Waves and applications 4th 1

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Unit-4Waves and applications

Mr. HIMANSHU DIWAKARAssistant ProfessorGETGI

Himanshu Diwakar, AP

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Learning objectives

• Faraday’s Laws• Various Induced Emf’s Directions and Magnitude• Practical applications


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Magnet is Moving, conductor is stationary

Faradays first LawStatement:

When a conductor is placed in a magnetic field, due to relative motion between the conductor and magnetic field an EMF is induced.

Exp.1 Exp.2

Conductor is Moving, Magnet is stationary


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Faraday’s Second law

Statement:

The magnitude of induced EMF is directly proportional to the rate of Change of Flux linkages.

Dynamically Induced EMF Statically Induced EMF Self Induced EMF Mutual Induced EMF


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Magnitude and direction of Dynamically induced EMF

Right hand Thumb rule

E=BLV SIN

E= Induced EMF in voltsB= Magnetic Flux Density in webersV= velocity m/sec

= Angle between flux and conductor position


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Dynamically induced EMF

• EMF produced due to space variation between the magnetic field and the conducto

AlternatorDC GeneratorHimanshu Diwakar, AP

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Negative sign indicates the opposition

Statically Induced EMF• EMF produced due to the time variation of flux linking with the stationary

conductor.

Self induced EMF Mutually induced EMF


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Lenz’s law

Working of Transformer Practical Transformer


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Displacement current


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Recall Ampere’s Law

encIsdB 0


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Imagine a wire connected to a charging or discharging capacitor. The area in the Amperian loop could be stretched into the open region of the capacitor. In this case there would be current passing through the loop, but not through the area bounded by the loop.


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If Ampere’s Law still holds, there must be a magnetic field generated by the changing E-field between the plates. This induced B-field makes it look like there is a current (call it the displacement current) passing through the plates.


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Properties of the Displacement Current

• For regions between the plates but at radius larger than the plates, the B-field would be identical to that at an equal distance from the wire.

• For regions between the plates, but at radius less than the plates, the Ienc would be determined as through the total I were flowing uniformly between the plates.


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Equation for Displacement current

dtdI E

d

0


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Modified Ampere’s Law (Ampere-Maxwell Law)

dtdIsdB

IIsdB

Eenc

encdenc

000

,00


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Maxwell’s Equations

h

09-18-2008Himanshu Diwakar, AP

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Content sField equationsEquation of continuity for time varying fieldsInconsistency of Ampere’s LawMaxwell’s equationsConditions at a Boundary surfaces


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The equations governing electric field due to charges at rest and the static magnetic field due to steady currents are

Contained in the above is the equation of continuity Time Varying Fields:From Faraday’s Law

In time varying electric and magnetic fields path of integration can be considered fixed. Faraday’s Law

becomes Hence 1st equation becomes

0.dsE

D. dvdaD .

JH daJdsH ..

0. B 0. daB

0. J 0.daJ

s

daBdtd

dtddsE ..

datBdsE

s

..

s

datBdsE ..

0 E

tBE


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Equation of continuity for Time-Varying Fields:

From conservation of charge concept

if the region is stationary

Divergence theorem

time varying form of equation of Continuity

Inconsistency of Ampere’s Law:Taking divergence of Ampere’s law hence Ampere’s law is not consistent for time varying equation of continuity.

(from Gauss’s Law)

displacement current density.

dVdtddaJ .

dVt

daJ .

dVt

JdV .

tJ

.

JH .).( 0J

Dt

J ..

0.

JtD

0.

daJtD


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Hence Ampere’s law becomes .Now taking divergence results equation of continuity

Integrating over surface and applying Stokes’s theorem

magneto motive force around a closed path=total current enclosed by the path.

Maxwell’s equations: These are electromagnetic equations .one form may be derived from the other with the help of Stoke’s

theorem or the divergence theorem

Contained in the above is the equation of continuity.

JtDH

daJtDdsH ..

JtDH

daJtDdsH ..

tBE

datBdsE ..

D. dVdaD .

0. B 0.daB

tJ

. dV

tdaJ

.


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Word statement of field equation:1. The magneto motive force (magnetic voltage)around a closed path is equal to the conduction current plus the

time derivative of electric displacement through any surface bounded by the path.

2. The electromotive force (electric voltage)around a closed path is equal to the time derivative of magnetic displacement through any surface bounded by the path

3. Total electric displacement through the surface enclosing a volume is equal to the total charge wihin the volume.

4. The net magnetic flux emerging through any closed surface is zero.

Interpretation of field equation:Using Stokes’ theorem to Maxwell’s 2nd equation

Again from Faraday’s law region where there is no time varying magnetic flux ,voltage

around the loop would be zero the field is electrostatic and irrational.

Again

there are no isolated magnetic poles or “magnetic charges” on which lines of magnetic flux can terminate(the lines of mag.flux are continuous)

dsEdaE ..da

dsEnE

.ˆ.

tBE

0 E0. B


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Boundary condition:1. E,B,D and H will be discontinuous at a boundary between two different media or at surface that

carries charge density σ and current density K2. Discontinuity can be deduced from the Maxwell’s equations

1. over any closed surface S 2.

3. for any surface S bounded by closed loop p

4.

From 1

Spf

Sp

S

Sf

daDdtdIdlH

daBdtddlE

daB

QdaD

enc

enc

..

..

0.

.

f

f

DD

aaDaD

21

21 ..1

2

f

D2

D1

a


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For metallic conductor it is zero for electrostatic case or in the case of a perfect conductor

normal component of the displacement density of dielectric = surface charge density of on the conductor.

Similar analysis leads for magnetic field

ED

snD 1

21 nn BB


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Electromagnetic Waves in homogeneous medium:The following field equation must be satisfied for solution of electromagnetic problem

there are three constitutional relation which determines

characteristic of the medium in which the fields exists.

Solution for free space condition:

In particular case of e.m. phenomena in free space or in a perfect dielectric containing no charge and

no conduction current

Differentiating 1st

JtDH

tBE

D.

0. B EJHBED

tDH

tBE

0. D0. B

tH

tH


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Also since and are independent of time

Now the 1st equation becomes on differentiating it

Taking curl of 2nd equation

( )

But

this is the law that E must obey .

lly for H

these are wave equation so E and H satisfy wave equation.

tH

tB

tE

tD

2

2

tE

tH

tH

tBE

tEE 2

2

tEEE 2

22.

EEE 2.

0.1. DE

2

22

tEE

2

22

tHH


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Uniform Plane wave propagation:If E and H are considered to be independent of two dimensions say X and Y

For uniform wave propagation differential equation equation for voltage or

current along a lossless transmission line.

General solution is of the form

reflected wave.

Uniform Plane Wave:

Above equation is independent of Y and Z and is a function of x and t only .such a wave is uniform plan

wave. the plan wave equation may be written as component of E

2

2

2

2

tE

xE

2

2

2

2

tE

xE yy

tvxftvxfE 0201

2

2

2

2

tE

xE

2

2

2

2

2

2

2

2

2

2

2

2

tE

xE

tE

xE

tE

xE

zz

yy

xx


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For charge free region

for uniform plane wave there

is no component in X direction be either zero, constant in time or increasing

uniformly with time .similar analysis holds for H . Uniform plane electromagnetic waves are

transverse and have components in E and H only in the direction perpendicular to direction of propagation

Relation between E and H in a uniform plane wave:For a plane uniform wave travelling in x direction

a)E and H are both independent of y and z

b)E and H have no x component

From Maxwell’s 1st equation

From Maxwell’s 2nd equation

0

0.1.

zE

yE

xE

DE

zyx

0xEx

xE

zxH

yxHH

zxE

yxEE

yz

yz

ˆˆ

ˆˆ

tDH

ytEz

tE

zxH

yxH zyyz ˆˆˆˆ

tBE

ztH

ytHz

xE

yxE yzyz ˆˆˆˆ


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Comparing y and z terms from the above equations

on solving finally we get

lly Since

The ratio has the dimension of impedance or ohms , called characteristic impedance or intrinsicimpedance of the (non conducting) medium. For space

tH

xE

tH

xE

tE

xH

tE

xH

zy

yz

zy

yz

y

z

z

y

yz

HE

HE

EH

22

22

zy

zy

HHH

EEE

HE

ohms

mhenry

v

v

v

v

377120

10361

/104

9

7


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ohmsv

vv 377

The relative orientation of E and H may be determined by taking their dot product and using above relation

In a uniform plane wave ,E and H are at right angles to each other.

electric field vector crossed into the magnetic field vector gives the direction in which the wave travells.

0. zyzyzzyy HHHHHEHEHE

222 ˆˆˆ HxHHxHEHExHE yzyzzy


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Thank you


Engineering

Waves and applications 4th 1