Download in a magnetized material

Lecture 8: Magnetostatics: Mutual
And Self-inductance; Magnetic
Fields In Material Media;
Magnetostatic Boundary Conditions;
Magnetic Forces And Torques
Lecture 8
1



To continue our study of magnetostatics
with mutual and self-inductance;
magnetic fields in material media;
magnetostatic boundary conditions;
magnetic forces and torques.
Lecture 8
2

Consider two magnetically coupled circuits
I2
S1
S2
C2
C1
I1
Lecture 8
3

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

The mutual inductance between two circuits is
the magnetic flux linkage to one circuit per unit
current in the other circuit:
 12 N 1 N 2 12
L12 

I1
I1
Lecture 8
5
12 N1 N 2 12 N1 N 2
L12 


I1
I1
I1
N1 N 2

I1
 A  dl
1
B

d
s
1
2

S2
2
C2
Lecture 8
6
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

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
Lecture 8
8


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
Lecture 8
9

For an isolated circuit, we call the self
inductance, inductance, and evaluate it using
 N 
L 
I
I
2
Lecture 8
10
iron core
I
N
S
air gap with
constant B
field
I
Lecture 8
11
N
I
• Magnetic dipole
can be viewed as a
pair of magnetic
charges by analogy
with electric dipole.
S
Lecture 8
12
N
B
S
Lecture 8
13



Electron orbiting nucleus
Electron spin
Nuclear spin
negligible
 A complete understanding of these atomic
mechanisms requires application of quantum
mechanics.
Lecture 8
14


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.
Lecture 8
15


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.
Lecture 8
16


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.
Lecture 8
17

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]
average
magnetic
dipole
moment
[Am2]
Lecture 8
18


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


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
Lecture 8
20


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.
Lecture 8
21

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 
Lecture 8
22
  B   0  J  J m    0 J   0  M
  B   0 M    0 J
 B

    M   J
 0

• define the magnetic field intensity as
Lecture 8
23

The general form of Ampere’s law in differential form becomes

The general form of Ampere’s law in integral form becomes
H  J
H

d
l

J

d
s

I
encl


C
S
Lecture 8
24

For some materials, the net magnetic dipole
moment per unit volume is proportional to the H
field
M  m H
magnetic
susceptibility
(dimensionless)
• the units of
both M and
H are A/m.
Lecture 8
25

Assuming that
we have
M  m H
B   0  H  M    0 1   m H   H

The parameter  is the permeability of the
material.
Lecture 8
26



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.
Lecture 8
27

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
Lecture 8
28




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.
Lecture 8
29



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.
Lecture 8
30


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.
Lecture 8
31


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.
Lecture 8
32




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.
Lecture 8
33



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.
Lecture 8
34

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
Lecture 8
35

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
Lecture 8
36



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.
Lecture 8
37



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.
Lecture 8
38



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.
Lecture 8
39



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.
Lecture 8
40




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).
Lecture 8
41
 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
Lecture 8
42
Ampere’s law
 H  J
B  0
Gauss’s law for magnetic
field
B  H
Constitutive relation
Lecture 8
43


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.
Lecture 8
44

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
ân
Lecture 8
45

The normal component of a solenoidal
vector field is continuous across a
material interface:
B1n  B 2 n

The tangential component of a
conservative vector field is continuous
across a material interface:
H 1t  H 2 t , J s  0
46
Lecture 8


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
aˆ n   H 1  H 2   J s
Lecture 8
47

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.
Lecture 8
48


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.
Lecture 8
49

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
Lecture 8
50

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 
Lecture 8
51


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.
Lecture 8
52

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
Lecture 8
53

Consider a small
rectangular
current carrying
loop in a region
permeated by a
magnetic field.
y
Fm1
B
I
W
x
Fm2
L
Lecture 8
54

Assuming a uniform magnetic field, the force on the upper wire is

The force on the lower wire is
F m1   aˆ z ILB
F m 2  aˆ z ILB
Lecture 8
55


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.
Lecture 8
56

The torque acting on a body with respect to a
reference axis is given by
T  rF
distance vector from the
reference axis
Lecture 8
57

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
Lecture 8
58

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
Lecture 8
59

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
Lecture 8
60

The magnetic energy stored in an inductor is
given by
1 2
Wm  LI
2
Lecture 8
61