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Transcript of 1 Chapter 8 The Steady State Magnetic Field The Concept of Field (Physical Basis ?) Why do Forces...
![Page 1: 1 Chapter 8 The Steady State Magnetic Field The Concept of Field (Physical Basis ?) Why do Forces Must Exist? Magnetic Field – Requires Current Distribution.](https://reader030.fdocuments.net/reader030/viewer/2022032702/56649ce65503460f949b3a01/html5/thumbnails/1.jpg)
1
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?
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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.
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From GSU Webpage
Magnetic Field – Concepts, Interactions and Applications
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Biot-Savart Law
dHIdL aR
4 R2
IdL R
4 R3
Magnetic Field Intensity A/m
H LI
4 R2
d a R Verified experimentally
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
Biot-Savart = Ampere’s law for the current element.
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I NK
d
The total current I within a transverse Width b, in which there is a uniform surface current density K, is Kb.
For a non-uniform surface current density, integration is necessary.
Alternate Forms H SK_x
4 R2
d aR H vJ_x
4 R2
d aR
Biot-Savart Law B-S Law expressed in terms of distributed sources
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H2
z1I
4 2
z12
3
2
d az a z1 az I
4
z1
2
z12
3
2
d a
H2I
2 a
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.
Biot-Savart Law
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Biot-Savart Law
HI
4 sin 2 sin 1 a
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8
Example 8.1
H2x8
4 0.3( )sin 53.1
180
1
a
H2x8
4 0.3( )sin 53.1
180
1
H2x 3.819
a must be refered to the x axis - which becomes az
H2y8
4 0.4( )1 sin 36.9
180
az
H2y8
4 0.4( )1 sin 36.9
180
H2y 2.547
H2 H2x H2y H2 6.366 az
1x 90
180 2x atan
0.4
0.3
1y atan0.3
0.4
2y 90
180
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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.
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Ampere’s Circuital Law
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.
LH_dot_
d I
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Ampere’s Circuital Law - Example
LH_dot_
d
0
2 H
d H 0
2 1
d I
HI
2
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HI
2 a b
a H 0 cH
I
2 a2
2 H I I
2b
2
c2
b2
HI
2
2
b2
c2
b2
H I
2
c2
2
c2
b2
Ampere’s Circuital Law - Example
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Ampere’s Circuital Law - Example
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Ampere’s Circuital Law - Example
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Ampere’s Circuital Law - Example
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CURL
The curl of a vector function is the vector product of the del operator with a vector function
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CURL
curl_H( )aN0SN
LH_dot_
d
SNlim
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CURL
Ampere’s Circuital LawSecond Equation of Maxwell
Third Equation
curl_H = x H
x H = J
x E = 0
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Illustration of Curl Calculation
CurlHy
Hzd
d zHy
d
d
axzHx
d
d xHz
d
d
ayxHy
d
d yHx
d
d
az
CurlH
ax
x
d
d
Hx
ay
y
d
d
Hy
az
z
d
d
Hz
CURL
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CURL
az
DelXHz 75DelXHzxH1y x y z( )d
d yH1x x y z( )d
d
ay
DelXHy 98DelXHyzH1x x y z( )d
d xH1z x y z( )d
d
ax
DelXHx 420DelXHxy
H1z x y z( )d
d zH1y x y z( )d
d
z 3y 2x 5Determine J at:
H1z x y z( ) 4 x2 y
2H1y x y z( ) y2 x zH1x x y x( ) y x x
2y
2
In a certain conducting region, H is defined by:
Example 1
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CURL
Example 2
H2x x y x( ) 0 H2y x y z( ) x2
z H2z x y z( ) y2 x
x 2 y 3 z 4
DelXHxy
H2z x y z( )d
d zH2y x y z( )d
d DelXHx 16 ax
DelXHyzH2x x y z( )d
d xH2z x y z( )d
d DelXHy 9 ay
DelXHzxH2y x y z( )d
d yH2x x y z( )d
d DelXHz 16 az
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Example 8.2
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Stokes’ Theorem
LH_dot_
d SDel H( )_dot_
d
S
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.
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Example 8.3Hr r 6 r sin
H r 0
H r 18 r sin cos
segment 1
r 4 0 0.1 0
segment 2
r 4 0.1 0 0.3
segment 3
r 4 0 0.1 0.3
dL 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)
LH
d H r
d H r sin
d H r
d
since H=0
0
0.3 H r r sin
d 22.249
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Magnetic Flux and Magnetic Flux Density
B 0 H 0 4 107
H
mpermeability in free space
SB_dot_
d
SB_dot_
d 0
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The Scalar and Vector Magnetic Potentials
H Del_Vm J 0
Vma
b
LH_dot_
d
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The Scalar and Vector Magnetic Potentials
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Derivation of the Steady-Magnetic-Fields Laws