Transcript Lecture 7 MAGNETOSTATICS

Lecture 8
MAGNETOSTATICS
Magnetic Fields
Fundamental Postulates of
Magnetostatics in Free Space
Prof. Viviana Vladutescu
Magnetic Fields
Magnetism and electricity have not been
considered distinct phenomena until Hans
Christian Oersted conducted an experiment that
showed a compass deflecting in proximity to a
current carrying wire
Produced by
-time varying electric fields
-permanent magnet (arises from quantum
mechanical electron spin/ can be considered
charge in motion=current )
-steady electric currents
H SI
A

m
If we place a wire with current I in the presence of
a magnetic field, the charges in the conductor
experience another force Fm
Fm ~ q, u, B
H
1

B
q –charge
u –velocity vector
B -strength of the field (magnetic flux density)
μr –relative permeability
μ –absolute permeability
7
μ0-permeability of the free space 0  4 10 ( H )
M=χmH- is the magnetization for linear and
homogeneous medium
Relative permeabilities for a variety
of materials
Material
μ/(H m-1)
μr
Application
Ferrite U 60
1.00E-05
8
UHF chokes
Ferrite M33
9.42E-04
750
Resonant circuit RM cores
Nickel (99% pure)
7.54E-04
600
-
Ferrite N41
3.77E-03
3000
Power circuits
Iron (99.8% pure)
6.28E-03
5000
-
Ferrite T38
1.26E-02
10000
Broadband transformers
Silicon GO steel
5.03E-02
40000
Dynamos, mains transformers
supermalloy
1.26
1000000
Recording heads
Lorentz’s Force equation
Fe  q E
Fm  qu  B
Fe  Fm  F
F  q(E  u  B)
Note: Magnetic force is zero for q moving in the
direction of the magnetic field (sin0=0)
When electric current is passed through a magnetic field a force
is exerted on the wire normal to both the magnetic field and the
current direction. This force is actually acting on the individual
charges moving in the conductor.
The magnetic force is exerting a torque on
the current carrying coil
  r  Fm  r  Fm sin  an
Cross product
Fundamental Postulates
of Magnetostatics in
Free Space
B  0
  B  0 J
 J  0
Law of conservation
of magnetic flux
B

d
s

0

s
There are no magnetic flow sources, and the
magnetic flux lines always close upon
themselves
Ampere’s circuital law




B

d
s


J

d
s


0
s
s
B

d
l


I
0

c
The circulation of the magnetic flux density in free space
around any closed path is equal to μ0 times the total current
flowing through the surface bounded by the path
Stoke’s Theorem




H

d
s

H

d
l


s
c
Note: For a closed
surface there will be
no surface bounding
external contour
Proof: Sum over
  H  s a
j
where
  H  lim
s 0
n
H

d
l

c
s
Note: The net contribution of all the common parts in the
interior to the total line integral is 0 and only the contribution
from the external contour C remains after summation


lim    H  d l    H  d l
s j 0
j 1  c
 c
N




H

ds

H

d
l


s
c
The maximum circulation of H per unit area as the area
shrinks to zero is equal to the current density through that area
Jds

H

d
l


c
H

d
l

I
enc

c
Two possible Amperian
paths around an infinite
length line of current.
Postulates of Magnetostatics
in Free Space
Differential Form
B  0
  B  0 J
Integral Form
B

d
s

0

s
 B  dl   I
0
c
Example
Given a 3.0 mm radius solid wire centered on the z-axis with
an evenly distributed 2.0 amps of current in the +az direction,
plot the magnetic field intensity H versus radial distance from
the z-axis over the range 0 ≤ r ≤ 9 mm.
The field from a
particular
line
of
current making up the
distributed current
The field from a second
line of current results in
the cancellation of ar
component
 H dL  I
enc
, where H  H a and dL  r d a ;
This will be true for each Amperian path.
I
AP1:
I enc   J dS, J =
a
2
a z , I enc
I
H
r a for r  a
2
2 a
AP2: Ienc = I,
H
I
2r
a for r  a
r
2r H  I enc
2
I
Ir 2
 2  r d r  d  2
a 0
a
0