Download The Steady Magnetic Field

Chapter 8
The Steady State Magnetic Field
The Concept of Field (Physical Basis ?)
Why do Forces Must Exist?
Magnetic Field – Requires Current Distribution
Effect on other Currents – next chapter
Free-space Conditions
Magnetic Field - Relation to its source – more complicated
Accept Laws on “faith alone” – later proof (difficult)
Do we need faith also after the proof?
1
Magnetic Field Sources
Magnetic fields are produced by electric currents, which can be macroscopic currents
in wires, or microscopic currents associated with electrons in atomic orbits.
2
Magnetic Field – Concepts, Interactions and Applications
From GSU Webpage
3
Biot-Savart Law
IdL  aR
dH
4  R
H





2
4  R

IdL  R
4  R
I
2
At any point P the magnitude of the magnetic
field intensity produced by a differential
element is proportional to the product of the
current, the magnitude of the differential
length, and the sine of the angle lying
between the filament and a line connecting
the filament to the point P at which the field is
desired; also, the magnitude of the field is
inversely proportional to the square of the
distance from the filament to the point P. The
constant of proportionality is 1/4
Magnetic Field Intensity A/m
3
dL  a R
Verified experimentally
Biot-Savart = Ampere’s law for the current element.
4
Biot-Savart Law
B-S Law expressed in terms of distributed sources
The total current I within a transverse
Width b, in which there is a uniform
surface current density K, is Kb.
I

 KdN

Alternate Forms
For a non-uniform surface current
density, integration is necessary.
H

K_x

d S aR

2
 4  R

H

J_x

d v  aR

2
 4  R

5
Biot-Savart Law
The magnitude of the
field is not a function
of phi or z and it
varies inversely
proportional as the
distance from the
filament.
The direction is of the
magnetic field
intensity vector is
circumferential.
H2








I
3

I
2  

2
4    z1

H2
2
2

d z1 az    a  z1 az

I 


4  






3

2

2
 z1
d z1 a
2

 a
6
Biot-Savart Law
H
I
4  
 



 sin  2  sin  1  a 
7
Example 8.1
H2x

4  ( 0.3) 


  1  a
  
180 

H2x 

4  ( 0.3) 


  1
 
180 

8

  sin  53.1
8
  sin  53.1

H2x  3.819
a must be refered to the x axis - which becomes
1x  90

180
0.3 
1y  atan 

 0.4 
2x  atan 
0.4 

 0.3 
2y  90

H2y

4  ( 0.4) 


    a 
 z
180  
H2y 

4  ( 0.4) 




180  
8
8
  1  sin  36.9
  1  sin  36.9
H2  H2x  H2y
az


H2  6.366
H2y  2.547
az
180
8
Ampere’s Circuital Law
The magnetic field in space around an electric current is proportional to the electric
current which serves as its source, just as the electric field in space is proportional to
the charge which serves as its source.
9
Ampere’s Circuital Law



H_dot_ d L
I
Ampere’s Circuital Law states that the line integral of H about any closed
path is exactly equal to the direct current enclosed by the path.
We define positive current as flowing in the direction of the advance of a
right-handed screw turned in the direction in which the closed path is
traversed.
10
Ampere’s Circuital Law - Example

 H_dot_ d L

H



2 
0
H  d 

H   

2 
1 d
I
0
I
2  
11
Ampere’s Circuital Law - Example
H
H
I
I 
 a
2  a
I
2 
H   
H
0
 c
2
  2  b2 

I  I 
 2 2
c b 
2    H
H
a b
2  

2
2
2
2
I
c 
 b
c b
2  

2
2
2
2
c b
12
Ampere’s Circuital Law - Example
13
Ampere’s Circuital Law - Example
14
Ampere’s Circuital Law - Example
15
CURL
The curl of a vector function is the
vector product of the del operator
with a vector function
16
CURL

 H_dot_ d L

( curl_H)aN
lim
SN  0
SN
17
CURL
curl_H =
x H
x H = J
Ampere’s Circuital Law
Second Equation of Maxwell
x E = 0
Third Equation
18
CURL
Illustration of Curl Calculation
CurlH
 d H  d H a 

z
y x
d
y
d
z


 ax

d
CurlH 
 dx
H
 x
ay
d
dy
Hy
 d H  d H a   d H  d H a
 x
z y 
y
x z
d
z
d
x
d
x
d
y






d 
dz 

Hz 
az
19
CURL
Example 1
In a certain conducting region, H is defined by:
2
H1x( x y  x)  y  x x  y
Determine J at:

2
x  5
2
H1y ( x y  z)  y  x z
y  2
2 2
H1z( x y  z)  4 x  y
z  3
DelXHx  420

ax
d
d
DelXHy  H1x( x y  z)  H1z( x y  z)
dz
dx
DelXHy  98

ay
d
d
H1y ( x y  z)  H1x( x y  z)
dx
dy
DelXHz  75

az
d
d
DelXHx  H1z( x y  z)  H1y ( x y  z)
dy
dz
DelXHz 
20
CURL
Example 2
H2x( x y  x)  0
x  2
y  3
2
H2y ( x y  z)  x  z
2
H2z( x y  z)  y  x
z  4
d
d
DelXHx  H2z( x y  z)  H2y ( x y  z)
dy
dz
d
d
DelXHy  H2x( x y  z)  H2z( x y  z)
dz
dx
d
d
DelXHz  H2y ( x y  z)  H2x( x y  z)
dx
dy
DelXHx  16

ax
DelXHy  9

ay
DelXHz  16

az
21
Example 8.2
22
Stokes’ Theorem
The sum of the closed line
integrals about the perimeter
of every Delta S is the same
as the closed line integral
about the perimeter of S
because of cancellation on
every path.

 H_dot_ d L


 ( Del  H)_dot_ d S

S
23
Hr r      6 r sin  
Example 8.3
H r      0
H r      18 r sin     cos  
segment 1
r  4
0    0.1 
r  4
  0.1 
  0
segment 2
0    0.3 
segment 3
r  4
dL
0    0.1 
  0.3 



dr ar  r d a  r sin     d a
First tem = 0 on all segments (dr = 0)
Second term = 0 on segment 2 ( constant)
Third term = 0 on segments 1 and 3 ( = 0 or constant)

 H dL



 H r d 

since H=0



0.3 

 H r sin    d 


 H r d 

H r       r sin     d   22.249
0
24
Magnetic Flux and Magnetic Flux Density
B
 0 H


 B_dot_ d S


 B_dot_ d S

0
7
4  10
H
permeability in free space
m
0
25
The Scalar and Vector Magnetic Potentials
H
Del_Vm
J
0
b
Vm

 H_dot_ d L

a
26
The Scalar and Vector Magnetic Potentials
27
Derivation of the Steady-Magnetic-Fields Laws
28