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Chapter 32 Maxwell’s equations; Magnetism in matter In this chapter we will discuss the following topics: - PowerPoint PPT Presentation
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Chapter 32
Maxwell’s equations; Magnetism in matter
In this chapter we will discuss the following topics:
-Gauss’ law for magnetism -The missing term from Ampere’s law added by Maxwell -The magnetic field of the earth -Orbital and spin magnetic moment of the electron -Diamagnetic materials -Paramagnetic materials -Ferromagnetic materials
(32 – 1)
The magnetic flux through each of five faces of a die (singular of ''dice'') is given by ΦB = ±N Wb, where N (= 1 to 5) is the number of spots on the
face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux (in Wb) through the sixth face of the die?
A.1 B.2 C.3 D.4 E.5
Fig.aFig.b
In electrostatics we saw that positive and negative charges
can be separated. This is not the case with magnetic poles,
as is shown in the figure. In fig.a we have a p
Gauss' Law for the magnetic field
ermanent bar
magnet with well defined north and south poles. If we
attempt to cut the magnet into pieces as is shown in fig.b
we do not get isolated north and south poles. Instead new
pole faces appear on the newly cut faces of the pieces and
the net result is that we end up with three smaller magnets,
each of which is a i.e. it has a north and a
south pole. This result can be expr
magnetic dipole
essed as follows:
The simplest magnetic structure that can exist is a magnetic dipole.
Magnetic monopoles do not exists as far as we know.
(32 – 2)
iB
ˆin
iΔAi
1 2 3
The magnetic flux through a closed surface
is determined as follows: First we divide
the surface into area element with areas
, , ,..., n
n
A A A A
BMagnetic Flux Φ
For each element we calculate the magnetic flux through it: cos
ˆHere is the angle between the normal and the magnetic field vectors
at the position of the i-th element. The inde
i i i i
i i i
B dA
n B
1 1
x runs from 1 to n
We then form the sum cos
Finally, we take the limit of the sum as
The limit of the sum becomes the integral:
cos
n n
i i i ii i
B
i
B dA
n
BdA B dA
SI magnetic flux un
2 T m known as the "Weber" (Wb)it :(32 – 3)
B B dA
Gauss' law for the magnetic field can be expressed
mathematically as follows: For any closed surface
Contrast this with Gauss' law for
cos
the electric field:
0B
encE
o
BdA B dA
qE dA
Gauss' law for the magnetic
field expresses the fact that there is no such a thing as a
" ". The flux of either the electric or
the magnetic field through a surface is proportional
magnetic charge
to the
net number of electric or magnetic field lines that either
enter or exit the surface. Gauss' law for the magnetic field
expresses the fact that the magnetic field lines are closed.
The number of magnetic field lines that enter any closed
surface is exactly equal to the number of lines that exit the
surface. Thus 0.B (32 – 4)
0 B B dA
Faraday's law states that: This law describes
how a changing magnetic field generates (induces) an electric
field. Ampere's law in its original form reads:
BdE dS
dt
Induced magnetic fields
. Maxwell using an elegant symmetry
argument guessed that a similar term exists in Ampere's law.
The new term is written in red :
This term, also known as "
Eo o
o enc
o enc
B dS i
B dS id
dt
M "
desrcibes how a changing electric field can generate a
magnetic field. The electric field between the plates of the
capacitor in the figure changes with time . Thus the elet
axwell's law of induction
E
ctric
flux through the red circle is also changing with and
a non-vanishing magnetic field is predicted by Maxwell's law
of induction. Experimentaly it was verified that
the predicted magnetic field
t
exists. (32 – 5)
,
Ampere's complete law has the form:
We define the displacement current
Using Ampere's law takes the form:
In t
he e
Ed o
Eo enc o o
d
o enc o d enc
dB dS i
dt
i
B dS i i
di
dt
The displacement current
xample of the figure we can show that
between the capacitor plates is equal to the
current that flows through the wires which
charge the capacitor plates.
di
i
Eo enc o o
dB dS i
dt
The electric flux through the capacitor plates .
1The displacement current
Eo o
Ed o o
o o o
qAE A
d q qi i
dt
(32 – 6)
,o enc o d encB dS i i
,
Consider the capacitor with
circular plates of radius
In the space between the capacitor
plates the term is equal to zero
Thus Ampere's law becomes:
We will use Ampere's law to
determ
o d encB dS
R
i
i
ine the magnetic field.
The calculation is identical to that of a magnetic field generated by a long wire
of radius . This calculation was carried out in chapter 29 for a point P at a distance
from the wire center. We w
R
r
ill repeat the calculation for points outside
as well as inside the capacitor plates. In this example is the distance
of the point P from the capacitor center C.
r R
r R r
(32 – 7)
B
di
r
RC
dS
P
We choose an Amperian loop that reflects the cylindrical symmetry of the problem.
The loop is a circle of radius that has its center at the capacitor platr
Magnetic field outside the capacitor plates :
,
e center C.
The magnetic field is tangent to the loop and has a constant magnitude .
cos 0 22o d
o d enc o d
i
B
B ds Bds B ds rB i i Br
(32 – 8)
B
dir
R
dS
P
C
We assume that the distribution of
within the cross-section of the capacitor plate is uniform.
We choose an Amperian loop is a circle of radius
( ) that
di
r
r R
Magnetic field inside the capacitor plates
,
2 2
, 2 2
2
2
2
has its center at C. The magnetic field is
tangent to the loop and has a constant magnitude .
cos 0
2
2
2
do enc
d enc d d
o do d
B
B ds Bds B ds rB i
r ri i i
R R
rrB i B r
R
i
R
R
2o di
R
r
B
O (32 – 9)
Below we summarize the four equations on which electromagnetic theory
is based on. We use here the complete form of Ampere's law as modified by
Maxwell:
: E
Maxwell's equations
Gauss' law for
:
These equations desc
ri
be a
g
0
enc
o
B
Eo enc o o
qE dA
B dA
dE dS
dt
dB dS
B
idt
Gauss' law for
Faraday's law :
Ampere's law :
roup of diverse phenomena and devices based
on them such as the magnetic compass,electric motors, electric generators,
radio, television, radar, x-rays, and all of optical effects.
All these in just four equations! (32 – 10)
In this section I will discuss a question which many of you may have.
Maxwell added just one term in one out of four equations, and all of a sudden
th
Eo enc o o
dB dS i
dt
A word of explanation :
e set is called after him. Why? The reason is that Maxwell manipulated
the four equations (with Ampere's law now containing histerm) and he got
solutions that described waves that could travel in vacu
8
um with a speed
1 3 10 m/s.
This happens to be the speed of light in vacuum measured a few years earlier
by Fizeau. It was natural for Maxwell to contemplate whether light,
whose nature was not
oo
v
clear could be such an electromagnetic wave.
Maxwell died soon after this and was not able to verify his hypothesis.
This task was carried out by Hertz who verified experimentally
the existance of electromagnetic waves.
(32 – 11)
N
S
Fig.b : Side view
horizontal
Compass needle
Earth has a magnetic field that can be approximated
as the field of a very large bar magnet that straddles the center of the planet. The
dipole axis does not coincide exactly
The magnetism of earth.
with the rotation axis but the two axes form
an angle of 11.5 , as shown in the figure. The direction of the earth's magnetic field
at any location is described by two angles:
(see
Field declination fig.a) is defined as the angle between the geographic north
and the horizontal component of the earth's magnetic field.
(see fig.b) is defined as the angle between the horizontal aField inclination nd the
earth's magnetic field.
N
S Fig.a : Top view
Geographic North
Compass needle
(32 – 12)
There are three ways in which electrons can generate
a magnetic field. We have already encountered the
first method. Moving electrons constitute a current
which according to Ampe
Magnetism and electrons
re's law generates a
magnetic field in its vicinity. An electron can also
generate a magnetic field because it acts as a magnetic
dipole. There are two mechanisms involved.
. An electron in an atom moves around the nucleus
as shown in the figure. For simplicity we assume a circular orbit of radius with
period . This constitutes an elect
r
T
Orbital magnetic dipole moment
2 2
ric current . The resulting2 / 2
magnetic dipole moment 2 2 2 2
In vector form: The negative sign is due to the negative charge
of t
2
he
orb or
orb orb
b
e e eviT r v r
e mvrev evr er i r L
r m meL
m
electron.
2
orb orb
eL
m
(32 – 13)
S
eS
m
In addition to the orbital angular momentum an electron
has what is known as " " or " " angular
momentum . Spin is a quantum relativistic effect. One
can give
S
Spin magnetic dipole moment
intrinsic spin
a simple picture by viewing the electron as a spinning
charge sphere. The corresponding magnetic dip
ole moment is
given by the equation:
Unlike classical mechanics in whi
S
eS
m
Spin quantization.
ch the
angular momemntum can take any value, spin and orbital
angular momentum can only have certain discreet values.
Furthermore, we cannot measure the vectors or but only
their projections
S
L
S L
along an axis (in this case defined by ).
These apparently strange rules result from the fact that at the
microscopic level classical mechanics do not apply and we must
use .
B
quantum mechanics
(32 – 14)
z- axis
SzS
B
z- axis
SzS
B
34
The quantized values of the spin angular momentum are:
The constant 6.63 10 J s is 2
known as " ". It is the yardstick by which
we can tell whethe
z S
hS m h
Spin quantization
Planck's constant
,
,
r a system is described by classical or by
1quantum mechanics. The term can take the values +
21
or . Thus the z-component of can take the values2
. The energy of the electron4
S
S z
S z
m
eh
m
,
24
The constant 9.27 10 J/T is known as4 4
the electron " " (symbol ). The electron energy
can be expressed as:
S S z
B
B
U B B
ehB ehU
m m
U U B
Bohr magneton
S
eS
m
2z S
hS m
, 4S z
eh
m
4
ehBU
m
(32 – 15)
Materials can be classified on the basis of their magnetic
properties into three categories: , , and .
Below we discuss briefly each catecory.
Magnetic Materials.
Diamagnetic paramagnetic ferromagnetic
2
3
Magnetic materials are characterized by
the magnetization vector defined as the magnetic moment per unit volume.
A m A
:
Di
m m
amag
netMV
M
Diamagneti
S
s
I unit for M
m.
netism occurs in materials composed of atoms that have electrons whose
magnetic moments are antiparallel in pairs and thus result in a zero net magnetic
moment. When we apply an external magnetic field , diamagnetic materials acquire
a weak magnetic moment which is directed opposite to . If is
inhomogeneous, the diamagnetic material is
to regines o
B
B Brepelled from regions of stronger
field
f weaker . All materials exhibit diamanetism but in
paramagnetic and ferromagnetic materials ths weak diamagnetism is masked
by the much stronger paramagnetism or ferromagnetism.
B
(32 – 16)
A model for a diamagnetic material is shown in the figure.
Two electrons move on identical orbits of radius with angular
speed . The electron in the top figure moves in the
whileo
r
counterclockwise that in the lower figure moves in the
direction. When the magnetic field 0 the
magnetic moments for each orbit are antiparallel and thus
the net magnetic moment 0. When a magnetic
B
clockwise
field
is applied, the top electron speeds up while the elecron in the
bottom orbit slows down. The corresponding angular speeds
are: , The magnetic dipole2 2
moment for
o o
B
Be Be
m m
22 2the electons is:
2 2
e eri r r
2
2
2 2 2
2 2 2 2 2 2
The negative sign indicates that are antiparall2
el
o o
netnet
er er Be er e
er
r Be
m m
Bm
(32 – 17)
. B
+v
F
BF
C e
+ω
.
-v
F
BFCe
-ω.
2
2 2
2 2 2 2
1/ 2
2 2
2
1
1 12 2
o B o
net B o
o o oo
o o oo o
net B o
F m F evB e rB
F F F m evB m r
Bem r m r e rB
m
Be Be Be
m m m
F F F m r evB m r
m
Top electron :
Bottom electron :
2 2 2
1/ 2
1
1 12 2
o o oo
o o oo o
Ber m r e rB
m
Be Be Be
m m m
(32 – 18)
. B
+v
F
BF
C e
+ω
.
-v
F
BFCe
-ω.
The atoms of paramagnetic materials
have a net magnetic dipole moment
in the absence of an external magnetic
field. This moment is the vector sum
of the electron magnetic moments.
Paramagnetism
In the presence of a magnetic field each dipole has energy cos . Here
is the angle between and . The potential energy is minimum when 0.
The magnetic field partially aligns the momen
U B
B U
t of each atom. Thermal motion
opposes the alignment. The alignment improves when the temperature is lowered
and/or when the magnetic field is large. The resulting magnetization is parallel
to the
M
field . When a paramagnetic material is placed in an inhomogeneous field
it moves in the region where is stronger.
B
B
(32 – 18)
Curie's Law
When the ratio is below 0.5 the magnetization of a paramagnetic material
follows
The constant is known as the Curie constant
When 0.5 Curie's law breaks down and a diffe
BM
T
BM C C
TB
T
Curie's law :
rent approach is required.
For very high magnetic fields and/or low temperatures, all magnetic moments
are parallel to and the magnetization
Here the ratio is the number of paramagnetic a
sat
NB M
VN
V
toms per unit volume.
B
M CT
(32 – 19)
Feromagnetism is exhibited by Iron, Nickel, Cobalt,
Gadolinium, Dysprosium and their alloys.
Ferromagnetism is abserved even in the absence of a
magnetic field (the familiar permanent ma
Ferromagnetism
gnets).
Ferromagnetism disappears when the temperature
exceeds the Curie temperature of the material.
Above its Curie temperature a ferromagnetic
material becomes paramagnetic.
Ferromagnetism is due to a quantum effect known as "exchange coupling" which
tends to align the magnetic dipole moments of neighboring atoms
The magnetization of a ferromagnetic material can be measured using a Rowland ring.
The ring consists of two parts. A prinary coil in the from of a toroid which generates
the external magnetic field . A secondary coil which measures the total magnetic fieloB d
. The amagnetic material forms the core of the torroid. The net field
Here is the contribution of the ferromagnetic core. is proportional to the
sample magnetization
o M
M M
B B B B
B B
M
(32 – 20)
Below the Curie temperature all magnetic moments
in a ferromagnetic material are perfectly aligned.
Yet the magnetization is not saturated. The reason
is that the ferromagnetic material
Magnetic domains
contains regions
" ". The magnetization is each domain is
saturated but the domains are aligned in such a
way so as to have at best a small net magnetic
moment. In the presence of an external ma
domains
gnetic
field two effects are observed:
The domains whose magnetization is aligned
with grow at the expence of those domains that
are not aligned.
The magnetization of the non-aligned dom
o
o
B
B
1.
2.
ains
turns and becomes parallel with . oB
(32 – 21)
If we plot the net field as function of the applied
field we get the loop shown in the figure known as
a " " loop. If we start with a unmagnetized
ferromagnetic material the cu
M
o
B
B
Hysteresis
hysteresis
rve follows the path from
point to point , where the magnetization saturates.
If we reduce the curve follows the path which is
different from the original path . Furtermore, even
when is
o
o
a b
B bc
ab
B switched off, we have a non-zero magnetic
field. Similar effects are observed if we reverse the
direction of . This is the familiar phenomenon of
permanent magnetism and forms the basis of magneticoB
data recording. Hysteresis is due to the fact that the
domain reorientation is not totally revesrsible and
that the domains do not return completely to their
original configuration.
(32 – 22)
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