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Magnetostatic Field:
Ampere Circuital Law
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In this chapter you will learn
Biot-Savats Law
Ampere Circuital Law
Magnetic flux density vector Magnetic potential vector and magnetic force
Magnetic circuit
ara ay s aw Maxwells Equation
Chapter 3 2
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Similar to Gausss law
Amperes law states that the line integral of theangen a componen s o aroun a c ose pa s e
same as the net current Ienc enclosed by the path
= encIdlH This is the integral form of Amperes Circuit Law
mpere s rcu aw s use w en we wan o
determine H when the current distribution is
Chapter 3 3
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Some symmetrical current distributions:
1. Infinite line current2. Infinite Sheet of current
3. Infinitely long coaxial transmission line
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Amperes Law for an Infinite Current
Filament
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Consider an infinite current sheet in the z = 0plane with
a uniform current density K = KyayA/m
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A l in Am eres Law we et
------------
.
resultant dH has only anx-component
Also H on one side of the sheet is the ne ative of that
on the other side.
------------(2)
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obtain
------------(3)
Compare eq (1) with eq (3), we get
8
u u e 0 n eq
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Thus now we can say
In general, for an infinite sheet of current density K A/m,
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Magnetic Field Inside Coaxial
Transmission Line
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two concentric cylinders having their axes along the z-axis
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The inner conductor has radius a and carries current I
while the outer conductor has inner radius b and- .
Since the current distribution is symmetrical, we apply
'
possible regions:
0 aa b,
+ a n d b + t .
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For region 0 a, we apply Ampere,s law to path L1
Since the current is uniformly distributed over the crosssection,
Chapter 3 14
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Thus
or
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For region a p b, we use L2
or
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For region b b + t, we use L3
J in this case is the current density (current per unit area)
of the outer conductor and is along a :
Chapter 3 17
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us
Then we can get H,
Chapter 3 18
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For region b + twe use L4
Chapter 3 19
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Putting it all together,
Chapter 3 20
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Plot of H against .
Chapter 3 21
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A toroid whose dimensions are shown below has N turns
and carries current I. determine H inside and outside the
toroid.
2
0
0
22
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A l Am eres circuit law to the Am erian ath which
is a circle of radius p. Since N wires cut through this
path each carrying current I, the net current enclosedby the path is NI. Hence,
aaNI
H
enc
+
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Outside the toroid, the current enclosed by the
Amperean path is
- =
Hence H = 0
Chapter 3 24
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J=
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I ==
=S
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Amperes Circuit Law in Differential
form: 3rd Maxwells Equation
If we apply Stokes theorem to eq 1, we obtain
)2.....(....................)( SHlH ddIenc ==
But since
)3......(.............................. = Senc dI SJ We compare (2) and (3) to obtain
=
Chapter 3 32
Third maxwells Equation: Amperes law in point form
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Chapter 3 33
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0J =
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Magnetic flux density, B is given by:
0=
is permeability of free space,0
70
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Magnetic flux through a surface S
IISN
S S
Unit is webers (Wb) and unit ofB2
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Magnetic Flux density Maxwells
Equation Magnetic flux lines always close upon
themselves - NOT POSSIBLE to have
charges)
exist Thus the total flux throu h a closed
surface is zero
Law of conservation of
magnetic flux orGausss
==
Chapter 3 37
field
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Apply divergence theorem.
rr rv
v = =
It means that magnetic field lines are alwayscontinuous
r
Fourth Maxwell's e uation
Chapter 3 38
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agne c po en a cou e sca ar m or vec or
Two identities
)4......(....................0)( = V
)5...(....................0)A( =
always hold for any scalar field Vand vector field A
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us as , we e ne e re a on e ween
and magnetic scalar potential, Vm as=
mV=H
Combine with 3rd Maxwell's e uation,0=Jif
0HJ === V
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Chapter 3 44
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vector magnetic potential, A (Wb/m) can be defined as
=
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0==
2
= xxJA 0
2 =
zz JA 02 =
=
= xxxx
dxIvdJA
)( 00 ar
Lv
M ti P i d L l
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Magnetic Poissons and Laplace
Equations
F M ti V t P t ti l t Bi
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From Magnetic Vector Potential to Bio-
Savart Law
aH
24 Rd =
Magnetic Vector Potential for Different
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Magnetic Vector Potential for Different
Source Distributions
=L R4
0A or ne curren
= S dS0K
A For surface charge
=
dv0 J
AFor volume charge
v R4
Magnetic Vector Potential inside a
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Magnetic Vector Potential inside a
Coaxial Transmission Line
Magnetic Vector Potential inside a
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Magnetic Vector Potential inside a
Coaxial Transmission Line
Magnetic Vector Potential inside a
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Magnetic Vector Potential inside a
Coaxial Transmission Line
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Thus we can replace B in equation = S
===LSS
So = dlAL
It is an alternative way of finding magnetic flux using
Chapter 3 53
magnetic potential vectorA
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Given the magnetic vector potential A = - /4 az Wb/m
Calculate the total magnetic flux crossing the surface
= 2, 1 2 m, 0 z 5 m
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Chapter 3 56
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Chapter 3 57
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Chapter 3 58
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xamp ecurren s r u on g ves r se o e vec or magne c
potential .2 2 4 /A a a a x y z x y y x xyz Wb m= +
-
(b)the flux through the surface defined by z = 1, 0 x 1,-1 4
Answer: (a)
(b) 20 Wb
220 40 3 / B a a a x y zWb m= + +
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