k
the mutual inductance coefficient
Lkk’ kk’/ Ik’ – schematically.
Due to the linear superposition principle, the total flux piercing the k th loop may be represented as Magnetic
flux from
Φ
Φ
L I
(5.68)
k
kk '
kk' k'
currents
k'
k'
For example, for the system of two currents, this expression is reduced to a clear analog of Eqs. (2.19): Φ L I MI ,
1
1 1
2
(5.69)
Φ MI L I .
2
1
2 2
For the even simpler case of a single current,
of a
single
LI ,
(5.70)
current
so that the magnetic energy of the current may be represented in several equivalent forms: 33 The SI unit of magnetic flux is called weber, abbreviated Wb – after Wilhelm Edward Weber (1804-1891), who in particular co-invented (with Carl Gauss) the electromagnetic telegraph. More importantly for this course, in 1856 he was the first (together with Rudolf Kohlrausch) to notice that the value of (in modern terms) 1/(00)1/2, derived from electrostatic and magnetostatic measurements, coincides with the independently measured speed of light c. This observation gave an important motivation for Maxwell’s theory.
Chapter 5
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L
1
1
U of a
2
2
U
I
Φ
I
Φ .
(5.71) single
2
2
2 L
current
These relations, similar to Eqs. (2.14)-(2.15) of electrostatics, show that the self-inductance L of a current loop may be considered a measure of the system’s magnetic energy. However, as we will see in Sec. 6.1, this measure is adequate only if the flux , rather than the current I, is fixed.
Now we are well equipped for the calculation of inductance coefficients for particular systems, having three options. The first one is to use Eq. (60) directly.34 The second one is to calculate the magnetic field energy from Eq. (57) as the function of all currents Ik in the system, and then use Eq. (59) to find all coefficients Lkk’. For example, for a system with just one current, Eq. (71) yields U
L
.
(5.72)
2
I / 2
Finally, if the system consists of thin wires, so that the loop areas Sk and hence the fluxes kk’ are well defined, we may calculate them from Eq. (65), and then use Eq. (67) to find the inductances.
Usually, the third option is simpler, but the first two one may be very useful even for thin-wire systems, if the notion of magnetic flux in them is not quite apparent. As an important example, let us find the self-inductance of a long solenoid – see Fig. 6a again. We have already calculated the magnetic field inside it – see Eq. (40) – so that, due to the field uniformity, the magnetic flux piercing each turn of the wire is just
BA nIA ,
(5.73)
1
0
where A is the area of the solenoid’s cross-section – for example R 2 for a round solenoid, though Eq.
(40), and hence Eq. (73) are valid for cross-sections of any shape. Comparing Eqs. (73) with Eq. (70), one might wrongly conclude that L = 1/ I = 0 nA (WRONG!), i.e. that the solenoid’s inductance is independent of its length. Actually, the magnetic flux 1 pierces each wire turn, so that the total flux through the whole current loop, consisting of N turns, is
N n 2 lAI ,
(5.74)
1
0
and the correct expression for the long solenoid’s self-inductance is
2
N 2 A
L
n lA
0
,
(5.75) L of a
I
0
l
solenoid
i.e. at fixed A and l, the inductance scales as N 2, not as N. Since this reasoning may seem not quite evident, it is prudent to verify this result by using Eq. (72), with the full magnetic energy inside the solenoid (neglecting minor fringe field contributions), given by Eq. (57) with B = const within the internal volume V = lA, and zero outside of it:
1
2
1
2
I
U
B Al
nI Al n lA .
(5.76)
0
2
2
2
2
0
2
0
0
Plugging this relation into Eq. (72) immediately confirms the result (75).
34 Numerous applications of that Neumann formula (derived in 1845 by F. Neumann) to electrical engineering problems may be found, for example, in the classical text by F. Grover, Inductance Calculations, Dover, 1946.
Chapter 5
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Essential Graduate Physics
EM: Classical Electrodynamics
This energy-based approach becomes virtually inevitable for continuously distributed currents.
As an example, let us calculate the self-inductance L of a long coaxial cable with the cross-section shown in Fig. 8, 35 and the full current in the outer conductor equal and opposite to that ( I) in the inner conductor.
a
b
0
c
I
Fig. 5.8. The cross-section of a coaxial cable.
I
Let us assume that the current is uniformly distributed over the cross-sections of both conductors. (As we know from the previous chapter, this is indeed the case if both the internal and external conductors are made of a uniform resistive material.) First, we should calculate the radial distribution of the magnetic field – which has only one, azimuthal component because of the axial symmetry of the problem. This distribution may be immediately found applying the Ampère law (37) to circular contours of radii within four different ranges:
2
2
a ,
for ρ a,
,
1
for a ρ b,
2 B I
I
(5.77)
0
contour
the
piercing
0
2 c 2 2 c 2 b, for b c,
,
0
for c .
Now, an easy integration yields the magnetic energy per unit length of the cable:
2
a
2
2
2
U
1
I
b
c
2
2
2
c
0
1
2
2
B d r
B d
d
d
d
l
2
4
2
a
( 2
2
c b )
0
0 0
0
a
b
(5.78)
2
2
b
c
c
c
I
0
1
2
ln
ln
.
2
2
2
2
2
a
c b c b
b
2
2
From here, and Eq. (72), we get the final answer:
L
2
b
c
2
c
c
1
0 ln
.
(5.79)
2
2
ln
2
2
l
2 a c b c b
b
2
Note that for the particular case of a thin outer conductor, c – b << b, this expression reduces to L
b
0
1
ln ,
(5.80)
l
2 a 4
where the first term in the parentheses is due to the contribution of the magnetic field energy in the free space between the conductors. This distinction is important for some applications because in 35 As a reminder, the mutual capacitance C between the conductors of such a system was calculated in Sec. 2.3.
Chapter 5
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superconductor cables, as well as the normal-metal cables as high frequencies (to be discussed in the next chapter), the field does not penetrate the conductor’s bulk, so that Eq. (80) is valid without the last term ¼ in the parentheses – for any b < c.
As the last example, let us calculate the mutual inductance between a long straight wire and a round wire loop adjacent to it (Fig. 9), neglecting the thickness of both wires.
y
I
2
R
I
Fig. 5.9. An example of the
1
I 1
x
mutual inductance calculation.
0
Here there is no problem with using the last approach discussed above, based on the direct calculation of the magnetic flux. Indeed, as was discussed in Sec. 1, the field B1 induced by the current I 1 at any point of the round loop is normal to its plane – e.g., to the plane of drawing of Fig. 9. In the Cartesian coordinates shown in that figure, Eq. (20) reads B 1 = 0 I 1/2 y, giving the following magnetic flux through the loop:
R
R
2 2
R x 1/ 2
I
dy
I R R R x
I R
0 1
0 1
2 21/2
1
0
1 1
2
1
1/2
Φ
dx
ln
dx
ln
d (5.81)
21
2
y
R
R
R
x
R
2
2
R x
1/ 2
1/ 2
2
2
2
1/ 2
0
0
1
.
1
This is a table integral equal to ,36 so that 21 = 0 I 1 R, and the final answer for the mutual inductance M L 12 = L 21 = 21/ I 1 is finite (and very simple):
M R ,
(5.82)
0
despite the magnetic field's divergence at the lowest point of the loop ( y = 0).
Note that in contrast with the finite mutual inductance of this system, the self-inductances of both its wires are formally infinite in the thin-wire limit – see, e.g., Eq. (80), which in the limit b/ a >> 1
describes a thin straight wire. However, since this divergence is very weak (logarithmic), it is quenched by any deviation from this perfectly axial geometry. For example, a fair estimate of the inductance of a wire of a large but finite length l >> a may be obtained from Eq. (80) by the replacement of b with l:
l 0 l
L
ln .
(5.83)
2
a
(Note, however, that the exact result depends on where from/to the current flows beyond that segment.) It turns out that a similar approximate result, with l replaced with 2 R in the front factor, and with R
under the logarithm, is valid for the self-inductance of a round loop with a << R. (A proof of this fact is a very useful exercise, highly recommended to the reader.)
36 See, e.g., MA Eq. (6.13), with a = 1.
Chapter 5
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EM: Classical Electrodynamics
5.4. Magnetic dipole moment, and magnetic dipole media
The most natural way of the magnetic media description parallels that for dielectrics in Chapter 3, and is based on the properties of magnetic dipoles – the notion close (but not identical!) to that of the electric dipoles discussed in Sec. 3.1. To introduce this notion quantitatively, let us consider, just as in Sec. 3.1, a spatially-localized system with a current distribution j(r ’), whose magnetic field is measured at relatively large distances r >> r’ (Fig. 10).
(
j '
r )
r
'
r
a
0
Fig. 5.10. Calculating the magnetic field of localized
V
currents at a distant point ( r >> a).
Applying the truncated Taylor expansion (3.5) of the fraction 1/r – r ’ to the vector potential given by Eq. (28), we get
1
1
A(r) 0
(
j '
r ) d 3 r'
r ' r
(
j r )
.
(5.84)
3
' d 3 r'
4 r
r
V
V
Now, due to the vector character of this potential, we have to depart somewhat from the approach of Sec. 3.1 and use the following vector algebra identity:37
f (j g) g(j f ) 3
d
0
r
,
(5.85)
V
that is valid for any pair of smooth (differentiable) scalar functions f(r) and g(r), and any vector function j(r) that, as the dc current density, satisfies the continuity condition j = 0 and whose normal component vanishes on the surface of the volume V. First, let us use Eq. (85) with f equal to 1, and g equal to any Cartesian component of the radius-vector r: g = rl ( l = 1, 2, 3). Then it yields (j n ) 3
3
d r j d r 0
,
(5.86)
l
l
V
V
so that for the vector as the whole
j r 3
d r 0
,
(5.87)
V
showing that the first term on the right-hand side of Eq. (84) equals zero. Next, let us use Eq. (85) again, but now with f = rl, g = rl’ (where l, l’ = 1, 2, 3); then it yields
r j r j d
r
,
(5.88)
l l '
l ' l
3
0
V
so that the l th Cartesian component of the second integral in Eq. (84) may be transformed as 37 See, e.g., MA Eq. (12.3) with the additional condition jn S = 0, pertinent to space-restricted currents.
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3
3
3
1 3
(r '
r ) j d r'
r '
r j d r'
r ( r' j r' j ) 3
d r'
l
l'
l' l
l'
l' l
l' l
V
V l' 1
2 l' 1
V
(5.89)
1 3
3
1
r ( r' j r' j ) d r'
r ( '
r ) 3
j d r'
.
2
l'
l' l
l' l'
l' 1
2
V
V
l
As a result, Eq. (84) may be rewritten as
m r
0
A(r)
,
(5.90)
3
4 r
Magnetic
where the vector m, defined as38
dipole and
its potential
1
m r (jr d 3
)
r ,
(5.91)
2 V
is called the magnetic dipole moment of a field source – that itself, within the long-rang approximation (90), is called the magnetic dipole.
Note a close analogy between the m defined by Eq. (91), and the orbital39 angular momentum of a non-relativistic particle with mass mk:
L r p r m v ,
(5.92)
k
k
k
k
k
k
where p k = mkv k is its linear momentum. Indeed, for a continuum of such particles with equal electric charges q, distributed with spatial density n, we have j = qnv, and Eq. (91) yields 1
3
nq
m
r j d r
r v d 3 r
,
(5.93)
2
2
V
V
while the total angular momentum of such a system of particles of equal masses m 0, is L nm r d 3
v r ,
0
V
so that we get a very straightforward relation
q
m
L .
(5.95) m vs. L
2 m 0
For the orbital motion, this classical relation survives in quantum mechanics for linear operators, and hence for eigenvalues of the observables. Since the orbital angular momentum is quantized in the units of the Planck constant , the orbital magnetic moment of an electron is always a multiple of the so-called Bohr magneton
e
,
(5.96) Bohr
B
2 m
magneton
e
where m e is the free electron mass.40 However, for particles with spin, such a universal relation between the vectors m and L is no longer valid. For example, the electron’s spin s = ½ gives a contribution of /2
to its mechanical angular momentum, but a contribution very close to B to its magnetic moment.
38 In the Gaussian units, the definition (91) is kept valid “as is”, so that Eq. (90) is stripped of the factor 0/4.
39 This adjective is used, especially in quantum mechanics, to distinguish the motion of a particle as a whole (not necessarily along a closed orbit!) from its intrinsic angular momentum, the spin – see, e.g., QM Chapters 3-6.
40 In the SI units, m e 0.9110-30 kg, so that B 0.9310-23 J/T.
Chapter 5
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EM: Classical Electrodynamics
The next important example of a magnetic dipole is a planar thin-wire loop, limiting area A (of arbitrary shape), and carrying current I, for which m has a surprisingly simple form,
m A
I ,
(5.97)
where the modulus of the vector A equals the loop’s area A, and its direction is normal to the loop’s plane. This formula may be readily proved by noticing that if we select the coordinate frame origin on the plane of the loop (Fig. 11), then the elementary component of the magnitude of the integral (91), 1
1
1
dm
r I dr I r dr I r 2 d , (5.98)
2
2
2
C
C
C
is just the elementary area dA = (1/2) rd( r) = r 2 d/2 – the equality already used in CM Eq. (3.40).
A
rd
C
r
dA
Fig. 5.11. Calculating the
magnetic dipole moment
0
of a planar current loop.
I
The comparison of Eqs. (96) and (97) allows a useful estimate of the scale of atomic currents, by finding what current I should flow in a circular loop of the atomic size scale (the Bohr radius) r B ~
0.510-10 m, i.e. of an area A ~ 10-20 m2, to produce a magnetic moment of the order of B.41 The result is surprisingly macroscopic: I ~ 1 mA – quite comparable to the currents driving the sound in your phone’s earbuds. Though due to the quantum-mechanical spread of electron's wavefunctions, this estimate should not be taken too literally, it is very useful for getting a gut feeling of how significant the atomic magnetism is, and hence why ferromagnets may provide such strong magnetic fields.
After these illustrations, let us return to the discussion of the general Eq. (90). Plugging it into (also general) Eq. (27), we may calculate the magnetic field of a magnetic dipole: 42
41 Another way to arrive at the same estimate is to take I ~ ef = e/2 with ~ 1016 s-1 being the typical frequency of radiation due to atomic interlevel quantum transitions.
42 Similarly to the situation with the electric dipoles (see Eq. (3.24) and its discussion), it may be shown that the magnetic field of any closed current loop (or any system of such loops) satisfies the following equality: B r 3
( ) d r 2 /
3 m
,
0
V
where the integral is over any sphere confining all the currents. On the other hand, as we know from Sec. 3.1, for a field with the structure (99), derived from the long-range approximation (90), such an integral vanishes. As a result, to get a coarse-grain description of the magnetic field of a small system located at r = 0, that would give the correct average value of the magnetic field, Eq. (99) should be modified as follows:
r
3 r
( m)
2
r
0
8
B
r
( )
m
m
,
cg
r
5
4
r
3
in a conceptual (though not quantitative) similarity to Eq. (3.25).
Chapter 5
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EM: Classical Electrodynamics
2
3r(r m) m r
Magnetic
0
B(r)
.
(5.99) dipole’s
5
4
r
field
The structure of this formula exactly replicates that of Eq. (3.13) for the electric dipole field – including the sign). Because of this similarity, the energy of a dipole of a fixed magnitude m in an external field, and hence the torque and the force exerted on it by a fixed external field, are given by expressions fully similar to those for an electric dipole – see Eqs. (3.15)-(3.19):43
Magnetic
U m
B ,
(5.100) dipole
ext
in external
and as a result,
field
τ m B ,
(5.101)
ext
F (m B ) .
(5.102)
ext
Now let us consider a system of many magnetic dipoles (e.g., atoms or molecules), distributed in space with an atomic-scale-averaged density n. Then we can use Eq. (90) generalized in an evident way for an arbitrary position r ’ of the dipole, and the linear superposition principle, to calculate the macroscopic vector potential A:
M(r ) (r r )
0
'
'
A(r)
d 3 r' ,
(5.103)
4
r ' 3
r
where M nm is the magnetization: the average magnetic moment per unit volume. Transforming this integral absolutely similarly to how Eq. (3.27) had been transformed into Eq. (3.29), we get:
' M '
r
0
( )
A(r)
3
d r' .
(5.104)
4
r '
r
Comparing this result with Eq. (28), we see that M is equivalent, in its magnetic effect, to the density jef of a certain effective “magnetization current”. Just as the electric-polarization charge ef discussed in Sec. 3.2 (see Fig. 3.4), the vector jef = M may be interpreted as the uncompensated part of the loop currents representing single magnetic dipoles m – see Fig. 12. Note, however, that since the atomic magnetic dipoles may be due to particles’ spins, rather than the actual electric currents due to the orbital motion, the magnetization current’s nature is not as direct as that of the polarization charge.
j
ef
Fig. 5.12. A cartoon illustrating the physical nature of
the effective magnetization current j
ef = M.
43 Note that the fixation of m and Bext effectively means that the currents producing them are fixed – please have one more look at Eqs. (35) and (97). As a result, Eq. (100) is a particular case of Eq. (53) rather than (54) – hence the minus sign.
Chapter 5
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Now, using Eq. (28) to add the possible contribution from the stand-alone currents j not included in the currents of microscopic magnetic dipoles, we get the general expression for the vector potential of the macroscopic field:
(
j r )
M(r )
0
' '
'
A(r)
d 3 r' .
(5.105)
4
r '
r
Repeating the calculations that have led us from Eq. (28) to the Maxwell equation (35), with the account of the magnetization current term, for the macroscopic magnetic field B we get
B
.
(5.106)
0 j M
Following the same reasoning as in Sec. 3.2, we may recast this equation as
H j,
(5.107)
where the field defined as
B
Magnetic
H
M ,
(5.108)
field H
0
for historic reasons (and very unfortunately) is also called the magnetic field.44 This is why it is crucial to remember that the physical sense of field H is very much different from field B. To understand this difference better, let us use Eq. (107) to bring Eqs. (3.32), (3.36), (29), and (107) together, writing them as the following system of macroscopic Maxwell equations (again, so far for the stationary case / t =
0):45
Stationary
E ,
0
H , j
macroscopic
(5.109)
Maxwell
D ,
B 0.
equations
These equations clearly show that the roles of the vector fields D and H are very similar: they both may be called “would-be fields” – meaning the fields that would be induced by the stand-alone charges and currents j, if the medium had not modified them by its dielectric and magnetic polarization.
Despite this similarity, let me note an important difference of signs in the relation (3.33) between E, D, and P, on one hand, and the relation (108) between B, H, and M, on the other hand. This is not just a matter of definition. Indeed, due to the similarity of Eqs. (3.15) and (100), including similar signs, the electric and magnetic fields both try to orient the corresponding dipole moments along the field.
Hence, in the media that allow such an orientation (and as we will see momentarily, for magnetic media it is not always the case), the induced polarizations P and M are directed along, respectively, the vectors E and B of the genuine (though macroscopic, i.e. atomic-scale-averaged) fields. According to Eq. (3.33), if the would-be field D is fixed – say, by a fixed stand-alone charge distribution (r) – such a polarization reduces the electric field E = (D – P)/0. On the other hand, Eq. (108) shows that in a magnetic media with a fixed would-be field H, the magnetic polarization making M parallel to B, 44 This confusion is exacerbated by the fact that in Gaussian units, Eq. (108) has the form H = B – 4M, and hence the fields B and H have the same dimensionality (and are formally equal in free space!) – though the unit of H has a different name ( oersted, abbreviated as Oe). Mercifully, in the SI units, the dimensionality of B and H is different, with the unit of H called the ampere per meter.
45 Let me remind the reader once again that in contrast with the system (36) of the Maxwell equations for the genuine (microscopic) fields, the right-hand sides of Eqs. (109) represent only the stand-alone charges and currents, not included in the microscopic electric and magnetic dipoles.
Chapter 5
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EM: Classical Electrodynamics
enhances the magnetic field B = 0(H + M). This difference may be traced back to the sign difference in the basic relations (1) and (2), i.e. to the fundamental fact that the electric charges of the same sign repulse, while the currents of the same direction attract each other.
5.5. Magnetic materials
In order to form a complete system, sufficient for the calculation of all fields from given (r) and j(r), the macroscopic Maxwell equations (109) have to be complemented with the constitutive relations describing the medium: D E, j E, and B H. The first two of them were discussed, in brief, in the last two chapters; let us proceed to the last one.
A major difference between the dielectric and magnetic constitutive relations D(E) and B(H) is that while a dielectric medium always reduces the external field, magnetic media may either reduce or enhance it. To quantify this fact, let us consider the most common case – linear magnetic materials in that M (and hence H) is proportional to B. For isotropic materials, this proportionality is characterized by a scalar – either the magnetic permeability defined by the following relation: B H
,
(5.110) Magnetic
permeability
or the magnetic susceptibility 46 defined as
M H .
(5.111) Magnetic
m
susceptibility
Plugging these relations into Eq. (108), we see that these two parameters are not independent, but are related as
1
( ) .
(5.112)
m
0
m vs.
Note that despite the superficial similarity between Eqs. (110)-(112) and the corresponding relations (3.43)-(3.47) for linear dielectrics:
D ,
E
P
,
E
1 ,
(5.113)
e 0
e 0
there is an important conceptual difference between them. Namely, while the vector E on the right-hand sides of Eqs. (113) is the actual (though macroscopic) electric field, the vector H on the right-hand side of Eqs. (110)-(111) represents a “would-be” magnetic field, in most aspects similar to D rather than E –
see, for example, Eqs. (109). This historic difference in the traditional form of the constitutive relations for the electric and magnetic fields is not without its physical reasons. Most experiments with electric and magnetic materials are performed by placing their samples into nearly-uniform electric and magnetic fields, and the simplest systems for their implementation are, respectively, plane capacitors (Fig. 2.3) and long solenoids (Fig. 6). The field in the former system may be most conveniently 46 According to Eqs. (110) and (112), i.e. in the SI units, m is dimensionless, while has the same dimensionality as 0. In the Gaussian units, is dimensionless: ()Gaussian = ()SI/0, and m is also introduced differently, as = 1
+ 4m, Hence, just as for the electric susceptibilities, these dimensionless coefficients are different in the two systems: (m )SI = 4(m)Gaussian. Note also that m is formally called the volumic magnetic susceptibility, to distinguish it from the atomic (or “molecular”) susceptibility defined by a similar relation, m H, where m is the induced magnetic moment of a single dipole – e.g., an atom. ( is an analog of the electric atomic polarizability – see Eq. (3.48) and its discussion.) In a dilute medium, i.e. in the absence of substantial dipole-dipole interactions, m = n, where n is the dipole density.
Chapter 5
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controlled by fixing the voltage V between its plates, which is proportional to the electric field E. On the other hand, the field provided by the solenoid may be fixed by the current I in it, and according to Eq.
(107), the field proportional to this stand-alone current is H, rather than B.47
Table 1 lists the approximate magnetic susceptibility values for several materials. It shows that in contrast to linear dielectrics whose susceptibility e is always positive, i.e. the dielectric constant =
e + 1 is always larger than 1 (see Table 3.1), linear magnetic materials may be either paramagnets (with m > 0, i. e. > 0) or diamagnets (with m < 0, i.e. < 0).
Table 5.1. Susceptibility (
m)SI of a few representative and/or important magnetic materials(a)
“Mu-me
tal” (75% Ni + 15% Fe + a few %% of Cu and Mo)
~20,000(b)
Perm
alloy (80% Ni + 20% Fe)
~8,000(b)
“Electrical” (or “transform
er”) steel (Fe + a few %% of Si)
~4,000(b)
Nickel
~100
Aluminum
+210-5
Oxygen (at am
bient conditions)
+0.210-5
Water
–0.910-5
Diamond
–210-5
Copper
–710-5
Bism
uth (the strongest non-superconducting diamagnet)
–1710-5
(a)
The table does not include bulk superconductors, which may be described, in a so-called coarse-scale approximation, as perfect diamagnets (with B = 0, i.e. formally with m = –1 and = 0), though the actual physics of this phenomenon is different – see Sec. 6.3 below.
(b)
The exact values of m >> 1 for soft ferromagnetic materials (see, e.g., the upper three rows of the table) depend not only on their composition but also on their thermal processing (“annealing”).
Moreover, due to unintentional vibrations, the extremely high values of
m of such materials may
decay with time, though they may be restored to the original values by new annealing. The reason for such behavior is discussed in the text below.
The reason for this difference is that in dielectrics, two different polarization mechanisms (schematically illustrated by Fig. 3.7) lead to the same sign of the average polarization – see the discussion in Sec. 3.3. One of these mechanisms, illustrated by Fig. 3.7b, i.e. the ordering of spontaneous dipoles by the applied field, is also possible for magnetization – for the atoms and molecules with spontaneous internal magnetic dipoles of magnitude m 0 ~ B, due to their net spins.
Again, in the absence of an external magnetic field the spins, and hence the dipole moments m0 may be disordered, but according to Eq. (100), the external magnetic field tends to align the dipoles along its direction. As a result, the average direction of the spontaneous elementary moments m0, and hence the direction of the arising magnetization M, is the same as that of the microscopic field B at the points of the dipole location (i.e., for a diluted media, of H B/0), resulting in a positive susceptibility m, i.e. in the paramagnetism, such as that of oxygen and aluminum – see Table 1.
47 This fact also explains the misleading term “magnetic field” for H.
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However, in contrast to the electric polarization of atoms/molecules with no spontaneous electric dipoles, which gives the same sign of e – 1 (see Fig. 3.7a and its discussion), the magnetic materials with no spontaneous atomic magnetic dipole moments have m < 0 – the effect called the orbital (or
“Larmor”48) diamagnetism. As the simplest model of this effect, let us consider the orbital motion of an atomic electron about an atomic nucleus as that of a classical particle of mass m 0, with an electric charge q, about an immobile attracting center. As classical mechanics tells us, the central attractive force does not change the particle’s angular momentum L m 0rv, but the applied magnetic field B (that may be taken uniform on the atomic scale) does, due to the torque (101) it exerts on the magnetic moment (95): L
d
q
τ m B
L B .
(5.114)
dt
2 m 0
The diagram in Fig. 13 shows that in the limit of a relatively weak field, when the magnitude of the angular momentum L may be considered constant, this equation describes the rotation (called the torque-induced precession 49) of the vector L about the direction of the vector B, with the angular frequency = – qB/2 m 0, independent of the angle . According to Eqs. (91) and (114), the resulting additional (field-induced) magnetic moment m q – q 2B/ m 0 has, irrespectively of the sign of q, a direction opposite to the field. Hence, according to Eq. (111) with H B/0, the susceptibility m
m/H is indeed negative. (Let me leave its quantitative estimate within this classical model for the reader’s exercise.) The quantum-mechanical treatment confirms this qualitative picture of the Larmor diamagnetism, giving only quantitative corrections to the classical result for m.50
B
L sin
L
dL / dt
v
0
m , q
0
Fig. 5.13. The torque-induced precession of a
m
classical charged particle in a magnetic field.
A simple estimate (also left for the reader’s exercise) shows that in atoms with spontaneous nonzero net spins, the magnetic dipole orientation mechanism prevails over the orbital diamagnetism, so that the materials incorporating such atoms usually exhibit net paramagnetism – see Table 1. Due to possible strong quantum interaction between the spin dipole moments, the magnetism of such materials is rather complex, with numerous interesting phenomena and elaborate theories. Unfortunately, all this physics is well outside the framework of this course, and I have to refer the interested reader to special literature,51 but still will mention some key facts.
48 Named after Sir Joseph Larmor who was the first (in 1897) to describe this effect mathematically.
49 For a detailed discussion of this effect see, e.g., CM Sec. 4.5.
50 See, e.g., QM Sec. 6.4. Quantum mechanics also explains why in most common ( s-) ground states, the average contribution (95) of the orbital angular momentum L to the net vector m vanishes.
51 See, e.g., D. J. Jiles, Introduction to Magnetism and Magnetic Materials, 2nd ed., CRC Press, 1998, or R. C.
O’Handley, Modern Magnetic Materials, Wiley, 1999.
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Most importantly, a sufficiently strong magnetic dipole-dipole interaction may lead to their spontaneous ordering, even in the absence of the applied field. This ordering may correspond to either parallel alignment of the dipoles ( ferromagnetism) or anti-parallel alignment of the adjacent dipoles ( antiferromagnetism). Evidently, the external effects of ferromagnetism are stronger, because this phase corresponds to a substantial spontaneous magnetization M even in the absence of an external magnetic field. (The corresponding magnitude of B = 0M is called the remanence field, B R.) The direction of the vector BR may be switched by the application of an external magnetic field, with a magnitude above a certain value H C called coercivity, leading to the well-known hysteretic loops on the [ H, B] plane (see Fig. 14 for a typical example) – similar to those in ferroelectrics, already discussed in Sec. 3.3.
Fig. 5.14. Experimental magnetization
curves of specially processed (cold-rolled)
electrical steel – a solid solution of ~10%
C and ~6% Si in Fe. (Reproduced from
www.thefullwiki.org/Hysteresis under the
Creative Commons BY-SA 3.0 license.)
Just as the ferroelectrics, the ferromagnets may also be hard or soft – in the magnetic rather than mechanical sense. In hard ferromagnets (also called permanent magnets), the dipole interaction is so strong that B stays close to B R in all applied fields below H C, so that the hysteretic loops are virtually rectangular. Hence, in lower fields, the magnetization M of a permanent magnet may be considered constant, with the magnitude B R/0. Such hard ferromagnetic materials (notably, rare-earth compounds such as SmCo5, Sm2Co17, and especially Nd2Fe14B), with high remanence fields (~1 T) and high coercivity (~106 A/m), have numerous practical applications.52 Let me give just two, most important examples.
First, permanent magnets are the core components of most electric motors. By the way, this venerable (~150-years-old) technology is currently experiencing a quiet revolution, driven mostly by the electric car development. In the most advanced type of motors, called permanent-magnet synchronous machines (PMSM), the remanence magnetic field B R of a permanent-magnet rotating part of the machine (called the rotor) interacts with the magnetic field of ac currents passed through wire windings in the external, static part of the motor (called the stator). The resulting torque may drive the rotor to extremely high speeds, exceeding 10,000 rotations per minute, enabling the motor to deliver several kilowatts of mechanical power from each kilogram of its mass.
As the second important example, despite the decades of the exponential ( Moore’s-law) progress of semiconductor electronics, most computer data storage systems (e.g., in data centers) are still based 52 Currently, the neodymium-iron-boron compound holds nearly 95% percent of the world permanent-magnet application market, due to its combination of high B R and HC with lower fabrication costs.
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on hard disk drives whose active media are submicron-thin layers of hard ferromagnets, with the data bits stored in the form of the direction of the remanent magnetization of small film spots. This technology has reached fantastic sophistication, with the recorded data density of the order of 1012 bits per square inch.53 (Only recently it started to be seriously challenged by solid-state drives based on the floating-gate semiconductor memories already mentioned in Chapter 3.) 54
In contrast, in soft ferromagnets, with their lower magnetic dipole interactions, the magnetization is constant only inside each of the spontaneously formed magnetic domains, while the volume and shape of the domains are affected by the applied magnetic field. As a result, the hysteresis loop’s shape of soft ferromagnets is dependent on the cycled field’s amplitude and cycling history – see Fig. 14. At high fields, their B (and hence M) is driven into saturation, with B B R, but at low fields, they behave essentially as linear magnetics with very high values of m and hence – see the top rows of Table 1.
(The magnetic domain interaction, and hence the low-field susceptibility of such soft ferromagnets are highly dependent on the material’s fabrication technology and its post-fabrication thermal and mechanical treatments.) Due to these high values of , soft ferromagnets, especially iron and its alloys (e.g., various special steels), are extensively used in electrical engineering – for example in the cores of transformers – see the next section.
Due to the relative weakness of the magnetic dipole interaction in some materials, their ferromagnetic ordering may be destroyed by thermal fluctuations, if the temperature is increased above some value called the Curie temperature T C, specific for each material. The transition between the ferromagnetic and paramagnetic phases at T = T C is a classical example of a continuous phase transition, with the average polarization M playing the role of the so-called order parameter that (in the absence of external fields) becomes different from zero only at T < T C, increasing gradually at the further temperature reduction.55
5.6. Systems with magnetic materials
Just as the electrostatics of linear dielectrics, the magnetostatics is very simple in the particular case when all essential stand-alone currents are embedded into a linear magnetic medium with a constant permeability . Indeed, let us assume that we know the solution B0(r) of the magnetic pair of 53 “A magnetic head slider [the read/write head – KKL] flying over a disk surface with a flying height of 25 nm with a relative speed of 20 meters/second [all realistic parameters – KKL] is equivalent to an aircraft flying at a physical spacing of 0.2 µm at 900 kilometers/hour. ” B. Bhushan, as quoted in the (generally good) book by G.
Hadjipanayis, Magnetic Storage Systems Beyond 2000, Springer, 2001.
54 The high-frequency properties of hard ferromagnets are also very non-trivial. For example, according to Eq.
(101), an external magnetic field Bext exerts torque = MBext on the spontaneous magnetic moment M of a unit volume of a ferromagnet. In some nearly-isotropic, mechanically fixed ferromagnetic samples, this torque causes the precession, around the direction of Bext (very similar to that illustrated in Fig. 13), of not the sample as such, but of the magnetization M inside it, with a certain frequency r. If the frequency of an additional ac field becomes very close to r, its absorption sharply increases – the so-called ferromagnetic resonance. Moreover, if
is somewhat higher than r, the effective magnetic permeability () of the material for the ac field may become negative, enabling a series of interesting effects and practical applications. Very unfortunately, I could not find time for their discussion in this series and have to refer the interested reader to literature, for example the monograph by A. Gurevich and G. Melkov, Magnetization Oscillations and Waves, CRC Press, 1996.
55 In this series, a quantitative discussion of such transitions is given in SM Chapter 4.
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the genuine (“microscopic”) Maxwell equations (36) in free space, i.e. when the genuine current density j coincides with that of stand-alone currents. Then the macroscopic Maxwell equations (109) and the linear constitutive equation (110) are satisfied with the pair of functions
Hr B0 r
,
Br H
r
B
.
(5.115)
0 r
0
0
Hence the only effect of the complete filling of a fixed-current system with a uniform, linear magnetic medium is the change of the magnetic field B at all points by the same constant factor /0 1
+ m, which may be either larger or smaller than 1. (As a reminder, a similar filling of a system of fixed stand-alone charges with a uniform, linear dielectric always leads to a reduction of the electric field E by a factor of /0 1 + e – the difference whose physics was already discussed at the end of Sec. 4.) However, this simple result is generally invalid in the case of nonuniform (or piecewise-uniform) magnetic samples. To analyze this case, let us first integrate the macroscopic Maxwell equation (107) along a closed contour C limiting a smooth surface S. Now using the Stokes theorem just as at the derivation of Eq. (37), we get the macroscopic version of the Ampère law (37): Macroscopic
H d I
Ampère
r
.
(5.116)
law
C
Let us apply this relation to a sharp boundary between two regions with different magnetic materials, with no stand-alone currents on the interface, similarly to how this was done for the field E in Sec. 3.4 – see Fig. 3.5. The result is similar as well:
H const
.
(5.117)
On the other hand, the integration of the Maxwell equation (29) over a Gaussian pillbox enclosing a border fragment (again just as shown in Fig. 3.5 for the field D) yields a result similar to Eq. (3.35): B const .
(5.118)
n
For linear magnetic media, with B = H, the latter boundary condition is reduced to H
const
.
(5.119)
n
Let us use these boundary conditions, first of all, to see what happens with a long cylindrical sample of a uniform magnetic material, placed parallel to a uniform external magnetic field B0 – see Fig.
15. Such a sample cannot noticeably disturb the field in the free space outside it, at most of its length: Bext = B0, Hext = 0Bext= 0B0. Now applying Eq. (117) to the dominating surfaces of the sample, we get Hint = H0.56 For a linear magnetic material, these relations yield Bint = Hint = (/0) B0.57 For the high-
media, this means that Bint >> B 0. This effect may be vividly represented as the concentration of the magnetic field lines in high- samples – see Fig. 15 again. (The concentration affects the external field 56 The independence of H on magnetic properties of the sample in this geometry explains why this field’s magnitude is commonly used as the argument in the plots like Fig. 14: such measurements are typically carried out by placing an elongated sample of the material under study into a long solenoid with a controllable current I, so that according to Eq. (116), H 0 = nI, regardless of the sample.
57 The reader is highly encouraged to carry out a similar analysis of the fields inside narrow gaps cut in a linear magnetic material, similar to that carried in Sec. 3.3 out for linear dielectrics – see Fig. 3.6 and its discussion.
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distribution only at distances of the order of (/0) t << l near the sample’s ends.) Such concentration is widely used in such practically important devices as transformers, in which two multi-turn coils are wound on a ring-shaped (e.g., toroidal, see Fig. 6b) core made of a soft ferromagnetic material (such as the transformer steel, see Table 1) with >> 0. This minimizes the number of “stray” field lines, and makes the magnetic flux piercing each wire turn of either coil virtually the same – the equality important for the secondary voltage induction – see the next chapter.
B0
B / B B
0
0
0
t l
~
t l
0
l
Fig. 5. 15. Magnetic field concentration in long, high- magnetic samples (schematically).
Samples of other geometries may create strong perturbations of the external field, extended to distances of the order of the sample’s dimensions. To analyze such problems, we may benefit from a simple, partial differential equation for a scalar function, e.g., the Laplace equation, because in Chapter 2 we have learned how to solve it for many simple geometries. In magnetostatics, the introduction of a scalar potential is generally impossible due to the vortex-like magnetic field lines. However, if there are no stand-alone currents within the region we are interested in, then the macroscopic Maxwell equation (107) for the field H is reduced to H = 0, similar to Eq. (1.28) for the electric field, showing that we may introduce the scalar potential of the magnetic field, m, using a relation similar to Eq. (1.33): H – .
(5.120)
m
Combining it with the homogenous Maxwell equation (29) for the magnetic field, B = 0, and Eq.
(110) for a linear magnetic material, we arrive at a single differential equation, (m) =0. For a uniform medium ((r) = const), it is reduced to our beloved Laplace equation:
2
0 .
(5.121)
m
Moreover, Eqs. (117) and (119) give us very familiar boundary conditions: the first of them
m const ,
(5.122a)
being equivalent to
const ,
(5.122b)
m
while the second one giving
m
const .
(5.123)
n
Indeed, these boundary conditions are absolutely similar for (3.37) and (3.56) of electrostatics, with the replacement .58
58 This similarity may seem strange because earlier we have seen that the parameter is physically more similar to 1/. The reason for this paradox is that in magnetostatics, the magnetic potential m is traditionally used to Chapter 5
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Let us analyze the geometric effects on magnetization, first using the (too?) familiar structure: a sphere, made of a linear magnetic material, placed into a uniform external field H0 B0/0. Since the differential equation and the boundary conditions are similar to those of the corresponding electrostatics problem (see Fig. 3.11 and its discussion), we can use the above analogy to reuse the solution we already have – see Eqs. (3.63). Just as in the electric case, the field outside the sphere, with 3
R
H r
(5.124)
m
0
cos ,
r R
0
2
2
r
0
is a sum of the uniform external field H0, with the potential – H 0 r cos – H 0 z, and the dipole field (99) with the following induced magnetic dipole moment of the sphere:59
0
3
m 4
R H .
(5.125)
0
20
On the contrary, the internal field is perfectly uniform, and directed along the external one:
3
H
3
B
H
3
H
r cos ,
that
so
,
. (5.126)
m
0
int
0
int
int
r R
0 2
H
2
B
H
2
0
0
0
0
0
0
0
Note that the field Hint inside the sphere is not equal to the applied external field H0. This example shows that the interpretation of H as the “would-be” magnetic field generated by external stand-alone currents j should not be exaggerated by saying that its distribution is independent of the magnetic bodies in the system. In the limit >> 0, Eqs. (126) yield H int/ H 0 << 1, B int/ H 0 = 30, the factor 3 being specific for the particular geometry of the sphere. If a sample is strongly stretched along the applied field, with its length l much larger than the scale t of its cross-section, this geometric effect is gradually decreased, and B int tends to its value H 0 >> B 0, as was discussed above – see Fig. 15.
Now let us calculate the field distribution in a similar, but slightly more complex (and practically important) system: a round cylindrical shell, made of a linear magnetic material, placed into a uniform external field H0 normal to its axis – see Fig. 16.
H
y sin
0
b
H
a
int
0
x cos
Fig. 5.16. Cylindrical magnetic shield.
describe the “would-be field” H, while in electrostatics, the potential describes the actual electric field E. (This tradition persists from the days when H was perceived as a genuine magnetic field.) 59 To derive Eq. (125), we may either calculate the gradient of the m given by Eq. (124), or use the similarity of Eqs. (3.13) and (99), to derive from Eq. (3.17) a similar expression for the magnetic dipole’s potential: 1 m cos
.
m
2
4
r
Now comparing this formula with the second term of Eq. (124), we immediately get Eq. (125).
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Since there are no stand-alone currents in the region of our interest, we can again represent the field H(r) by the gradient of the magnetic potential m – see Eq. (120). Inside each of three constant-
regions, i.e. at < b, a < < b, and b < (where is the 2D distance from the cylinder's axis), the potential obeys the Laplace equation (121). In the convenient, polar coordinates (see Fig. 16), we may, guided by the general solution (2.112) of the Laplace equation and our experience in its application to axially-symmetric geometries, look for m in the following form:
H b ' /
b
0
1
cos ,
for
,
(5.127)
m
a b /
a
b
1
1
cos
,
for
,
H cos
,
for a
.
int
Plugging this solution into the boundary conditions (122)-(123) at both interfaces ( = b and
= a), we get the following system of four equations:
H b b ' / b a b b / b,
a a b a H a
0
1
1
1
/
1
1
,
int
(5.128)
H b b H a b b
a b a H
0
' / 2
0
1
0
/ 2
1
1
,
/ 2
1
1
,
0
int
for four unknown coefficients a 1, b 1, b 1 ’, and H int. Solving the system, we get, in particular: 2
H
1
int
c
0
(5.129)
H
a b
c
/ ,
with
.
2
c
0
0
According to these formulas, at > 0, the field in the free space inside the cylinder is lower than the external field. This fact allows using such structures, made of high- materials such as permalloy (see Table 1), for passive shielding60 from unintentional magnetic fields (e.g., the Earth's field) – the task very important for the design of many physical experiments. As Eq. (129) shows, the larger is , the closer is c to 1, and the smaller is the ratio H int/ H 0, i.e. the better is the shielding, for the same a/ b ratio. On the other hand, for a given magnetic material, i.e. for a fixed parameter c, the shielding is improved by making the ratio a/ b < 1 smaller, i.e. the shield thicker. On the other hand, as Fig. 16 shows, smaller a leaves less space for the shielded samples, calling for a compromise.
Note that in the limit /0 , both Eq. (126) and Eq. (129), describing different geometries yield H int/ H 0 0. Indeed, as it follows from Eq. (119), in this limit the field H tends to zero inside magnetic samples of virtually geometry. (The formal exception is the longitudinal cylindrical geometry shown in Fig. 15, with t/ l 0, where Hint = H0 for any finite , but even in it, the last equality holds only if t/ l << 0/.)
Now let us discuss a curious (and practically important) approach to systems with relatively thin, closed magnetic cores made of several sections of high- magnetic materials, with the cross-section areas Ak much smaller than the squared lengths lk of the sections – see Fig. 17. If all k >> 0, virtually all field lines are confined to the interior of the core. Then, applying the macroscopic Ampère law (116) to a contour C that follows a magnetic field line inside the core (see, for example, the dashed line in Fig.
17), we get the following approximate expression (exactly valid only in the limit
2
k/0, lk / Ak ):
60 Another approach to the undesirable magnetic fields' reduction is the "active shielding" – the external field’s compensation with the counter-field induced by controlled currents in specially designed wire coils.
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B
H dl
l H l k NI .
(5.130)
l
k
k
k
C
k
k
k
However, since the magnetic field lines stay in the core, the magnetic flux k BkAk should be the same ( ) for each section, so that Bk = / Ak. Plugging this condition into Eq. (130), we get Magnetic
NI
l
Ohm law
k
Φ
,
where
.
(5.131)
k
and
R
R
A
k
k
k
reluctance
k
I
I
C
N
l
N
k
Fig. 5.17. Deriving the “magnetic Ohm law” (131).
k
0
Ak
Note a close analogy of the first of these equations with the usual Ohm law for several resistors connected in series, with the magnetic flux playing the role of electric current, while the product NI, the role of the voltage applied to the chain of resistors. This analogy is fortified by the fact that the second of Eqs. (131) is similar to the expression for the resistance R = l/ A of a long, uniform conductor, with the magnetic permeability playing the role of the electric conductivity . (To sound similar, but still different from the resistance R, the parameter R is called reluctance.) This is why Eq. (131) is called the magnetic Ohm law; it is very useful for approximate analyses of systems like ac transformers, magnetic energy storage systems, etc.
Now let me proceed to a brief discussion of systems with permanent magnets. First of all, using the definition (108) of the field H, we may rewrite the Maxwell equation (29) for the field B as
B
,
(5.132)
0
H M ,0
as
i.e.
H M
While this relation is general, it is especially convenient in permanent magnets, where the magnetization vector M may be approximately considered field-independent.61 In this case, Eq. (132) for H is an exact analog of Eq. (1.27) for E, with the fixed term –M playing the role of the fixed charge density (more exactly, of /0). For the scalar potential m, defined by Eq. (120), this gives the Poisson equation
2 M ,
(5.133)
m
similar to those solved, for quite a few geometries, in the previous chapters.
61 Note that in this approximation, there is no difference between the remanence magnetization M R B R/0, of the magnet and its saturation magnetization M S lim H[ B( H)/0 - H].
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In the case when M is not only field-independent but also uniform inside a permanent magnet’s volume, then the right-hand sides of Eqs. (132) and (133) vanish both inside the volume and in the surrounding free space, and give a non-zero effective charge only on the magnet’s surface. Integrating Eq. (132) along a short path normal to the surface and crossing it, we get the following boundary conditions:
H
(5.134)
n
Hn
Hn
M
M cos ,
free
in
space
magnet
in
n
where is the angle between the magnetization vector and the outer normal to the magnet’s surface.
This relation is an exact analog of Eq. (1.24) for the normal component of the field E, with the effective surface charge density (or rather /0) equal to M cos.
This analogy between the magnetic field induced by a fixed, constant magnetization and the electric field induced by surface electric charges enables one to reuse the solutions of quite a few problems considered in Chapters 1-3. Leaving a few such problems for the reader's exercise (see Sec. 7), let me demonstrate the power of this analogy on just two examples specific to magnetic systems. First, let us calculate the force necessary to detach the flat ends of two long, uniform rod magnets, of length l and cross-section area A << l 2, with the saturated remanent magnetization M0 directed along their length – see Fig. 18.
M 0
A
M 0
F
?
min
Fig. 5.18. Detaching two magnets.
l
l
Let us assume we have succeeded to detach the magnets by an infinitesimal distance << A 1/2, l.
Then, according to Eqs. (133)-(134), the distribution of the magnetic field near this small gap should be similar to that of the electric field in a system of two equal by opposite surface charges with the surface density proportional to M 0. From Chapters 1-3, we know the properties of such a system very well: within the gap, the field is virtually constant, uniform, proportional to , and independent of . For its magnitude in the magnetic case, Eq. (134) gives simply H = M 0, and hence B = μ 0 M 0. (Just outside of the gap, the field is very low, because due to the condition A << l 2, the effect of the similar effective charges at the "outer" ends of the rods on the field near the gap t is negligible.) From here, we can readily calculate F min as the force exerted by this field on the effective surface
"charges". However, it is even easier to find it from the following energy argument. Since the magnetic field energy localized inside the magnets and near their outer ends cannot depend on , this small detachment may only alter the energy inside the gap. For this part of the energy, Eq. (57) yields: B 2
M
0
0 2
U
V
A .
(5.135)
2
2
0
0
The gradient of this potential energy is equal to the attraction force F = –( U), trying to reduce Δ U by decreasing the gap, with the following magnitude:
U
2
M A
0
0
F
.
(5.136)
2
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The magnet detachment requires an equal and opposite external force. For a typical permanent magnet, with 0 M 0 B R ~ 1T, the force corresponds to a ratio F/ A close to 4105 Pa, a few times the normal atmospheric pressure.
Now let us consider the situation when similar long permanent magnets (such as the magnetic needles used in magnetic compasses) are separated, in otherwise free space, by a larger distance d >> A 1/2 – see Fig. 19. For each needle (Fig. 19a), of a length l >> A 1/2, the right-hand side of Eq. (133) is substantially different from zero only in two relatively small areas at the needle’s ends. Integrating the equation over each area, we see that at distances r >> A 1/2 from each end, we may reduce Eq. (132) to
H q (r r ) q (r r ) , (5.137)
m
m
where r are ends’ positions, and q m M 0 A, with A being the needle’s cross-section area. This equation is completely similar to Eq. (3.32) for the electric displacement D, for the particular case of two equal and opposite point charges, i.e. with = q(r – r+) – q(r – r+), with the only replacement q q m.
Since we know the resulting electric field all too well (see, e.g., Eq. (1.7) for E D/0), we may immediately write a similar expression for the field H:
1
r r
r r
Hr
q
.
(5.138)
m
4
3
3
r r
r r
(a)
(b)
H
q
q
m
m
qm
M
q
0
m
q
q
m
m
Fig. 5.19. (a) “Magnetic charges” at the ends of a thin permanent-magnet needle and (b) the result of its breaking into two parts (schematically).
The resulting magnetic field B(r) = 0H(r) exerts on another “magnetic charge” q’ m, located at some point r ’, the force F = q’ mB(r ’).62 Hence if two ends of different needles are separated by an intermediate distance R ( A 1/2 << R << l, see Fig. 19b), we may neglect one term in Eq. (138), and get the following “magnetic Coulomb law” for the interaction of the nearest ends:
R
0
F
q q'
.
(5.139)
4
m
m
3
R
The “only” (but conceptually, crucial!) difference between this interaction and that of the electric point charges is that the two “magnetic charges” (quasi-monopoles) of a magnetic needle cannot be fully 62 The simplest way to verify this (perhaps, obvious) expression is to check that for a system of two “charges”
q’ m, separated by vector a, placed into a uniform external magnetic field Bext, it yields the potential energy (100) with the correct magnetic dipole moment m = q ma – cf. Eq. (3.9) for an electric dipole.
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EM: Classical Electrodynamics
separated. For example, if we break a needle in the middle in an attempt to bring its two ends further apart, two new “point charges” appear – see Fig. 19b.
There are several solid-state systems where more flexible structures, similar in their magnetostatics to the needles, may be implemented. First of all, certain (“type-II”) superconductors may carry so-called Abrikosov vortices – flexible tubes with field-suppressed superconductivity inside, each carrying one quantum 0 = / e 210-15 Wb of the magnetic flux. Ending on superconductor’s surfaces, these tubes let their magnetic field lines spread into the surrounding free space, essentially forming magnetic monopole analogs – of course, with equal and opposite “magnetic charges” q m on each end of the tube – just as Fig. 19a shows. Such flux tubes are not only flexible but also stretchable, resulting in several peculiar effects – see Sec. 6.4 for more detail. Another recently found example of such paired quasi-monopoles is spin chains in the so-called spin ices – crystals with paramagnetic ions arranged into a specific (pyrochlore) lattice – such as dysprosium titanate Dy2Ti2O7.63 Let me emphasize again that any reference to magnetic monopoles in such systems should not be taken literally.
In order to complete this section (and this chapter), let me briefly discuss the magnetic field energy U, for the simplest case of systems with linear magnetic materials. In this case, we still may use Eq. (55), but if we want to operate only with macroscopic fields, and hence only stand-alone currents, we should repeat the manipulations that have led us to Eq. (57), using j not from Eq. (35), but from Eq.
(107). As a result, instead of Eq. (57) we get
2
2
B
B
H
Magnetic field
U ur 3
d r, with u
H
,
(5.140) energy:
2
2
2
linear medium
V
This result is evidently similar to Eq. (3.73) of electrostatics.
As a simple but important example of its application, let us again consider a long solenoid (Fig.
6a), but now filled with a linear magnetic material with permeability . Using the macroscopic Ampère law (116), just as we used Eq. (37) for the derivation of Eq. (40), we get
H In , and hence B In
,
(5.141)
where n N/ l, just as in Eq. (40), is the winding density, i.e. the number of wire turns per unit length.
(At = 0, we immediately return to that old result.) Now we may plug Eq. (141) into Eq. (140) to calculate the magnetic energy stored in the solenoid:
2
H
nI 2 lA
U uV
lA
,
(5.142)
2
2
and then use Eq. (72) to calculate its self-inductance:64
U
L
n 2
lA
(5.143)
I 2 / 2
We see that L V, so filling a solenoid with a high- material may allow making it more compact while preserving the same value of inductance. In addition, as the discussion of Fig. 15 has 63 See, e.g., L. Jaubert and P. Holdworth, J. Phys. – Cond. Matt. 23, 164222 (2011), and references therein.
64 Admittedly, we could get the same result simpler, just by arguing that since the magnetic material fills the whole volume of a substantial magnetic field in this system, the filling simply increases the vector B at all points, and hence its flux , and hence L / I by the factor /0 in comparison with the free-space value (75).
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shown, such filling reduces the fringe fields near the solenoid's ends, which may be detrimental for some applications, especially in physical experiments striving for high measurement precision.
However, we still need to explore the issue of magnetic energy beyond Eq. (140), not only to get a general expression for it in materials with an arbitrary dependence B(H), but also to finally prove Eq.
(54) and explore its relation with Eq. (53). I will do this at the beginning of the next chapter.
5.7. Exercise problems
5.1. DC current I flows around a thin wire loop bent into the form of a plane equilateral triangle with side a. Calculate the magnetic field in the center of the loop.
5.2. A circular wire loop, carrying a fixed dc current, has been placed
inside a similar but larger loop, carrying a fixed current in the same direction
– see the figure on the right. Use semi-quantitative arguments to analyze the
I
mechanical stability of the coaxial and coplanar position of the inner loop
I'
with respect to its possible angular, axial, and lateral displacements relative to
the outer loop.
5.3. Two straight, plane, parallel, long, thin conducting strips of width w,
I
separated by distance d, carry equal but oppositely directed currents I – see the figure on the right. Calculate the magnetic field in the plane located in the w
I
middle between the strips, assuming that the flowing currents are uniformly
d
distributed across the strip widths.
5.4. For the system studied in the previous problem, but now only in the limit d << w, calculate: (i) the distribution of the magnetic field in space,
(ii) the vector potential of the field,
(iii) the magnetic force (per unit length) exerted on each strip, and
(iv) the magnetic energy and self-inductance of the loop formed by the strips (per unit length).
z
R
d / 2
5.5. Calculate the magnetic field distribution near the center of the system of
I
two similar, plane, round, coaxial wire coils, carrying equal but oppositely directed 0
currents – see the figure on the right.
d / 2
I
z
R
5.6. The two-coil-system considered in the previous problem, now carries d / 2
equal and similarly directed currents – see the figure on the right.65 Calculate what I
should be the ratio d/ R for the second derivative 2 B
0
z/ z 2 to equal zero at z = 0.
d / 2
I
65 This system (called the Helmholtz coils), producing a highly uniform field near its center, is broadly used in physical experiment.
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EM: Classical Electrodynamics
j
5.7. DC current of a constant density j flows along a round cylindrical wire of
r
R
radius R, with a round cylindrical cavity of radius r cut in it. The cavity’s axis is parallel to that of the wire but offset from it by a distance d < R – r (see the figure on d
0
the right). Calculate the magnetic field inside the cavity.
I
5.8. Calculate the magnetic field’s distribution along the axis of a straight
R
solenoid (see Fig. 6a, partly reproduced on the right) with a finite length l, and N
l
round cross-section of radius R. Assume that the solenoid has many ( N >> 1, l/ R) I
wire turns, uniformly distributed along its length.
5.9. A thin round disk of radius R, carrying an electric charge of a constant areal density , rotates around its axis with a constant angular velocity . Calculate:
(i) the induced magnetic field on the disk’s axis,
(ii) the magnetic moment of the disk,
and relate these results.
5.10. A thin spherical shell of radius R, with charge Q uniformly distributed over its surface, rotates about its diameter with a constant angular velocity . Calculate the distribution of the magnetic field everywhere in space.
5.11. A sphere of radius R, made of an insulating material with a uniform electric charge density
, rotates about its diameter with a constant angular velocity . Calculate the magnetic field distribution inside the sphere and outside it.
5.12. The reader is hopefully familiar with the classical Hall effect in the usual rectangular Hall bar geometry – see the left panel of the figure below. However, the effect takes a different form in the so-called Corbino disk – see the right panel of the figure below. (Dark shading shows electrodes, with no appreciable resistance.) Analyze the effect in both geometries, assuming that in both cases, the conductors are thin and planar, have a constant Ohmic conductivity and charge carrier density n, and that the applied magnetic field B is uniform and normal to conductors’ planes.
I
I
I
I
R 2
B
w
R
1
B
l
5.13.* The simplest version of the famous homopolar (or “unipolar”) motor is a thin, round conducting disk, placed into a uniform magnetic field normal to its plane, with dc current passed between the disk’s center and a sliding electrode (“brush”) on its rim – see the figure below.
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EM: Classical Electrodynamics
(i) Express the torque rotating the disk via its radius R, the magnetic field B, and the current I.
B
(ii) If the disk is allowed to rotate about its axis, and the motor is
I
driven by a battery with e.m.f. V, calculate its stationary angular velocity
, neglecting friction and the electric circuit’s resistance.
(iii) Now assuming that the current circuit (battery + wires + V
I
contacts + disk itself) has a non-zero resistance R, derive and solve the
equation for the time evolution of , and analyze the solution.
5.14. Current I flows in a thin wire bent into a plane round loop of radius R. Calculate the net magnetic flux through the plane in which the loop is located.
5.15. A wire with a round cross-section of radius a has been bent into a round loop of radius R
>> r. Prove the formula for its self-inductance, mentioned at the end of Sec. 5.3 of the lecture notes: L =
0 R ln( cR/ a), with c ~ 1.
5.16. Prove that:
(i) the self-inductance L of a current loop cannot be negative, and
(ii)any inductance coefficient Lkk’, defined by Eq. (60), cannot be larger than ( LkkLk’k’)1/2.
5.17. Calculate the mutual inductance of two similar thin-wire
h
square-shaped loops, offset by distance h in the direction normal to their planes – see the figure on the right.
a
a
5.18.* Estimate the values of magnetic susceptibility due to
(i) orbital diamagnetism, and
(ii) spin paramagnetism,
for a medium with negligible interaction between the induced molecular dipoles. Compare the results.
Hints: For Task (i), you may use the classical model described by Eq. (114) – see Fig. 13. For Task (ii), assume the mechanism of ordering of spontaneous magnetic dipoles m0, with a magnitude m 0
