Magnetism
INEL 4151Dr. Sandra Cruz-Pol
Electrical and Computer Engineering Dept.UPRM ch 7
http://www.treehugger.com/files/2008/10/spintronics-discover-could-lead-to-magnetic-batteries.php
http://videos.howstuffworks.com/hsw/11955-magnetism-introduction-to-magnetism-video.htm
Applications
MotorsTransformersMRIMore…
http://videos.howstuffworks.com/hsw/18034-electricity-and-magnetism-magnetic-levitation-video.htm
HB H= magnetic field intensity [A/m]
B= magnetic field density [Teslas]
In free space the permeability is:H/m 104 7 o
Magnetic FieldBiot-Savart Law
• States that:
Example
aI
H ˆcoscos4 12
For an infinite line filament with current I (1=180o and 2=0o):
a
IH ˆ
2
2
1
a
PE. 7.1 Find H at (0,0,5)
• Due to current in (figure):where 1=90o and
10A 1 y
z
x
1
(0,0,5)
aI
H ˆcoscos4 12
22
22
2
25
11cos
zyx
l
aaa
aaa
ˆ2
ˆˆ
ˆˆˆ
2
ˆˆ xy aa
2
ˆˆ0
25
2
54
1 yx aaH
m
mAˆˆ30 yx aaH
Circular loop• Defined by• Apply Biot-Savart:
• Only z-component of H survives due to symmetry:
0,922 zyx
dl
R
z
y
dHz
dH
xzadahd
h
d
aaa
Rld
ˆˆ
0
00
ˆˆˆ
2
z
2
02/322
2
4
ˆ
h
adIH z
2/322
2
2
ˆ
h
aI z
Ampere’s Law
• Simpler• Analogous to Gauss Law for Coulombs• For symmetrical current distributions
Ampere’s Law SdJIldH enc
IIldH enc
adld
a
IH
2
We define an Amperian path where H is constant.
Infinitely long coaxial cable SdJIldH enc
Four cases: 1) For <a 2
2
2ˆ
a
Iadd
a
II zenc
z
2HdH
22 a
IH
Infinitely long coaxial cable SdJIldH enc
Four cases: 2) For a<<b Iadda
a
II zz
a
enc
2
02
0
ˆˆ
z
2HdH
2
IH
Infinitely long coaxial cable SdJIldH enc
Four cases: 3) For b<<b+c
2
022
ˆb
zencbcb
ddIIaddJI
z
2HdH
bcc
bIH
21
2 2
22
2
22
2 cbc
bIIIenc
Infinitely long coaxial cable SdJIldH enc
Four cases: 4) For >b+c
0 IIIenc
20 H
0H
Sheet of current distribution
0ˆ2
1
0ˆ2
1
zaK
zaKH
xy
xy
0ˆ
0ˆ
zaH
zaHH
xo
xo
bKIldH yenc
K [A/m]
ba
x
z
y
Cross section is a Line!
The H field on the Amperian path is given by:
ldHldH
1
4
4
3
3
2
2
1
bH
bHabHa
o
oo
2
))(()(0))(()(0 naKH ˆ
2
1
The H field is given by:
2
4
1
3
PE. 7.5 Sheet of currentPlane y=1 carries a current K=50 az mA/m.
Find H at (0,0,0).
naKH ˆ2
1
xyz aaaH ˆ25ˆˆ502
1
K =50 mA/m
-x
y
z
Toroidal inductors can have higher Q factors and higher inductance than similarly constructed solenoid coils. This is due largely to the smaller number of turns required when the core provides a closed magnetic path. The magnetic flux in a toroid is largely confined to the core, preventing its energy from being absorbed by nearby objects, making toroidal cores essentially self-shielding.
A toroidA circular ring-shaped magnetic core of iron powder, ferrite, or
other material around which wire [N- loops] is coiled to make an inductor. Toroidal coils are used in a broad range of applications, such as high-frequency coils and transformers.
NIH
SdJIldH enc
2
elsewhere0
core theinside2 l
NINIH
Fields stay inside core, no interference.
Magnetic Flux Density, B• The magnetic flux is defined as:
which flows through a surface S.• The total flux thru a closed surface in a
magnetic field is:
[Wb] S
SdB
0S
SdB
0 vS
dvBSdB
0 B
Monopole doesn’t exist.
vD
Maxwell’s Equations for Static Fields
Differential formDifferential form Integral FormIntegral FormGaussGauss’’ss Law for Law for EE field.field.
GaussGauss’’ss Law for Law for HH field. Nonexistence field. Nonexistence of monopole of monopole
FaradayFaraday’’ss Law; E Law; E field is conserved.field is conserved.
AmpereAmpere’’ss Law Law
vD
0 B
0 E
JH
v
v
s
dvSdD
0s
SdB
0 ldEL
sL
SdJldH
Magnetic Scalar and Vector Potentials, Vm & A
When J=0, the curl of H is =0, then recalling the vector identity:
• We can define a Magnetic Scalar Potential as:
• The magnetic Vector Potential A is defined:
0J if
mVH
VH 0
AB
The magnetic vector potential, A, is
AB
L
Roo R
alIdHB
24
ˆ
L
o
R
IdlA
4
It can be shown that:
2
11
RR
Substituting into equation for Magnetic Flux:
SS
SdASdB
L
ldA
L
ldA
The magnetic vector potential A is used in antenna theory.
This is another way of finding magnetic flux.
P.E. 7.7 A current distribution causes a magnetic vector potential of:
Find :• B at (-1,2,5)Answer:• Flux thru surface z=1, 0≤x≤1, -1≤y ≤4
Answer :
zxyzyxyxyxA ˆ4ˆˆ 22
AB
SS
ldASdB
[Wb] 20
[T] ˆ3ˆ40ˆ20 zyxB