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2
Basic concepts
This section introduces the magnetization M and the two magnetic fields B and H. These are vectors which are
defined at every position r. Methods of calculating B and H are introduced.
Units and dimensions in magnetism are discussed.
2.1
Moments and fields
2.1.1. Magnetization.
This is the basic magnetic quantity in a solid.
m
I
Fig 2.1 All magnetism is due to circulating currents
m=IA
(2.1)
m magnetic dipole moment, I electric curent, A area vector. Units of m are A m2.
The magnetisation M of a solid is the magnetic dipole moment per unit volume.
M = m/V
Units of M are A m-1.
(2.2)
2.1.2. B and H fields
B
Fig 2.2. The magnetic field B produced by the current loop is divergenceless (solenoidal)
B is known as the magnetic flux density or B-field. Units are Tesla (T)
There are no magnetic 'poles' to act as sources or sinks of B (not like E)


(Maxwell's Eqn)
(2.3a)

The vector operator means (∂/∂x, ∂/∂y, ∂/∂z).
is the divergence (div) of a vector
∂/∂x + ∂/∂y + ∂/∂z
It can be written in integral form as
SB.dA = 0
(2.3b)
The relation between magnetic flux density and current density j (units A m-2) in the
steady state is given by another of Maxwell's equations which can also be written in
point form or integral form. In point form,
µoj
(2.4a)
Here µ0 is the magnetic constant
µ0 = 4π 10-7 T m A-1.
(2.5)
is the rotation (curl) of a vector
is given by the expression
ex(∂By/∂z - ∂By/∂y) + ey(∂Bz/∂x - ∂Bx/∂z) + ez(∂Bx/∂y - ∂By/∂x)
In integral form,
I
loopB.dl = µ0I
loo p
(2.4b)
Equation 2.4 is known as Ampere's law.
The difficulty wiTh (2.4a) is that the current density j is made up of contributions from
currents in external circuits j0 (which we can measure) and from the atomic currents jm that
create the magnetization of a solid (which we cannot measure).
The relation between jm and M is
jm
From (2.4a)
(2.6)
µo(j0 + jm)
µ0 - M)µoj0
Define µ0 - M)H', or
B = µ0(H' + M)
(2.7)
Then we can retain Ampere's law for the field H', which depends only on the measurable
currents j0.
'j0
loopH'.dl = I0
(2.8a)
(2.8b)
Eq 2.7 does not mean that H' is created only by the conduction currents. Any magnet creates
an H-field both in the space around it, and within its own volume.
We write
H' = Hd + H0
(2.9)
where H0 is created by the conduction currents j0 and Hd is the field created by the magnet. It
is known as the stray field outside the magnet, and the demagnetizing field inside the magnet.
In free space, the distinction between the B-field and the H-field is trivial, they are
proportional, though measured in different units; from (2.7),
B = µ0H'
(2.10)
hence the magnetic constant µ0 is known as the permeability of free space. Inside the magnet
the B-field and the H-field are quite different, and oppositely directed.
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Fig 2.3 Illustration of M, B and H' for a magnet. The relation (2.7) is illustrated at a point P.
2.1.2 External Fields.
An external field H which can be created by conduction currents or other magnets or
both creates a torque  on a magnet as it lowers its energy by aligning with the field.
Whenever H interacts with matter, µ0 comes into the equation.
In free space  = µ0 H
µ0mH sin;
 = µ0mH
= mB
(2.11)
 = -µ0m H
= -m B
(2.12)
The corresponding energy
E = -µ0mH cos;
m



H
Fig 2.4 A magnet in an external field
The relation between the internal field in a magnet H' and the external field H is
H' = Hd + H
(2.13)
Only in the case of a uniformly-magnetized ellipsoid is there a simple expression for Hd.
Hd = - N M
(2.14)
N is generally a 3 x 3 matrix with Nxx + Nyy + Nzz = 1 , but it reduces to a scalar demagnetizing
factor when M is along one of the principal axes, x, y, z.
It is common practice to use a demagnetizing factor to obtain approximate internal fields in
samples of other shapes, which may not be uniformly magnetized.
N
Examples.
Long needle, M parallel to axis,a
0
Long needle, M perpendicular to axis
1/2
Sphere
1/3
Thin film, M parallel to plane
Thin film, M perpendicular to plane
1
Toroid, M perpendicular to r
0
General ellipsoid of revolution
0
Nc = ( 1 - Na)/2
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Fig 2.5 Demagnetizing factors for general ellipsoids.
For ellipsoids of revolution with axes (a,c,c) explicit formulae are
Na = 2/(1-2){(1-2)-1/2 sinh-1[(1-2)1/2/] - 1}
(2.15a)
for prolate (cigar-shaped) ellipsoids with = c/a < 1 and
Na = 2/(1-2){1 - (1-2)-1/2 sinh-1[(1-2)1/2/]}
(2.15b)
for oblate (cigar-shaped) ellipsoids with  > 1. For nearly-spherical shapes ( ≈ 1)
Na = 1/3 - 1/15(
The state of magnetization of a sample depends on H, M = M(H). H is the independent
variable, also known as ‘magnetizing force’.
M
M
H
H'
Fig 2.6 Magnetization of a sphere in the external and internal fields
The magnetization of a uniform soft ellipsoid adjusts itself so that H' = 0 up to saturation,
hence the external susceptibility r defined as
r = M/H
is 1/N.
(2.16)
From (2.13) and (2.14)
H' = H - NM
(2.17)
Usually in paramagnets and diamagnets, the difference between the external susceptibility
(M/H) and the internal susceptibility (M/H') is negligible. Not so in ferromagnets. There
1/rint =1/r -N
(2.18)
A related quantity is the permeability, defined for a paramagnet, or a soft ferromagnet in
small fields as
µ = B/H
(2.19)
from (2.7)
µ = µ0(1 + r)
(2.20)
The relative permeability
µr = µ/µ0 = (1 + r)
(2.21)
2.2
Magnetic field calculations
From (2.2a) and (2.7), it follows that ’
Any divergence of M in the material produces a stray field.

There are three main methods for calculating magnetic fields created by a magnet.
¨
Dipole sum*
H(r) = 1/4π(r)r/r5 - M/r3]d3r
¡
Current sheet
H(r) = 1/4πjm(r-r’)/|r-r’|3d3r’ + 1/4πjs(r-r’)/|r-r’|3d2r’
¬
Pole distribution.
H(r) = -1/4πM(r-r’)/|r-r’|3d3r’ + 1/4πM.n(r-r’)/|r-r’|3d2r’
P
P

r
MdV

P
n
js


¨
¡
¬
For uniform magnetization, only the surface contributions count, jm = 0, js = M
Also m = -M = 0; s - M. . Here  is the unit vector normal to the surface.
* Equivalent expressions for the field of a dipole m are
H = 1/4π[(mr)r/r5 - m/r3] ;
H = 1/4πr3[2mcoser + msine];
(Hx, Hy, Hz) = (m/4πr3)(3sincoscos, 3sincossin, 3cos2 - 1).
Potential can be exploited in these calculations.
Provided there are no conduction currents present, H’ can be deduced from the the magnetic
scalar potential m; since 0 for any scalar ,
H = m
(2.22)
The analogy is with electrostatics. There is supposed to be a distribution of positive
and negative magnetic poles, which act as sources of H. There is a surface pole
density s = M.,, and a volume pole density m = -The potential of at distance
r from a magnetic pole p is p/4πr. The potential of a dipole m is m.r/4πr3.Units of
mare amps. The potential for a general, nonuniform magnetiztion distribution is
m = -1/4πM/|r-r’|d3r’ + 1/4πM.n/|r-r’|d2r’
(2.23)
Then equation ¬ above follows from (2.22) and (2.23).
Since = 0, it follows from (2.7) and (2.22) that msatisfies Poisson’s equation
m=  



(2.24)
More generally applicable is the magnetic vector potential A. There is no restriction
regarding conduction currents. The flux density at any point can be written as
B = A
(2.25)
This definition is consistent with the vector identity a  0, true for any vector a.
Units of A are T m. The equipotentials of A are the 'lines of force'. The vector
potential for a general, nonuniform magnetiztion distribution is
A = µ0/4πM/|r-r’|d3r’ + µ0/4πM/|r-r’|d2r’
(2.26)
Then equation ¡ above follows from (2.25) and (2.26), since jm = M js = M
The vector identity a = (a) - a, and the Coulomb gauge a = 0 give Poisson’s
equation for the vector potential
A = -µ0(jm + j)
(2.27)
Boundary Conditions: At any interface, it follows from (2.2) and (2.4) that the perpendicular
component of B and the parallel component of H are continuous (see Fig 2.3, 2.7).

H||
Fig.2.7. Boundary conditions for B and H.
2.3 Units
2.3.1 SI Units
We use SI throughout with the Sommerfeld convention (2.7). Engineers prefer the
Kenelly convention, both are consistent, compatible SI units.
B = µ0H + J
(2.19)
J is the polarization, with units of Tesla, like B. J = µoM.
At room temperature, for iron Js = 2.16 T
-1
M = 1.71 MA m
The international system is based on five fundamental units kg, m, s, K, and A.
Derived units include the newton (N) = kg.m/s2, joule (J) = N.m, coulomb (C) = A.s,
volt (V) = J/C, tesla (T) = J/Am2 = Vs/m2, weber (Wb) = V.s = T.m2 and hertz (Hz) =
s-1.
Recognized multiples are in steps of 10±3, but a few exceptions are admited such as
cm (10-2 m) and Å (10-10 m). Multiples of the meter are fm (10-15), pm (10-12), nm
(10-9), µm (10-6), mm (10-3) m (10-0) and km (103).
Flux density B is measured in telsa (also mT, µT). Magnetic moment is measured in
A.m2 so the magnetization and magnetic field are measured in A/m. From (2.12) it is
seen hat an equivalent unit for magnetic moment is J/T, so magnetization can also be
expressed as J/(T.m3). , the magnetic moment per unit mass in J/(T.kg) is the
quantity most usually measured in practice. µ0 is exactly 4π.10-7 T.m/A.
The SI system has two compelling advantages for magnetism: (i) it is possible to
check the dimensions of any expression by inspection and (ii) the units are directly
related to the practical units of electricity.
2.3.2 cgs Units
Much of the primary literature on magnetism is still written using cgs units, or a
confusing mixture where large fields are quoted in tesla and small ones in oersted!
Fundamental cgs units are cm, g and s. The electromagnetic unit of current is
equivalent to 10 A. The electromagnetic unit of potential is equivalent to 10 nV. The
electromagnetic unit of magnetic dipole moment (emu) is equivalent to 1 mA.m2.
Derived units include the erg (10-7 J) so that an energy density such as K1 of 1 J/m3
is equivalent to 10 erg/cm3.
The convention relating flux density and magnetization is
B = H+ 4πM
where the flux density or induction B is measured in gauss (G) and field H in oersted
(Oe). Magnetic moment is usually expresed as emu, and magnetization is therefore in
emu/cm3, although 4πM is frequently considerd a flux-density expression and quoted
in kilogauss. µ0 is numerically equal to 1 G/Oe, but it is normally omitted from the
equations. The most useful conversion factors between SI and cgs units in magnetism
are
B
1 T kG
H
1 kA/m  12.57 (≈12.5) Oe
m
1 J/T 1000 emu
(BH)max1 MJ/m3 MG.Oe

1 J/T.kg 1 emu/g
M
1 kA/m 1 emu/cm3
1 G  0.1 mT
1 Oe ≈A m-1
1 emu 1 mJ/T
1 MG.Oe kJ/m3
The dimensionless susceptibility M/H is a factor 4π larger in SI than in cgs.
Table 2.1: Magnetic units.
_________________________________________________________
SI relation
cgs relation
_________________________________________________________
B
B
M
J
H
Hmd
= µo(H + M) = µoH + J
B = H + 4 πM = H + I
=1T
B
= 10 kG
= 1 kA/m
M = 1 emu/cm3
=1T
4πM = 10 kG
= 1 kA/m
H
= 4π Oe
= -DM
Hmd = - 4πD M = - N M
(0 ≤ D ≤ 1)
(0 ≤ D ≤ 1, 0 ≤ N ≤ 4π)
2
m
= 1 J/T (A.m )
m = 1000 emu (erg/G)

= 1 J/Tkg 
= 1 emu/g

= ∂M/∂H = 1

= ∂M/∂H = 4π
(BH)max = 1 MJ/m3
(BH)max = 40π MG.Oe
E
= -Vµ0H.M
(J)
E = -VH.M
(erg)
2
2

= µ0m/4πr (A)

m/r (Oe.cm)
_________________________________________________________
Table 2.2: Energy units.
______________________________________________________
Unit
J
eV
K
_________________________________________________________________
J
= 1
6.242  1018
7.243  1022
-19
eV
= 1.6022 10
1
11.60  103
K
= 1.381  10-23
8.617  10-5
1
-24
-5
T
= 9.274  10
5.788  10
0.6717
mc2
= 8.188 10-14
0.5110  106
5.930  109
-21
-2
kJ/mole = 1.6605 10
1.036  10
120.3
kcal
= 4184
2.611  1022
3.030  1026
-18
Ry=
2.180  10
13.606
157.9  103
1/cm
= 1.986  10-23
12.389  10-5
1.4388
-34
-15
Hz
= 6.6261 10
4.136  10
4.799  10-11
______________________________________________________
2.4 Dimensions
In the SI system, the basic quantities are mass (m), length (l), time (t), charge (q) and
temperature (). Any other quantity has dimensions which are a combination of the
dimensions of these five basic quantities, m, l, t, q and . In any relation between a
combination of physical properties, all the dimensions must balance.
Table 2.3 Dimnsions
Mechanical
Quantity
area
volume
velocity
acceleration
density
energy
momentum
angular momentum
moment of inertia
force
power
pressure
stress
elastic modulus
frequency
diffusion coefficient
viscosity (dynamic)
viscosity (kinematic)
Planck’s constant
symbol
A
V
v
a
m
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
1
0
1


unit
J
J.K-1
J.K-1.kg-1
J.K-1
W.m-1.K-1
J.mol-1.K-1
m l t q
1 2 -2 0
1 2 -2 0
0 2 -2 0
1 2 -2 0
1 1 -3 0
1 2 -2 0
k
J.K-1

E
p
L
I
F
p
P
S
K

D


h
l
t
2
3
1
1
-3
2
1
2
2
1
2
-1
-1
-1
0
2
-1
2
2
0
0
-1
-2
0
-2
-1
-1
0
-2
-3
-2
-2
-2
-1
-1
-1
-1
-1
q 
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
unit
m2
m3
m.s-1
m.s-2
kg.m-3
J
kg.m.s-1
kg.m2.s-1
kg.m2
N
W
Pa
N.m-2
N.m-2
s-1
m2.s-1
N.s.m-2
m2.s-1
J.s
Thermal
Quantity
enthalpy
entropy
specific heat
heat capacity
thermal conductivity
Sommerfeld
coefficient
Boltzmann’s constant
symbol
H
S
C
c
1
2 -2

0
-1
-1
-1
-1
-1
0 -1
Electrical
Quantity
current
current density
potential
electromotive force
capacitance
resistance
resistivity
conductivity
dipole moment
electric polarization
electric field
electric displacement
electric flux
permitivity
thermopower
mobility
symbol
I
j
V

C
R


p
P
E
D


S
µ
unit
A
A.m-2
V
V
F

.m
S.m-1
C.m
C.m-2
V.m-1
C.m-2
C
F.m-1
V.K-1
m2V-1s-1
m
0
0
1
1
-1
1
1
-1
0
0
1
0
0
-1
1
-1
l
m
0
0
-1
0
1
1
1
0
1
1
0
0
1
1
1
0
l
0
-2
2
2
-2
2
3
-3
1
-2
1
-2
0
-3
2
0
t
-1
-1
-2
-2
2
-1
-1
1
0
0
-2
0
0
2
-2
1
q 
1 0
1 0
-1 0
-1 0
2 0
-2 0
-2 0
2 0
1 0
1 0
-1 0
1 0
1 0
2 0
-1 -1
1 0
t
-1
-1
-1
-1
-1
-1
0
0
0
-1
-1
-1
-2
-2
-2
0
q 
1 0
1 0
1 0
1 0
-1 0
-1 0
-2 0
0 0
-2 0
-1 0
1 0
1 0
0 0
0 0
0 0
-1 0
Magnetic
Quantity
magnetic moment
magnetisation
specific moment
magnetic field strength
magnetic flux
magnetic flux density
inductance
susceptibility (M/H)
permeability (B/H)
magnetic polarisation
magnetomotive force
magnetic ‘charge’
energy product
anisotropy energy
exchange coefficient
Hall coefficient
symbol unit
m
A.m2
M
A.m-1

A.m2.kg-1
H
A.m-1

Wb
B
T
L
H

µ
H.m-1
J
T
F
A
qm
A.m
(BH)
J.m-3
K
J.m-3
A
J.m-1
RH
m3.C-1
2
-1
2
-1
2
0
2
0
1
0
0
1
-1
1
1
3
Examples:
1)
Kinetic energy of a body; E = (1/2)mv2
[E] = [ 1, 2,-2, 0, 0]
2)
Lorentz force on a moving charge; F = qvxB
[F] = [ 1, 1,-2, 0, 0]
3)
[m] = [ 1, 0, 0, 0, 0]
[v2] = 2[ 0, 1,-1, 0, 0]
[ 1, 2,-2, 0, 0]
[q] =
[ 0, 0, 0, 1, 0]
[v] =
[B] =
[ 0, 1,-1, 0, 0]
[ 1, 0,-1,-1, 0]
[ 1, 1,-2, 0, 0]
Domain wall energy w = AK (w is an energy per unit area)
[w] = [EA-1]
[AK] =1/2[ AK]
= [ 1, 2,-2, 0, 0] [A]=1/2[ 1, 1,-2, 0, 0]
-[ 1, 1,-2, 0, 0] []=1/2[ 1,-1,-2, 0, 0]
= [ 1, 0,-2, 0, 0]
[ 1, 0,-2, 0, 0]
4)
Magnetohydrodynamic force on a moving conductor f = vxBxB (f is a force
per unit volume)
[f] = [ FV-1]
= [ 1, 1,-2, 0, 0]
-[ 0, 3, 0, 0, 0]
[ 1,-2,-2, 0, 0]
[] = [-1,-3, 1, 2, 0]
[v] =
[ 0, 1,-1, 0, 0]
[B2] = 2[ 1, 0,-1,-1, 0]
[ 1,-2,-2, 0, 0]
5)
Flux density in a solid B = µ0(H + M). (Note that quantities added or
subtracted in a bracket must have the same dimensions)
[B]= [ 1, 0,-1,-1, 0]
[µ0] =
[ 1, 1, 0,-2, 0]
[M],[H] = [ 0,-1,-1, 1, 0]
[ 1, 0,-1,-1, 0]
6)
Maxwell’s equation xH = j + dD/dt.
[xH] = [Hr-1]
= [ 0,-1,-1, 1, 0]
-[ 0, 1, 0, 0, 0]
= [ 0,-2,-1, 1, 0]
[j] = [ 0,-2,-1, 1, 0]
[dD/dt] = [Dt-1]
= [ 0,-2, 0, 1, 0]
-[ 0, 0, 1, 0, 0]
= [ 0,-2,-1, 1, 0]