Chapter 08 b [Compatibility Mode]

8/14/2019 Chapter 08 b [Compatibility Mode]

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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

Chapter 3 BEE 3113ELECTROMAGNETIC FIELDS THEORY 4


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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

Chapter 3 6


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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)

Chapter 3 7


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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,

9


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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 .

Chapter 3 13


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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

Chapter 3 15


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For region a p b, we use L2

or

Chapter 3 16


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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

Chapter 3 35


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Magnetic flux through a surface S

IISN

S S

Unit is webers (Wb) and unit ofB2

Chapter 3 36


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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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39


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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

Chapter 3 43


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Chapter 3 44


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vector magnetic potential, A (Wb/m) can be defined as

=

Chapter 3 45


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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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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

Chapter 3 54


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Chapter 3 55


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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= + +

Chapter 3 59

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Chapter 08 b [Compatibility Mode]