Download DE-MYSTERFYING MAGNETISM

DE-MYSTERFYING MAGNETISM
Electrical properties of materials
free charges  electric displacement D
electric dipoles bound surface charges  polarization P
 electric field E
E
1
0
( D  P)
Magnetic properties of materials
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 N.A-1.m-1]

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
p2/em/magnetism_1.doc
same direction  m &  r scalars
M rarely in same direction
 m &  r tensors
1
What is an electric field? What creates an electric field?
What is a magnetic field?
A moving charge experiences a force in a
magnetic field.
+q
F  B q v sin 
+I
F  B I L sin 
F  qv  B
F
out of page
dF  i dl  B
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
I
magnetic fields
right hand screw rule
currents
B
B
p2/em/magnetism_1.doc
I
2
FUNDAMENTAL LAWS GOVERNING MAGNETISM
B
Ampere’s Law – line integral
permeability of free space
dl  0 itotal
0  4 107 N.A -2
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  i f
if
free currents – does not depend upon medium
Faraday’s Law and Magnetic flux
Magnetic flux
Faraday’s Law
 B   B dA
emf 
E
dl  
[T.m2]
dB
dt
(generation of electricity by
time varying magnetic fields)
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
p2/em/magnetism_1.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 the magnet 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 are same outside the magnet (same field patterns)
HH
B
6
S
N
HFe
1
H

2
1
2
5 4
Bair
H Fe 
0
Hair
2
3
3
Circulation loop: square side L
5
6
dl   H Fe dl   H air dl   H air dl   H Fe dl  0
1
2
4
5
6
3
5
5
2
4
H Fe dl   H Fe dl    H air dl   H air dl
 H Fe   H air
p2/em/magnetism_1.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
B-f ield lines –
f orm continuous loops
M
M 0
H
N pole
Bound surf ace currents (right hand screw rule)

M

B
Interaction between magnetic fields
Like poles repel
Unlike poles attract
p2/em/magnetism_1.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
north pole attracts
south pole
 Bar magnet will attract
the iron that was initially
un-magnetized
Explain what happens in the following diagrams when a magnet is placed on a ramp.
Fe ramp
Cu ramp
plastic ramp
Uniformly magnetized sphere
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.
p2/em/magnetism_1.doc
6
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 (if = 0)
 H  dl
if
In the air gap H air 
 H iron (2 r  a )  H air (a )  0
B
o
or B = o Hair this 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.
In the iron (The H-fields point in different directions)
a
a
H iron  
H air  
B
2 r  a
(2 r  a) o
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.
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.
N
Hair
Hiron
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.
p2/em/magnetism_1.doc
7
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?
p2/em/magnetism_1.doc
8
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?
p2/em/magnetism_1.doc
9
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
r
Hair
coil windings
Magnetic field of electromagnet confined to region inside the solenoid’s coil
 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
.
.
.
.
.
.
.
.
4
1
X
X
X
X
X
X
3
Circulation loop:
square of length L
2
Current i
into page
Cross-section through electromagnet
p2/em/magnetism_1.doc
10
 H  dl
 n Li  0  H L  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)?
p2/em/magnetism_1.doc
11
CURRENTS AND MAGNETIC FIELDS
Magnetic field surrounding a long straight wire
Apply Ampere’s Law to the circumference of a circle of radius r.
p2/em/magnetism_1.doc
12