Download EMP 3

EMP 3
DE-MYSTERFYING MAGNETISM
ELECTRICAL PROPERTIES OF MATERIALS
What is an electric field? What creates an electric field?
free charges
 electric displacement D
electric dipoles  bound surface charges  polarization P
E
electric field E
1
0
( D  P)
MAGNETIC PROPERTIES OF MATERIALS
What is a magnetic field? What creates a magnetic field?
free currents  H-field, magnetic field, magnetic field intensity H [A.m-1]
magnetic dipoles bound surface currents  magnetization M
[A.m-1]
B-field, magnetic induction, magnetic flux density B [T]

B  0 H  M

B  0 H  M


M  m H
B   H  r  0 H
r  1   m
magnetic susceptibility  m - not a simple number – can depend upon history of
sample
 m < 0 small  diamagnetic materials
 m > 0 small  paramagnetic materials
 m > 0 large  ferromagnetic materials
Isotropic materials: B H
M
Non-isotropic materials: B H
ijcooper/physics/p2/em/emp_03.doc
same direction  m &  r scalars
M rarely in same direction
 m &  r tensors
1
What is a magnetic field?
A moving charge experiences a force in a magnetic
field.
+q
F  B q v sin 
+I
I
I
F  B I L sin 
B
F  qv  B
F
out of page
dF  i dl  B
B
I
B
right hand palm rule
B
B
I
What creates a magnetic field?
Moving charges  currents  magnetic fields
Orbital motion and spin of electrons in atoms  permanent magnets
Biot-Savart Law
dB 
0 i dl  r
4 r3
I
dl
magnetic fields
right hand screw rule
currents
B
F
permeability of free space
µ0 = 410-7 T.m.A-1 (N.A-2 H.m-1)
ijcooper/physics/p2/em/emp_03.doc
2
FUNDAMENTAL LAWS GOVERNING MAGNETISM
B
Ampere’s Law – line integral
dl  0 itotal
itotal
depends on free currents and medium (not simply the current through a wire); total
current passing through the loop defined by the integration)
itotal
= Ni
number of turns N
(magnetic devices have many turns)
i   J dA
H
dl  N i f
if free currents – does not depend upon medium
.
.
ifree
.
.
. .
dl
.
H
.
Faraday’s Law and Magnetic flux
Magnetic flux
 B   B dA
[T.m2]
Faraday’s Law ((generation of electricity by time varying magnetic fields)
emf 
E
dl  
dB
dt
Gauss’s Law for Magnetism
Total magnetic flux through any closed surface is zero
 B
dA  0
No magnetic poles – 2 poles of a magnet
B-field lines form continuous loops
B-field lines are closed
ijcooper/physics/p2/em/emp_03.doc
3
BAR MAGNETS (permanent magnet)
There are no free currents - the magnet is magnetized all by itself if = 0
H
dl  0  H-field inside and the H-field outside point in opposite directions
 B  dA  0  the magnetic field lines for B must be continuous, the lines just keep
going on (there are no magnetic monopoles).
Inside the magnet: H 
1
BM
o
lines of H point in a direction opposite to M and B .
Outside the magnet : M  0
B and H have the same field pattern
HH
B
6
S

2
1
2
4
N
HFe
1
H
5
0
Hair
2
3
Circulation loop: square side L
3
5
6
2
4
5
dl   HFe dl   Hair dl   Hair dl  
1
Bair
H air 
6
3
5
5
2
4
HFe dl  0
H Fe dl   H Fe dl    Hair dl   Hair dl
H Fe   H air
ijcooper/physics/p2/em/emp_03.doc
4
Gauss’s Law f or magnetism
Cylindrical
Gaussian
surface
Binside
Aoutside
Ainside
Boutside
 B
dA   Binside A  Boutside A  0
 Binside  Boutside
B-f ield lines –
f orm continuous loops
B
M
M 0
H
N pole
im
Bound surf ace currents i m (right hand screw rule)

M
 B
Interaction between magnetic fields
Like poles repel
Unlike poles attract
ijcooper/physics/p2/em/emp_03.doc
5
Why does a magnet stick to a piece of iron?
un-magnetized piece of iron
Bar magnet bought near
un-magnetized piece of iron
B
N
N
N
 Bar magnet will attract
the iron that was initially
un-magnetized
north pole attracts
south pole
Consider a magnetic field B to be non-uniform in
which the B field points in a direction orthogonal
to the plane of the current loop. There is a net force
that pulls the magnetic dipole m towards the
region of high magnetic field.
Proof
The forces all pull the current elements
outwards. The force on each current element
is
F = BiL
Forces F3 and F4 cancel.
F1 > F2 since the B-field is non uniform, the
B-field larger on the left than the right. So
there is a net force that pulls the magnetic
dipole towards the region of larger magnetic
field.
B
B large
B small
m
F4
F1
F2
F3
i
Explain what happens in the following diagrams when a magnet is placed on a ramp.
Fe ramp
ijcooper/physics/p2/em/emp_03.doc
Cu ramp
plastic ramp
6
Uniformly magnetized sphere
B-field
H-field
B-field continuous loops (no beginning or end)
The H-field lines start where the M lines end and finish where M start.
H-field has de-magnetizing effect since H and M are in opposite direction.
HORSE SHOE MAGNETS (permanent magnets)
A permanent iron magnet is in the form of circular disk with a radius, r and a small gap in
it of width, a. For the case when r >> a, discuss the H-field, B-field and magnetization
for this example of a horse shoe magnet.
Circulation loop f or circulation integration
used in applying Ampere’s Law
N
Use Amperes’s Law for a loop around the permanent magnetic (i = 0)
 H  dl
if
 H iron (2 r  a )  H air (a )  0
The H-field in the iron, Hiron must point in the opposite direction to the H-field in the air,
Hair.
ijcooper/physics/p2/em/emp_03.doc
7
Ampere’s Law
 B
dA  0 The B-field field is perpendicular to the plane surfaces of
the ring, and the perpendicular component of the B field is constant at an interface, so B
is constant throughout the ring, B = Bair = Biron
In the air gap H air 
B
o
or B = o Hair
In the iron
H iron  
a
a
H air  
B
2 r  a
(2 r  a) o
N
The H-field inside the magnet is in the opposite
direction to the magnetization and has a demagnetizing effect.
This corresponds to a points on the hysteresis loop
H > 0 & B < 0 or H < 0 and B > 0.
Hair
Hiron
For soft materials, the de-magnetizing effect is usually sufficient to bring the material
back to B = 0 (M = 0) i.e., an un-magnetized state.
This is why a horse-shoe magnet is stored with an iron keeper. Then, the B-field, H-field
and magnetization all point in the same direction.
ijcooper/physics/p2/em/emp_03.doc
8
ELECTROMAGNETS – ROWLAND RINGS
A Rowland ring is a toroidal ring with many windings around its circumference. For an
iron Rowland ring with N windings and a mean radius r, what is the B-field and the
magnetic flux inside the ring?
Apply Ampere’s Law about the circumference of length L= 2 r
 H .dl  N i f
 H L  N if
 H
N if
L

N if
2 r
Assume that the iron in the Rowland ring is operated in the linear region so that
B   H   r 0 H   r 0
Ni
Ni
  r 0
L
2 r
The magnetic flux is
m   B  dA  B A 
r 0 A N i
L

r 0 A N i
2 r
where A is the cross-sectional area of the ring
What are the directions of the fields B, H and M?
ijcooper/physics/p2/em/emp_03.doc
9
How would the results be different if a small
gap of length d was in the Rowland ring?
if
if
d
Apply Ampere’s Law about the
circumference of length L
 H . dl  N i

f
H Fe ( L  d )  H gap d  N i f
Assume the B-field is confined to the gap,
then by Gauss’s Law
B  BFe  Bgap  o H gap  o r H Fe
 H gap 
B
o
o N i f
Ld
r
m 
B
H Fe 
B
o  r
B-field not limited to the maximum value of the magnetization
d
o A N i f
Ld
r
d
What are the directions of the fields B, H and M?
ijcooper/physics/p2/em/emp_03.doc
10
ELECTROMAGNETS – INFINITE SOLENOID
Assume the magnetic field is totally enclosed within the coil.
Apply Ampere’s Circulation law to the three loops
H
E
A
B dA  N I
G
I
D

L
K
I
J
F
C
B
Loop ABCD  B = 0
Loop EFGH  BEF L+0 - BGH L+0 = 0 
Loop IJKL
B uniform inside coil
N
 B  0 I  0 n I
L
 0 + 0 + BL + 0 = NI
ELECTROMAGNETS – rod inside a coil
Assume that the
electromagnet is very
long. The relative
permeability of its
iron core is µr. The
electromagnet coil
current is i and the
number of winding
per metre is n.
iron core
BFe
gap region
Bgap
HFe
Hgap
i
Give expressions for
B, H and M in the air,
in the gap region
between the coil
windings and the iron
core and inside the
iron core.
Bair
Hair
r
coil windings
Magnetic field of electromagnet confined to region inside the solenoid’s coil
ijcooper/physics/p2/em/emp_03.doc
11
 Bair = 0
Hair = 0 Mair = 0.
The H-field is simply determined by the current i in the coil windings
 HFe = Hgap = H
M Fe  m H   r 1 H
Mgap = 0
Apply Ampere’s Law to a loop 1234
BH M
Current i
out of page
.
.
.
.
.
.
.
.
2
1
X
X
X
X
X
X
3
Circulation loop:
square of length L
4
Current i
into page
Cross-section through electromagnet
 H  dl
 n Li  H L  0  0  0  n Li  H  ni
Bgap  o H
BFe  r Bg ap  r 0 H  r 0 n i
M Fe  m H   r 1 H   r  1 ni
What makes a strong electromagnet?
Why is the iron core important (what is a typical value for µr)?
ijcooper/physics/p2/em/emp_03.doc
12