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MAGNETOSTATICS
Mutual And Self-inductance; Magnetic
Fields In Material Media;
Magnetostatic Boundary Conditions;
Magnetic Forces And Torques
1
Objectives

To continue our study of magnetostatics
with mutual and self-inductance; magnetic
fields in material media; magnetostatic
boundary conditions; magnetic forces and
torques.
2
Lecture 8
Flux Linkage

Consider two magnetically coupled circuits
I2
S1
S2
C2
C1
I1
3
Lecture 8
Flux Linkage (Cont’d)

The magnetic flux produced I1 linking the
surface S2 is given by
12   B1  d s 2
S2

If the circuit C2 comprises N2 turns and the
circuit C1 comprises N1 turns, then the total
flux linkage is given by
12  N1 N 2 12  N1 N 2  B1  d s 2
S2
4
Lecture 8
Mutual Inductance

The mutual inductance between two
circuits is the magnetic flux linkage to one
circuit per unit current in the other circuit:
12 N1 N 2 12
L12 

I1
I1
5
Lecture 8
Neumann Formula for Mutual
Inductance
12 N1 N 2 12 N1 N 2
L12 


I1
I1
I1
N1 N 2

I1
B

d
s
1
2

S2
A

d
l
1
2

C2
6
Lecture 8
Neumann Formula for Mutual
Inductance (Cont’d)
N1 N 2
L12 
I1
 A  dl
1
2
C2
0 N1 N 2
d l1  d l 2



4 C C R12
1
2
 0 I1 d l 1
A1 

4 C R12
1
7
Lecture 8
Neumann Formula for Mutual
Inductance (Cont’d)

The Neumann formula for mutual
inductance tells us that
 L12
= L21
 the mutual inductance depends only on the
geometry of the conductors and not on the
current
8
Lecture 8
Self Inductance


Self inductance is a special case of mutual
inductance.
The self inductance of a circuit is the ratio of
the self magnetic flux linkage to the current
producing it:
11 N1 11
L11 

I1
I1
2
9
Lecture 8
Self Inductance (Cont’d)

For an isolated circuit, we call the self
inductance, inductance, and evaluate it
using
 N 
L 
I
I
2
10
Lecture 8
Generation of Magnetic Field
iron core
I
N
S
air gap with
constant B
field
I
11
Lecture 8
Equivalent of a Magnetic Dipole
N
I
• Magnetic dipole
can be viewed as a
pair of magnetic
charges by analogy
with electric dipole.
S
12
Lecture 8
Forces Exerted on a Magnetic
Dipole in a Magnetic Field
N
B
S
13
Lecture 8
Current Loops (Magnetic
Dipoles) in Atoms
Electron orbiting nucleus
 Electron spin
 Nuclear spin
negligible

 A complete understanding of these atomic
mechanisms requires application of quantum
mechanics.
14
Lecture 8
Current Loops (Magnetic
Dipoles) in Atoms (Cont’d)


In the absence of an applied magnetic field, the
infinitesimal magnetic dipoles in most materials
are randomly oriented, giving a net macroscopic
magnetization of zero.
When an external magnetic field is applied, the
magnetic dipoles have a tendency to align
themselves with the applied magnetic field.
15
Lecture 8
Magnetized Materials
A material is said to be magnetized when
induced magnetic dipoles are present.
 The presence of the induced magnetic
dipoles modifies the magnetic field both
inside and outside of the magnetized
material.

16
Lecture 8
Permanent Magnets
Most materials lose their magnetization
when the external magnetic field is
removed.
 A material that remains magnetized in the
absence of an applied magnetic field is
called a permanent magnet.

17
Lecture 8
Magnetization Vector

The magnetization or net magnetic
dipole moment per unit volume is given by
M  Nm
[A/m]
Number of
dipoles per unit
volume [m-3]
18
average
magnetic
dipole
moment
[Am2]
Lecture 8
Magnetic Materials


The effect of an applied electric field on a
magnetic material is to create a net magnetic
dipole moment per unit volume M.
The dipole moment distribution sets up induced
secondary fields:
B  B app  B ind
Total field
Field in free space
due to sources
19
Field due to
induced magnetic
dipoles
Lecture 8
Volume and Surface
Magnetization Currents


A magnetized material may be represented as an
equivalent volume (Jm) and surface (Jsm)
magnetization currents.
These charge distributions are related to the
magnetization vector by
J m   M
J sm  M  aˆ n
20
Lecture 8
Volume and Surface
Magnetization Currents (Cont’d)


Magnetization currents are equivalent currents that
account for the effect of the magnetized material, and
are analogous to equivalent volume and surface
polarization charge densities in a polarized dielectric.
If the magnetization vector is constant throughout a
magnetized material, then the volume magnetization
current density is zero, but the surface magnetization
current is nonzero.
21
Lecture 8
Ampere’s Law in Magnetic Media

Ampere’s law in differential form in free space:
  B  0 J

Ampere’s law in differential form in a
magnetized material:
  B  0  J  J m 
22
Lecture 8
Magnetic Field Intensity
  B   0  J  J m    0 J   0  M
  B   0 M    0 J
 B

    M   J
 0

• define the magnetic field intensity as
23
Lecture 8
General Forms of Ampere’s
Law

The general form of Ampere’s law in
differential form becomes
 H  J

The general form of Ampere’s law in integral
form becomes
H

d
l

J

d
s

I
encl


C
S
24
Lecture 8
Permeability Concept

For some materials, the net magnetic
dipole moment per unit volume is
proportional to the H field
M  m H
magnetic
susceptibility
(dimensionless)
25
• the units of
both M and
H are A/m.
Lecture 8
Permeability Concept (Cont’d)

Assuming that
we have
M  m H
B  0 H  M   0 1   m H   H

The parameter  is the permeability of the
material.
26
Lecture 8
Permeability Concept (Cont’d)



The concepts of magnetization and magnetic dipole
moment distribution are introduced to relate
microscopic phenomena to the macroscopic fields.
The introduction of permeability eliminates the
need for us to explicitly consider microscopic
effects.
Knowing the permeability of a magnetic material
tells us all we need to know from the point of view
of macroscopic electromagnetics.
27
Lecture 8
Relative Permeability

The relative permeability of a magnetic
material is the ratio of the permeability of
the magnetic material to the permeability
of free space

r 
0
28
Lecture 8
Diamagnetic Materials




In the absence of applied magnetic field, each
atom has net zero magnetic dipole moment.
In the presence of an applied magnetic field, the
angular velocities of the electronic orbits are
changed.
These induced magnetic dipole moments align
themselves opposite to the applied field.
Thus, m < 0 and r < 1.
29
Lecture 8
Diamagnetic Materials (Cont’d)
Usually, diamagnetism is a very miniscule
effect in natural materials - that is r  1.
 Diamagnetism can be a big effect in
superconductors and in artificial
materials.
 Diamagnetic materials are repelled from
either pole of a magnet.

30
Lecture 8
Paramagnetic Materials


In the absence of applied magnetic field, each
atom has net non-zero (but weak) magnetic
dipole moment. These magnetic dipoles
moments are randomly oriented so that the net
macroscopic magnetization is zero.
In the presence of an applied magnetic field, the
magnetic dipoles align themselves with the
applied field so that m > 0 and r > 1.
31
Lecture 8
Paramagnetic Materials (Cont’d)
Usually, paramagnetism is a very miniscule
effect in natural materials - that is r  1.
 Paramagnetic materials are (weakly)
attracted to either pole of a magnet.

32
Lecture 8
Ferromagnetic Materials




Ferromagnetic materials include iron, nickel and
cobalt and compounds containing these elements.
In the absence of applied magnetic field, each atom
has very strong magnetic dipole moments due to
uncompensated electron spins.
Regions of many atoms with aligned dipole
moments called domains form.
In the absence of applied magnetic field, the
domains are randomly oriented so that the net
macroscopic magnetization is zero.
33
Lecture 8
Ferromagnetic Materials (Cont’d)
In the presence of an applied magnetic
field, the domains align themselves with
the applied field.
 The effect is a very strong one with m >>
0 and r >> 1.
 Ferromagnetic materials are strongly
attracted to either pole of a magnet.

34
Lecture 8
Ferromagnetic Materials (Cont’d)

In ferromagnetic materials:
 the
permeability is much larger than the
permeability of free space
 the permeability is very non-linear
 the permeability depends on the previous
history of the material
35
Lecture 8
Ferromagnetic Materials
(Cont’d)
 In ferromagnetic materials, the relationship B = H
can be illustrated by means of a magnetization
curve (also called hysteresis loop).
B
remanence
(retentivity)
H
coercivity
36
Lecture 8
Ferromagnetic Materials (Cont’d)
Remanence (retentivity) is the value of B
when H is zero.
 Coercivity is the value of H when B is
zero.
 The hysteresis phenomenon can be used
to distinguish between two states.

37
Lecture 8
Antiferromagnetic Materials



Antiferromagnetic materials include chromium
and manganese.
In antiferromagnetic materials, the magnetic
moments of individual atoms are strong, but
adjacent atoms align in opposite directions.
The macroscopic magnetization of the material
is negligible even in the presence of an applied
field.
38
Lecture 8
Ferrimagnetic Materials



Ferrimagnetic materials include oxides of iron,
nickel, or cobalt.
The magnetic moments of adjacent atoms are
aligned opposite to each other, but there is
incomplete cancellation of the moments because
they are not equal.
Thus, there is a net magnetic moment within a
domain.
39
Lecture 8
Ferrimagnetic Materials (Cont’d)



In the absence of applied magnetic field, the
domains are randomly oriented so that the net
macroscopic magnetization is zero.
In the presence of an applied magnetic field, the
domains align themselves with the applied field.
The magnetic effects are weaker than in
ferromagnetic materials, but are still substantial.
40
Lecture 8
Ferrites




Ferrites are the most useful ferrimagnetic
materials.
Ferrites are ceramic material containing
compounds of iron.
Ferrites are non-conducting magnetic media
so eddy current and ohmic losses are less
than for ferromagnetic materials.
Ferrites are often used as transformer cores at
radio frequencies (RF).
41
Lecture 8
Fundamental Laws of
Magnetostatics in Integral Form
 H  dl   J  d s
C
Ampere’s law
S
Gauss’s law for magnetic
field
B

d
s

0

S
B  H
Constitutive relation
42
Lecture 8
Fundamental Laws of
Magnetostatics in Differential Form
Ampere’s law
 H  J
B  0
Gauss’s law for magnetic
field
B  H
Constitutive relation
43
Lecture 8
Fundamental Laws of
Magnetostatics


The integral forms of the fundamental laws are
more general because they apply over regions of
space. The differential forms are only valid at a
point.
From the integral forms of the fundamental laws
both the differential equations governing the
field within a medium and the boundary
conditions at the interface between two media
can be derived.
44
Lecture 8
Boundary Conditions

Within a homogeneous
medium, there are no
abrupt changes in H or
B. However, at the
interface between two
different media (having
two different values of
, it is obvious that one
or both of these must
change abruptly.
1
2
45
ân
Lecture 8
Boundary Conditions (Cont’d)

The normal component of a solenoidal vector
field is continuous across a material interface:
B1n  B2 n

The tangential component of a conservative
vector field is continuous across a material
interface:
H1t  H 2t , J s  0
46
Lecture 8
Boundary Conditions (Cont’d)
The tangential component of H is
continuous across a material interface,
unless a surface current exists at the
interface.
 When a surface current exists at the
interface, the BC becomes

aˆn  H 1  H 2   J s
47
Lecture 8
Boundary Conditions (Cont’d)

In a perfect conductor, both the electric
and magnetic fields must vanish in its
interior. Thus,
Bn  0
aˆ n  H  J s
• a surface current must
exist
• the magnetic field just
outside the perfect
conductor must be
tangential to it.
48
Lecture 8
Overview of Magnetic Forces and
Torques


The experimental basis of magnetostatics is the
fact that current carrying wires exert forces on
one another as described by Ampere’s law of
force.
A number of devices are based on the forces
and torques produced by static magnetic fields
including DC electric motors and electrical
instruments such as voltmeters and ammeters.
49
Lecture 8
Magnetic Forces on Moving
Charges

The force on a charged particle moving
with velocity v in a magnetostatic field
characteristic by magnetic flux density B is
given by
F m  qv  B
50
Lecture 8
Lorentz Force Equation

The force on a charged particle moving
with velocity v in a region where there
exists both a magnetostatic field B and an
electrostatic field E is given by
F  q E  v  B 
51
Lecture 8
Lorentz Force Equation (Cont’d)
The Lorentz force equation can be used to
obtain the equations of motion for charged
particles in various devices including
cathode ray tubes (CRTs), microwave
klystrons and magnetrons, and cyclotrons.
 The Lorentz force equation also explains
the Hall effect in conductors and
semiconductors.

52
Lecture 8
Magnetic Force on CurrentCarrying Conductors

When a current carrying wire is placed in a
region permeated by a magnetic field, it
experiences a net magnetic force given by
F m   Id l  B
C
53
Lecture 8
Torque on a Current Carrying
Loop

Consider a small
rectangular current
carrying loop in a
region permeated by a
magnetic field.
y
Fm1
B
I
W
x
Fm2
L
54
Lecture 8
Torque on a Current Carrying
Loop (Cont’d)

Assuming a uniform magnetic field, the force on
the upper wire is
F m1  aˆ z ILB

The force on the lower wire is
F m2  aˆ z ILB
55
Lecture 8
Torque on a Current Carrying
Loop (Cont’d)
The forces acting on the loop have a
tendency to cause the loop to rotate about
the x-axis.
 The quantitative measure of the tendency
of a force to cause or change rotational
motion is torque.

56
Lecture 8
Torque on a Current Carrying
Loop (Cont’d)

The torque acting on a body with respect
to a reference axis is given by
T  rF
distance vector from the
reference axis
57
Lecture 8
Torque on a Current Carrying
Loop (Cont’d)

The torque acting on the loop is
W
W
T  aˆ y  F m1  aˆ y  F m 2
2
2
 aˆ x ILWB  aˆ z ILW  B
magnetic dipole
moment of loop
58
Lecture 8
Torque on a Current Carrying
Loop (Cont’d)

The torque acting on the loop tries to align
the magnetic dipole moment of the loop
with the B field
holds in general
regardless of
loop shape
T  m B
59
Lecture 8
Energy Stored in Magnetic Field

The magnetic energy stored in a region
permeated by a magnetic field is given by
1
1
2
Wm   B  H dv    H dv
2V
2V
60
Lecture 8
Energy Stored in an Inductor

The magnetic energy stored in an inductor
is given by
1 2
Wm  LI
2
61
Lecture 8