chapter 1 faradays law

8/12/2019 Chapter 1 Faradays Law

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KEEE3213

Electromagnetic Theory

Dr. Wan Nor Liza Mahadi

Room RB 10

Email: [email protected]


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Syllabus

Time varying fields and Maxwellsequations

Uniform Plane Waves

Transmission Lines

Waveguides

Antenna and Antenna Arrays


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Reference Books

Fundamentals of EngineeringElectromagnetics- David K.Cheng,

Addison Wesley** main ref.

Engineering Electromagnetics- WilliamHyatt

Electromagnetics with applications,

John D.Krauss, Mc Graw Hill


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Chapter 1:

Time-VaryingFields and Maxwells

Equations


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The magnetic field produced by

a conductor


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The right hand rule


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Overview

Fundamental governing equations for

electrostatic model

For linear and isotropic (not necessarily

homogeneous) media, E and D are relate

by the constitutive relation


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For the magnetostatic model, the

fundamental governing differential

equations are

The constitutive relation for B and Hin

linear and isotropic media is


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Faradays Law of

Electromagnetic Induction Fundamental postulate for

electromagnetic induction

Applying the Stokestheorem, we get


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A Stationary Circuit in a Time-

Varying Magnetic Field Faradays law of electromagnetic

induction

=

Electromotive force induced in a

stationary closed circuit is equal to the

negative rate of increase of the

magnetic flux linking the circuit.

Known as t rans former emf


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=

The above equation states that the electromagnetic

force induced in a stationary closed circuit is equal

to the negative rate of increase of the magnetic flux

linking the circuit.

The negative sign is an assertion that the induced

emf will cause a current in to flow in the closed

loop in such direction as to oppose the change inthe linking magnetic flux, known as Lenzslaw


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Q1: An inductor is formed by winding N turns of a thinconducting wire into a circular loop of radius a. The inductor

loop is in the x-y plane with it center at the origin, and it is

connected to a transistor R, as shown in Figure Q1 below. In In

the presence of a magnetic field given by B = B0( 2 + 6 ) sint, where is the angular frequency, find:

(i) the magnetic flux linking a single turn of the inductor,

(ii) the transformer emf, given that N=20, B0= 0.2T, a= 10

cm, and is 103rad/s.

Figure Q1: Circular loop with N turns in

the x-y plane


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Solutions to Q1


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Introduction

A transformer in electrical engineering is a magneticdevice which consists of two or more multiturn coils

wound on a common core. Its major use is in the

supply and distribution of alternating current (a.c.)

electrical power.


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Transformer

Conditions of an ideal transformer(Assuming noleakage flux, etc)

Current relation for an ideal transformer


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Voltage relation for an idealtransformer

Resistance transformation by an ideal

transformer

(R1)eff = vi/i1


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A moving conductor in a static magnetic

field

When a conductor moves with a velocity u in astatic(non time varying) magnetic field B, a force

Fm = qu x B will cause the freely moveable

electrons in the conductor to drift toward one end of

the conductor and leave the other end positivelycharged.


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To an observer moving with the conductorthere is no apparent motion, and the

magnetic force per unit charge Fm/q = u x B

can be interpreted as an induced electric

field acting along the conductor and

producing a voltage

21= 21 .


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If the moving conductor is a part of closedcircuit C, then the emf generated around

the circuit is

= This is known as f lux cut t ing or mot ional

emf


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Q2: Figure below shows a conducting sliding bar oscillates

over two parallel conducting rails in a sinusoidally varying

magnetic field

B = aZ5cos t ( mT )

The position of sliding bar is given by x = 0.35(1-cos t)

(m), and the rails are terminated in a resistance R = 0.2.

Determine the current i.


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A moving circuit in a time-varying

magnetic field Lorentzs force equation

General form of Faradays law

Another form of Faradays law


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Maxwells Equations


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Equation of continuity


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Differential (operator) form ofMaxwells equations (the four

Maxwells equations are not all

independent)


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is the volume density of freecharges

J is the density of free currents, whichmay comprise of both convection

current

(u) and conduction current ()

Integral form of Maxwells


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Integral form of Maxwell s

equations

Take the surface integral of both sidesof the curl equations over an open

surface s with a contour c and apply

Stokesstheorem;Type equation here.

. = +

.

ds

Faradays law

Amperes circuital

law


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Taking the volume integral of bothsides of the divergence equations over

a volume v with a closed surface s

and using divergence theorem;

Gausss law

No isolated magnetic

charge


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The integral forms of Maxwellsequations describethe behaviour of electromagnetic field quantities in

all geometric configurations.

The differential forms of Maxwells equations areonly valid in regions where the parameters of the

media are constant or vary smoothly i.e. in regions

where (x,y, z, t),(x,y, z, t) and (x,y, z, t) do not

change abruptly.


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Electromagnetic Boundary

Conditions

Boundary condition for tangential

component of E

Boundary condition for tangential

component of H


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Boundary condition for normalcomponent of D

Boundary condition for normal

component B


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Interface between two lossless media

Can be specified by ,, with = 0 .

Usually no free charges and no surfacecurrents at the interface between two

lossless media.


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Interface between dielectric and perfect

conductor

Etand B nare continuous Htand Dnare discontinuous by an amount equal to

surface current Jsand surface charge density s

Potential Functions


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Potential Functions

If the above eqn is substituted in the

differential form of Faradays law, we get


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In time-varying case, E is a function of both scalar

electric potential V and vector magnetic potential A

Unit V/m


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Lorentz condition for potentials

Wave equation for vector potential A

Which is the nonhomogeneous waveeqn for vector potential A


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Wave potential for scalar potential V

Which is the nonhomogeneous waveequation for scalar potential V


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Solution of wave equation

Finding retarded scalar potential V from

charge distribution

Finding retarded vector potential A from

current distribution

Unit ( Wb/m)

Ti h i


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Time harmonics

electromagnetics Time-harmonic Maxwells equations in

terms of vector field phasors(E,H) and

source phasors ( , )in a simple (linear,isotropic and homogeneous medium)


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Phasor form for retarded vector

potential


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Th l t ti


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The electromagnetic

spectrum Maxwells equations in source-free

nonconducting media


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Homogeneous wave equation for E

Homogeneous wave equation for H


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Homogeneous Helmholtzs equationfor phasor Es

Homogeneous Helmholtzs equation

for phasor Hs

chapter 1 faradays law

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