. We also touch briefly on conduction in metals for the sake of completeness.
7.3.1 Polar Molecules (Langevin-Debye Formula)
A polar molecule has a permanent dipole moment. If the nearest neighbor inter- action energies are small, a material made of such molecules will normally have the dipoles oriented randomly (to maximize entropy) in the absence of an electric field. In an exceedingly strong field all the dipoles will align with the electric field, giving a maximum polarization P =np(n is the number density of molecules and
pis the dipole moment of each). At field strengths normally encountered, thermal randomizing will oppose the alignment to some extent. The average polarization may be found from thermodynamics.
According to Boltzmann statistics, the probability of finding a molecule in a state of energy W is proportional to e−W/kT. We consider only the energy of the dipole in the electric field, W = −p·E =−pEcosθ. The mean value of pmust just be the component along E, the perpendicular components averaging to zero.
Hence the mean polarization is
pcosθ=
pcosθe+pEcosθ/kTdΩ
e+pEcosθ/kTdΩ
=p
cothpE kT −kT
pE
(7–34)
a result known as the Langevin formula.
The low field limit of the polarization is readily found. We abbreviate x = pE/kT to write