answer 1(b) (2011)
TRANSCRIPT
1
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Chapter 7
Magnetostatic field
Part 2
3
. .enc
L
I H dl J dS
Ampere’s
path
Current
Surface
.enc vQ D dS dv
4
. ( ) .enc
L
I H dl H dS J dS
J H
5
Application of Ampere’s law
(1) An infinite line current
2
0
.
2
2
enc
L
I H dl
H d H
IH
a a
a
(2) Infinite sheet of current
1
2nH k a
6
(3) Infinitely Long Coaxial Transmission Line
Example in your book
7
Plane y=1 carries current K=50 az mA/m (K:surface current density ) find H at (a) (0,0,0) (b) (1,5,-3)
1 1( ) (50 ) ( ) 25 mA/m
2 2
1 1( ) (50 ) ( ) 25 mA/m
2 2
n z y x
n z y x
a H k
b H k
a a a a
a a a a
8
Plane X=10 carries 100mA/m along az while line X=1,y=-2 carries 20∏ mA along az . Find H at (4,3,2)
9
1
2 φ x y
x y
x y y x
φ
3y x
2 x y
1 2
1 1(0.1 ) ( ) 0.05 A/m
2 2
IH = , (4,3,2) (1, 2,2) 3 5
2
3 5 , | |= 34 ,
34
3 5 3 5( )
34 34
3 520 10H = ( ) 1.47 0.889
2 ( 34) 34
n z x y
z
z
H k
Htot H H
a a a a
a a a
a aa a a
a a a aa a
a aa a
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A hollow conductor cylinder inner radius a and outer radius b carries current I along az find H everywhere.
enc
enc
H.dl=I =0
I H.dl = J.dS
(amp. path) (current surf.)
(1)for ρ < a
(2)for a < ρ < b
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2 2
2
2 2
2 2 2 2
0
2 2
2 2
2 2
2 2
amp. path= H.dl (2 ) ...........(1)
I I(current surf.)= J.dS= . .
S b a
I I[ a ]................(2)
b a b a
(1) (2)
I(2 ) [ a ]
b a
I a[ ]
2 b a
a
H
dS d d
d d
H
H
a
12
2 2
2 2
amp. path= H.dl (2 ) ...........(1)
I I(current surf.)= J.dS= . [ a ]................(2)
S b a
(1) (2)
(2 )
I
2
H
dS b
H I
H
(3) for ρ > b
a
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14
Consider the two-wire transmission line whose cross section is in fig. Each wire is of radius 2 cm and the wires are separated 10 cm. I=5A , find H at (5cm,0)
1
1
2
1 1
tot 1 2
H2
5 , , ( ) ( )
5H 15.915 A/m
2 (0.05)
2
H2
5 , , ( ) ( )
515.915 A/m=H
2 (0.05)
H =H +H 31.83 A/m
x z x y
y y
x z x y
y y
y
I
cm
I
cm
H
for(1)
a
a a a a a a
a a
for( )
a
a a a a a a
a a
a
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Magnetic flux Density
2
7
,( / ) ( )
: of free space
4 10 H/m
magnetic flux: .
o
o
o
B H W m or Tesla
permeability
B dS Webers
16
For a magnetic flux density equal to 1 tesla, a force of 1 newton must act on a
wire of length 1 meter carrying 1 ampere of current.
A Newton : the force required to accelerate a 1 kg weight at one meter per
second squared .
The most powerful superconducting electromagnets only produce magnetic
fields of around 20T.
A gauss , another unit for measuring magnetism, is 1/10,000th of a tesla.
A large loudspeaker magnet generates 1T.
tesla
flux line: closed ,no beginning or end.
doesn't cross each other
17
In an electrostatic field, the flux passing through a closed
surface is the same as the charge enclosed .
It is possible to have an isolated electric charge
it is not possible to have isolated magnetic poles
(or magnetic charges).
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flux passing through a closed surface is the same as the
charge enclosed .
.D dS Q . 0B dS Gauss's law for electrostatic fields Gauss's law for magnetostatic fields
0B law of conservation of magnetic flux
Maxwell’s equations for static fields
enc
enc
D d S Q
B d S 0
E d L 0
H d L I
vD
B 0
E 0
H J
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If H=yax-xay A/m on plane Z=0 (a)Determine the current density(b) Verify Ampere’s law by taking circulation of H around the
edge of the rectangle: Z=0 , 0<X<3 , -1<y<4
2
4 3
y=1 0
1 2 3 4
3 3 3
1 1y= 1
x=0 x=0 x=0
3y=
. ( ) .
2 A/m
. 2 30 A ......(1)
H.dl= + + +
(1) H .dl = ( ).dx = dx= 3
(2) ( ).dy = 3dy= 15
enc
L
z
enc
x
x y x
x y yx
I H dl H dS J dS
J H
I J dS dxdy
x
y x
(a)
a
(b)
a a a
a a a
4 4
y= 1 1
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0 0
43 3
1 1
0y=4 4
(3) ( ).dx = 4dx = 12
(4) ( ).dy = (0)dy=0
(1) (2) (3) (4) 30A as in (1)
x y xy
x x
x y yx
y
y x
y x
a a a
a a a
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Magnetic scalar potentials (Vm)
Vector Magnetic potentials (A)
POTENTIALS
mH V
B A
. ( ). .S S L
B dS dS A dl A
Wb/m
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An infinitely long wire carries 2A along az find:(a)B at (-3,4,7)(b)Flux through the square loop 0<ρ<6 , 0<z<4 , Φ=90
1
1
o o
4 6 4 6
o
0 2 0 2
o
( )
H2
4 3 4 3 4 34 3 , , ( ) ( )
5 5 5
4 32H 50.93 38.2 A/m
2 (5) 5
B= H= ( 50.93 38.2 ) T
( )
.2 2
ln( )|2
y x y x x y
y x z
x y
x y
x y
z z
a
I
b
IIB dS d dz d dz
I
a
a a a a a aa a a a a
a aa a
a a
a a
6 46
2 0
z | 1.76 10 Wb
24
Which of the following fields represents an electrostatic or magnetostatic field in free space
( ) cos( ) ( )
sin( ) 0
0
neither electrostatic nor magnetostatic field
0 is ES
0
0 is MS
0
x
x za A y ax y e
A ya ax
A
A
CC
C
CC
C
a a
vD
B 0
E 0
H J
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The magnetic vector potential A=15e-ρ sin Φ az Wb/m
(a)Find H at (3,pi/4,-10)(b) find the flux through ρ=5 , 0<Φ<pi/2 , 0<Z<10
o
10 / 2
0 0
15cos 15 sin
(3, / 4, 10) 0.176 0.528 T
BH= (140 420 ) kA/m
15. cos 1.011 Wb
.
S z
L
B A e e
B
B dS e d dz
A dl
(a) a a
a a
a a
other solution :
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