Maxwells equation and Electromagnetic Waves

1/11/2017 1 Dr A K Mishra, Academic Coordinator,

JIT Jahangirabad

Engineering Physics II

By

Dr. A. K. Mishra

Associate Professor

Jahangirabad Institute of Technology, Barabanki

Maxwell`s Equations and Electromagnetic Waves

•Electromagnetism was developed by Michel faraday in 1791-1867and latter

James Clerk Maxwell (1831-1879),put the law of electromagnetism in he form

in which we know today. these laws are called Maxwells equation.

Scalar field: A scalar field is defined as that region of space whose each point

is associated with scalar function ie. A continuous function which gives the

value of a physical quantity like as flux, potential, temperature,etc.

Vector field: A vector field is specified by a continuous vector point function

having magnitude and direction both changes from point to point in given

region of field. The method of presentation of a vector field is called vector

line.

1/11/2017 Dr A K Mishra, Academic Coordinator,

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Gradient , Divergence and curl

• The rate of change of scalar and vector fields is denoted by a common operator called Del,or nebla is used which is written as

If is a differentiable scalar function, its gradient is defined as

grad

grad is a vector whose magnitude at any point is equal to the rate of change of at a point along a normal to the surface at the point.


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zk

y j

x i

z)y,(x,

)z

k y

j x

i (

z

k y

j x

i

Gauss Divergence theorem

(Relation between surface and volume integration )

According to this theorem , the flux of a vector field over any closed

surface S is equal to the volume integral of the divergence of the vector

field over the volume enclosed by the surface S.


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F

)1.....(....................dv F div sd . v

sF

Stokes Theorem ( Relation between surface and volume integration)

• The surface integral of the curl of a vector

field taken over an surface S is equal to the

line integral of around the closed curve.


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A

A

...(2)....................ld . A s.d ) A x (

ld . A s.d ) A Curl (

s

s

Fundamental laws of electricity and

magnetism

• Gauss law of electrostatics

i.e electric flux from a closed surface is equal to the charge enclosed by the surface.

Gauss law of magnetostatics:

i.e the rate of change of magnetic flux from a closed surface is always equal to zero.


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(3)....................q

s.d

0

E

0

1

.......(4).................... 0 sd .

B

Continued……………..

• Faradays law of electromagnetic induction: the rate of change of magnetic flux in a closed circuit induces an e.m.f which opposes the cause,i.e

• Amperes law :

the line integral of magnetic flux is equal to times the current enclosed by the current loop.


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5).........(.................... d

- dt

e

.....(6).................... I ld .0

B

0

Equation of continuity

• Electric current is defined as the rate of flow of charge i.e

If dq charge is enclosed in a volume dv and is leaving a surface area ds then we have

where J is the current density and is the volume charge density .therefore eq (1 ) becomes


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....(1).............................. dt

dq - i

vs

dV q and sd . J i

Continued……………


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equation. continuity called is 0 t

J

surfacearbitrary an for 0 )dVt

J (div

dVt

- dV J div

become 2equation re therefodV J div - sd .

get we2equation of L.H.Son theoremdivergence gaus using

2).........(.......... dVt

- sd .

dV dt

d - sd .

v

vv

v

v

v

div

J

J

J

s

s

s

Displacement current

• According to Maxwell not only current in the

conductor is Produces a magnetic field but

changing electric field in vacuum or in

dielectric is also Produces a magnetic field.

Means changing electric field is equivalent to

current produces same magnetic effect as A

conventional current in a conductor. This

equivalent current is called displacement


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Maxwells Electromagnetic

equations:(diffrantial form)


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density Charge

nal)(conventiodensity Current J

intensity field Magnetic B

intensity field Electric E

nt vectordisplaceme Electric D

wheret

D J H Curl or

t

D J H.

t - E Curl or

t - E.

0 B Div or 0 B.

D Div or .

BB

D

Maxwell's Electromagnetic

equations:(Integral form)


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meaning usual their have symbols s.d B t

- ld .

S.d )t

D J( .

0 sd .

.ds D or dv sd .

s

s

sv

E

ldH

B

Ds

Derivation of Maxwells Equation

• Maxwells First Equation ( ):

When a dielectric is placed in a uniform electric field , its molecule get

polarized. Thus ,a dielectric in an electric field contains two type of

charges- free charge and bound charge . if and be the free and

bound charge densities respectively, at appoint in a small volume element

dv, then for such a medium, Gausss law may be expressed as


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D . or D

div

p

dvsdE

dvsdE

s

s

)P div - ( 1

.

thereforeon.polarizati electric is P where,P div -

density charge bound nowthe

space. free ofy permitivit theis where

..(1)..............................) ( 1

.

v0

p

0

vp

0

Continued…………..

• Using Gauss divergence theorem on left hand side of the above equation,

we get


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equation. required theis this D .

D div or 0 - D div

have wefunction,arbitrary an for , therefore

0 dv ) - D (divor dV dv D div

.ent vectordisplacema electric theis D ) P (

dV dV ) P (

dv dv P div dv E div

dv dv P div dv E div

dV P div 1

- dV 1

dV E div .

vv

0

vv

0

v v

0

v

v vv

0

v0v0

Ebut

Ediv

or

sdEs v

Derivation of Maxwell's Second Equation

• The net magnetic flux through any closed surface is always zero.

Using Gauss theorem

The above expression shows that monopole or an isolated pole

can not exist to serve as a source. this law is also known as Gauss law in

magnetostatics. where V is the volume enclosed by surface S.

Hence ,for an arbitrary surface div B = 0 or


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.......(1).................... 0 .B s

sdB

0 dV B div

dV B div .B

v

vs

sd

0 B .

0 B .

Derivation of Maxwell's third Equation (faraday

law of electromagnetic induction)

• According to faraday law of electromagnetic induction,induced emf

around a closed circuit is equal to the negative time rate of change of

magnetic flux i.e.

if B is the magnetic field induction, then the magnetic flux linked with

the area ds

On combining the equation (1) and (2) we get

according to definition the induced emf is related to the corresponding

field as


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)1....(....................t

- B

e

)2....(....................sd . B s

B

(3).............................. sd . B - s

e

(4).............................. l.d E - l

e

Continued………..

Therefore from (3) and (4) will give


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t

B - E x or

t

B - E Curl

0 t

B sd E Curl function,arbitraryban For

0 sdt

B sd E Curl

sd t

B - sd E Curl

have we, Thus

sd E Curl - ld .

get weside, handleft on the theoremstoks theusing now

sd t

- ld . or

ds) B. ( t

- ld .

s

ss

sc

sc

sc

E

BE

E

Maxwells fourth equation

(modified amperes law)

• According to ampere law

Using stokes theorem on left hand side of the above expression, we get


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s.d J l.d B

sd J I formula thesin

l.d

0

0

gu

IB

J div H Curl

surface,arbitrary an For 0 sd ) J - (Curlor

sd J sd . H Curl

B properties dielectric from now sd J sd .

B

s.d J s.d B Curl 1

sd J sd B

s

s

0s

0

s0

s0

H

Curl

Curl

s

s

s

Continued…………………

• Taking div on both side we get ,

This shows amperes law applicable for static charges, therefore Maxwell's

suggested that ampere law must be modified by adding a quantity having dimension as

that of current, produced due to polarization of charges. this physical quantity is called

displacement current (Jd).thus modified ampere law becomes


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(static)constant

0 t

Hence

0 t

J div

have weequation,continuty from 0 J div Since

) calculus vector (from 0 H curl divbut

J div H Curl div

Continued………


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t

D J H Curl

becomes law Ampere modifie

t

D or ) ( div

D) (div t

div since D div

t - J div

div - J div

div J div 0

) 0 H Curl (div div J div 0

) (J div H curl div

get weside,both on divergence taking J H

JJ

j

J

J

J

J

J

dd

d

d

d

d

d

d

Therefore

t

Ddiv

but

But

Curl

Electromagnetic Energy (Pointing theorem)

• This is the analysis of transportation of energy from one place to

another due to propagation of electromagnetic waves.

Maxwell third and fourth equation in differential form are as follows


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(4).................... t

D . E J. E )H x ( E and

.(3).......... B

H - ) E x ( H get we

E, with (2)eqn and H with (1)eqn ofproduct scalar Taking

)........(2.................... t

D J H x

...(1).............................. B

- E x

t

t

Continued…………………

• Subtracting eqn (5) from eqn (4)


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J. E - )E . (E

2

1

)H . (H

2

1 - ) H x E ( .

J. E - )(

2

1

)(

2

1 - ) H x E ( .

J. E - t

E) (

t

H)( H - ) H x E ( .

Becomes

(6)equation theso , E D and H B medium,linear afor

......(6).......... J. E - t

D E

t

B . H - ) H x E ( .

becomesequation above The

) H x ( . E - ) E x ( . H ) H x E ( .

identity vector theusing

t

D . E - J . E -

t

B . H - ) H x ( E - ) E x ( H

EH22

tt

tt

Now

Continued………………. • hence


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sv

v

s v

.....(7).......... .ds ) H x E ( - dV ) H . B .D E ( 2

1

t )J. E(

)J. E() H . (B 2

1 ) D . (E

2

1

t

t - s)d H x (

get weside, handleft on the theoremdivergence gauss sin

)J. E() H . (B 2

1 ) D . (E

2

1

t - .dV ) H x E ( .

get weS, Surface aby bounded v volumeaover gintegratin

J. E -) H . (B 2

1 ) D . (E

2

1

t - ) H x E ( .

J. E - )D . (E

2

1

t

)B . (H

2

1 - ) H x E ( .

dV

dVdVE

gu

dVdV

t

v

vs

v

Continued…………

• Equation (7) is known as Pointing theorem. each term has its own physical significance which is as follows:

• The term is the generalized statement of Joules law and

represent the total power dissipated in volume V.

• The first term on right hand side of the equation is the sum of energy stored in electric field (E.D)and in magnetic field (B.H) or the total energy stored in electromagnetic field. therefore this term represents the rate of change in energy stored in volume V.

• The last term represents the law of conservation of energy, hence it represents

• The rate at which the energy is carried out of volume V across its boundry surface by electromagnetic waves.


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dVJEv

).(

Continued…………

• Thus the pointing theorem state that the work done on the

charge by an electromagnetic force is equal to the decrease in

energy stored in the field, less than the energy which flowed out

through the surface. it is also called the energy conservation law

in electrodynamics.

• The energy per unit time, per unit area transported by

electromagnetic field is called the pointing vector and is given by


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)H x E(1

H x E

0

Sor

S

Electromagnetic waves in free space

and its solution

• For free space or vacuum, Maxwell,s equations are as follows:


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) t

B (- x E x x

get we(3),equation of curl the

)4..(..........t

E B x

......(3).......... t

B - E x

..(2).................... 0 B .

...(1).................... 0 E .

0o

Taking

Continued……………….. • Using identity


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.......(5).................... E or

(4)equation using t

E .

(1)equation using )E x ( t

- E . - 0

t

B - x E) . ( - )E . (

get we,C)B. A( - B)C. A( )C x B( x

2

2

00

2

00

2

2

t

E

t

E

A

Continued……………….. • Equation (5)is a wave equation for electric field in free space, similarly, for

magnetic field, we have


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0

0

0

0

00

002

2

2

2

2

2

2

00

2

4

1 x

4

1

4 x 4

1 v

1

1or

1

space. freein vector magnetic and electric ofn propogatio of

velocity gives (7)equation with (6)(5)or equation Compairing

)7.(.................... 1

y

as,given is velocity v

a with gpropagatin wavea ofequation theNow

)........(6.......... B

v

v

v

B

B

t

t

Continued………………..

• Now,


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light. ofvelocity

with space freein propogates wavesneticelectromag hence

light) of(velocity c m/s x 3

x 9 x 1

x 9 4

1 and m -

A

Wb

4

10

1010

1010

8

9

7-

9

0

7-0

v

v

Depth of Penetration: (Skin Depth):

• It has been observed that an electromagnetic waves shows exponential

damping with distance due to various dissipative effect in the medium. In

conductors the rate of attenuation (loss of amplitude with distance) is very

high and the electromagnetic waves get almost attenuated after traversing

a quite distance.

Skin depth : Describe the conducting behavior in electromagnetic field and in radio

communication. it is defined as the depth for which the strength of

electric field associated with the electromagnetic waves reduces to

times to the initial value.


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e

1

Continued………………..

In terms of attenuation constant


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0.368

1.0

E

Distance travelled X

The amplitude of electric field

of an electromagnetic waves

is decreased by a factor

therefore according to

definition of skin depth we

should have

eax

Continued………………


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frequency. oft independen isdepth skin the, dielectricfor Thus,

medium. theofy permitivit theis , where

2

asgiven bemay

depthskin theinsulatorsor dielectric goodor conductorspoor For

frequencyin increase with decreasesdepth skin thus f

1

f 2

2

2

1

isdepth skin theTherefore,

ty.conductivi theis and medium theofindex

refractive is frequency,angular is where2

have weconductors good a

)1.(....................constant n Attenuatio

1 depth skin .

1 x or 1 x

e

1 e

1-

For

ei

ex