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,
JIT Jahangirabad 2
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.
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 3
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.
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 4
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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JIT Jahangirabad 5
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.
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 6
(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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JIT Jahangirabad 7
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 8
....(1).............................. dt
dq - i
vs
dV q and sd . J i
Continued……………
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JIT Jahangirabad 9
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 10
Maxwells Electromagnetic
equations:(diffrantial form)
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 11
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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JIT Jahangirabad 12
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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JIT Jahangirabad 13
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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JIT Jahangirabad 14
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 15
.......(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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JIT Jahangirabad 16
)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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JIT Jahangirabad 17
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 18
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 19
(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………
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 20
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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JIT Jahangirabad 21
(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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JIT Jahangirabad 22
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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 23
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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JIT Jahangirabad 25
)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:
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 26
) 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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JIT Jahangirabad 27
.......(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
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 28
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,
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 29
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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JIT Jahangirabad 30
e
1
Continued………………..
In terms of attenuation constant
1/11/2017 Dr A K Mishra, Academic Coordinator,
JIT Jahangirabad 31
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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JIT Jahangirabad 32
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
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