4 slides

Dr. Rakhesh Singh Kshetrimayum
4. Electrodynamic fields
Dr. Rakhesh Singh Kshetrimayum
2/16/20131 Electromagnetic FieldTheory by R. S. Kshetrimayum

4.1 Introduction
Electrodynamics
Faraday’s law
Maxwell’s
equations
Wave
equations
2/16/2013Electromagnetic FieldTheory by R. S. Kshetrimayum2
Lenz’s law Integral
form
Boundary
conditions
Fig. 4.1 Electrodynamics: Faraday’s law and Maxwell’s equations
Differential
form
Phasor
form

4.1 Introduction
We will study Faraday’s law of electromagnetic induction
(the 4th Maxwell’s equations)
Faraday showed that a changing magnetic field creates an
electric field
He was the first person who connected these two forcesHe was the first person who connected these two forces
(magnetic and electric forces)
Maxwell, in 19th century, combined the works of his
predecessors (the four Maxwell’s equations)

4.1 Introduction
Since everything in nature is symmetric, he postulated that a
changing electric field should also produce magnetic field
Whenever the dynamic fields interact with a media interface,
their behavior is governed by electromagnetic boundary
conditionsconditions
We will discuss how to obtain Helmholtz wave equations
from the two Maxwell’s curl equations and
generate electromagnetic (EM) waves

4.2 Faraday’s law of electromagnetic4.2 Faraday’s law of electromagnetic4.2 Faraday’s law of electromagnetic4.2 Faraday’s law of electromagnetic
inductioninductioninductioninduction
2
v
I I
dl
v
r
tB
r
tB
r
sB
r
Fig. 4.2 (a) Case I (b) Case II and (c) Case III
1
(a) (b) (c)

A major advance in electromagnetic theory was made by
Michael Faraday in 1831
He discovered experimentally that a current was induced in a
conducting loop when the magnetic flux linking that loop
changes
changes
The possible three cases of changing magnetic flux linkage is
depicted in Fig. 4.2

Case I: A stationary circuit in a time varying magnetic
field
It has been observed experimentally that when both the loop
and magnet is at rest
But the magnetic field is time varying (denoted by Bt in Fig.
But the magnetic field is time varying (denoted by Bt in Fig.
4.2(a)), a current flows in the loop
Mathematically:
sd
t
B
sdEsd
t
B
dt
d
ldE
r
r
rrr
r
rr
•
∂
∂
−=•×∇⇒•
∂
∂
−=−=•= ∫∫∫∫
φ
ξ

It is for any arbitrary surface
Hence
The emf induced in a stationary closed circuit is equal to the
t
B
E
∂
∂
−=×∇⇒
The emf induced in a stationary closed circuit is equal to the
negative rate of increase of magnetic flux linking the circuit
The negative sign is due to Lenz’s law

Case II: A moving conductor in a static magnetic field
When a conductor moves with a velocity in a static magnetic
field (denoted by Bs in Fig. 4.2(b)), a force
BvqF m ×=
will cause the freely movable electron in the conductor to
drift toward one end of the conductor and
leave the other end positively charged
BvqF m ×=

The magnetic force per unit charge
can be interpreted as an induced electric field acting along
the conductor and
mF
v B
q
= ×
ur
r ur
the conductor and
producing a voltageV21
2 2
21
1 1
( )V E dl v B dl= • = × •∫ ∫
r rr rr

Case III: A moving circuit in a time varying magnetic
field
When a charge moves with a velocity in a region where
both an electric field and a magnetic field exists
The electromagnetic force on q is given by Lorentz’s force
The electromagnetic force on q is given by Lorentz’s force
field equation,
)( BvEqF ×+=

Hence, when a conducting circuit
with contour C and
surface S
moves with a velocity in presence of magnetic and electric
field, we obtain,
field, we obtain,
The line integral in the LHS is the emf induced in the moving
frame of reference
dlBvds
t
B
dlE
CSC
•×+•
∂
∂
−=• ∫∫∫ )(

The first term in the right side represents transformer emf
due to the variation of magnetic field
The second term represents the motional emf
due the motion of the circuit in magnetic field
Faraday’s law of electromagnetic induction shows that
Faraday’s law of electromagnetic induction shows that
there is a close connection between electric and magnetic fields
In other words, a time changing magnetic field produces an
electric field

Lenz’s law
Fig. 4.3 Lenz’s law

The induced current creates a magnetic flux
which prevents the variation of the magnetic flux generating the
induced emf
For example,
the loop L is drawn to the coil c,
the loop L is drawn to the coil c,
the magnetic flux through the loop increases
The current induced in the loop in this case is in the
counterclockwise direction

4.3 Maxwell’s Equations
The theory of EM field was mathematically completed by
Maxwell
It was James Clerk Maxwell who has combined the previous
works of
Carl Federick Gauss,
Carl Federick Gauss,
Andre MarieAmpere and
Michael Faraday
into four laws

which completely explains the entire electromagnetic
phenomenon in nature
except quantum mechanics
4.3.1 Electrodynamics before Maxwell
Table 4.1 Electrodynamics before Maxwell (next page)

Equation
No.
Equations Laws
1. Gauss law for electric field
0
v
E
ρ
ε
∇• =
ur
2. Gauss law for magnetic field
3. Ampere’s law
4. Faraday’s law
0B∇• =
r
0B Jµ∇× =
ur r
B
E
t
∂
∇× = −
∂
ur
ur

There is fatal inconsistency in one of these formulae
It has to do with the rule that the divergence of curl is always
zero
Apply this rule to 4, we get,
This is consistent
LHS is zero
since it is the divergence of curl of electric field vector
( ) 0)( =
∂
•∇∂
−=×∇•∇
t
B
E

RHS is also zero from Gauss’s law for magnetic field which
states that divergence of a magnetic flux density vector is
zero
Let us apply this rule to 3, we get,
( ) ( )JJB
rr
•∇=•∇=×∇•∇ 00)( µµ

The RHS is zero for
steady state currents only
whereas the LHS is zero for all cases
since it is the divergence of curl of magnetic flux density vector
One fundamental question is that
One fundamental question is that
what is happening in between the parallel plates of the capacitor
while
charging or discharging

From our circuit analysis,
no current flows between the two plates since they are
disconnected
In accordance with Gauss theorem,
for time varying electric flux density and
for time varying electric flux density and
fixed or static closed surface
( )∫∫ ∂
∂
=•
∂
∂
⇒=•
t
q
sdD
t
qsdD
rrrr

So the term on the LHS of the above equation is
rate of change of electric flux or
it can be also termed as displacement current
we can add this extra term in theAmpere’s law and we get,
D∂rr
The 2nd term on the RHS can also be termed as
displacement current density
Both terms on the RHS together is known as the total
current.
t
D
JH
∂
∂
+=×∇
rr

4.3.2 Maxwell’s equations in integral form
Maxwell equations are the elegant mathematical expression
of decades of experimental observations
of the electric and
magnetic effects
of charges and
currents
by many scientists viz.,
Gauss,
Ampere and
Faraday

Maxwell’s own contribution indeed is
the last term of the third equation (refer toTable 4.2)
But this term had profound impact on the electromagnetic
theory
It made evident for the first time that
It made evident for the first time that
varying electric and magnetic fields could produce or generate
each other
and these fields could propagate indefinitely through free
space,
far from the varying charges and currents
where they are originated

Previously the fields had been envisioned
bound to the charges and currents giving rise to them
Maxwell’s new term displacement current freed them
to move through space in a self-sustaining fashion, and
even predicted their velocity of motion was
even predicted their velocity of motion was
the speed of light
Electrodynamics after Maxwell can be represented by the
four Maxwell’s equations in integral form (listed inTable 4.2)
Table 4.2 Electrodynamics after Maxwell (next page)

Equation numbers Equations Laws
1.
Gauss law for electric
field
2.
Gauss law for magnetic
field
dvsdD v∫ ∫=• ρ
rr
0∫ =• sdB
rr
2. field
3.
Ampere’s law
4.
Faraday’s law
0∫ =• sdB
sd
t
D
JldH
r
r
rrr
•







∂
∂
+=• ∫∫
sd
t
B
ldE
r
r
rr
•







∂
∂
−=• ∫∫

The relations for linear, isotropic and non-dispersive
materials can be written as:
where ε is the permittivity and
,D E B Hε µ= =
r r r r
where ε is the permittivity and
µ is the permeability of the material
Suppose, we restrict ourselves to time-independent situations
That means nothing is varying with time and
all fields are stationary

If the fields are stationary (electric and magnetic field are
constant with time),
Maxwell’s equations reduces to four groups of independent
equations
∫ =• qsdD
rr
.1 0.2 ∫ =• sdB
rr
IldH∫ =•
rr
.3 0.4 ∫ =• ldE
rr

In this case, the electric and magnetic fields are independent
of one another
These are laws for electrostatics and magnetostatics
4.3.3 Differential form of Maxwell’s equations
We can obtain the 4 laws of Maxwell’s equations in
We can obtain the 4 laws of Maxwell’s equations in
differential form listed below from the integral form of
Maxwell’s equations by applying
Divergence and
Stokes theorems

2. 0B∇• =
r
D∂
r
r r
ε
ρv
E =•∇
r
.1
3.
D
H J
t
∂
∇× = +
∂
r r
4.
B
E
t
∂
∇× = −
∂
ur
ur

Equation 1 means that static or dynamic charges in a given
volume are responsible for a diverging electric field
Equation 2 means that there is no physical medium which
makes a magnetic field diverge
Equation 3 means either a current flow through a medium or
Equation 3 means either a current flow through a medium or
a time-varying electric field produces a spatially curling
magnetic field
Equations 4 means that a spatially varying (curling) electric
field will cause or produce a time-varying magnetic field

Maxwell equation include the continuity equation
4.3.4Time harmonic fields and Maxwell’s equations in phasor
form
0=•







∂
∂
+∫ sd
t
D
J
r
r
r
form
For time harmonic fields
where ω is the angular frequency of the time varying field
( , , , ) ( , , ) j t
F x y z t F x y z e ω
=
r r

where c1 is a constant of integration
Since we are interested only in time varying quantities, we
1
( , , , )
( , , , ) ( , , , ); ( , , , )
F x y z t
F x y z t j F x y z t F x y z t dt c
t j
ω
ω
∂
= = +
∂ ∫
r
r r r
Since we are interested only in time varying quantities, we
can take, c1=0
2
2
2
; ωωωω −=×=
∂
∂
≡
∂
∂
jj
t
j
t

Maxwell’s equations in phasor form
If we take the time variation explicitly out, then we can write
1. E j Bω∇× = −% %
JDjH
~~~
.2 +=×∇ ω
JDjH
~~~
.2 +=×∇ ω
υρ=•∇ D
~
.3
0
~
.4 =•∇ B

Some points to be noted on perfect conductors and magnetic
fields:
A perfect conductor or metal can’t have time-varying
magnetic fields inside it
At the surface of a perfect metal there can be no component
At the surface of a perfect metal there can be no component
of a time-varying magnetic field that is normal to the surface
Time-varying currents can only flow at the surface of a
perfect metal but not inside it
4.3.5 Electromagnetic boundary conditions

Table 4.3 Electromagnetic boundary conditions for time varying fields
Sl. No. Scalar form Vector form
1. Et1 = Et2
2. Ht1-Ht2=Js
3. Bn1=Bn2
0)
~~
(ˆ 21 =−× EEn
sJHHn
r
=−× )
~~
(ˆ 21
0)
~~
(ˆ 21 =−• BBn
3. Bn1=Bn2
4.
5. Jn1=Jn2
6.
0)(ˆ 21 =−• BBn
snn DD ρ=− 21
sDDn ρ=−• )
~~
(ˆ 21
0)
~~
(ˆ 21 =−• JJn
1 1
2 2
t
t
J
J
σ
σ
= 1 1
2 2
ˆ
J
n
J
σ
σ
 
× = 
 
%
%

The tangential component of the electric field is continuous
across the boundary between two dielectrics
The tangential component of the magnetic field is
discontinuous across the boundary between two magnetic
materials by the surface current density flowing along the
materials by the surface current density flowing along the
boundary
The normal component of the magnetic flux density is
continuous across the boundary between two magnetic
materials

The normal component of the electric flux density is
discontinuous across the boundary between two dielectrics
by the surface charge density at the boundary
It states that the normal component of electric current
density is continuous across the boundary
density is continuous across the boundary
The ratio of the tangential components of the current
densities at the interface is equal to the ratio of the
conductivities

4.4Wave equations from Maxwell’s equations
4.4.1 Helmholtz wave equations
In linear isotropic medium, the two Maxwell curl equations
in phasor form are
HjE
rr
ωµ−=×∇
( )H j E J j E E j Eωε ωε σ ωε σ∇× = + = + = +
r r r r r r

To solve for electric and magnetic fields,
we can take curl on the first equation and
eliminate the curl of magnetic field vector in the RHS of the
first equation by using the second equation given above
2
( )E E E E∇×∇× = ∇×∇× = ∇ ∇• −∇
r r r r

For a charge free region,
Hence,
0E∇• =
r
( )2 2
( )E E j H j j E Eωµ ωµ ωε σ γ∴∇×∇× = −∇ = − ∇× = − + = −
r r r r r
Hence,
Similarly,
2 2
0E Eγ∇ − =
r r
2 2
0H Hγ∇ − =
r r

These two equations are known as Helmholtz equations or
wave equations
γ=α+jβ is a complex number and it is known as propagation
constant
The real part α is attenuation constant, it will show how fast
The real part α is attenuation constant, it will show how fast
the wave will attenuate
whereas the imaginary part β is the phase constant
4.4.2 Propagation in rectangular coordinates

The Helmholtz vector wave equation can be solved in any
orthogonal coordinate system by substituting the appropriate
Laplacian operator
Let us first solve it in Cartesian coordinate systems
Let us assume that the electric field is polarized along the x-
Let us assume that the electric field is polarized along the x-
axis and it is propagating along the z-direction
( ) ( ), , , ,xE x y z t E z t x=
r )

Therefore, the wave equation now reduces to
Hence, the solution of the wave equation is of the form
2
2 2 2
2
0 0
E
E E E
z
γ γ
∂
∇ − = ⇒ − =
∂
r
r r r
where superscript + and – for E0 the arbitrary constants
denote for wave propagating along + and – z-axis
respectively
( ) ( ) ( )0 0
, , z z
xE x y z E z x E e E e xγ γ+ − − +
= = +
r ) )

Assuming that the electric field propagates only in the
positive z-direction and of finite value at infinity
then the arbitrary constant for wave traveling along –z axis
must be equal to zero, which means
( ) ( ), , z
E x y z E z x E e xγ+ −
= =
r ) )
Dropping the superscript + in the constant E0 of the above
expression, we can write
( ) ( ) 0
, , z
xE x y z E z x E e xγ+ −
= =
r ) )
( ) 0
, , z
E x y z E e xγ−
=
r )

Let us put the time dependence now by multiplying the
above expression by ejωt, which gives
The real part of this electric field becomes
( ) ( ) ( ) ( )0 0
, , , , z j t z j z j t
xE x y z t E z t x E e e x E e e e xγ ω α β ω− − −
= = =
r ) ) )
The real part of this electric field becomes
( ){ } ( ){ } ( )0
Re , , , Re , cosz
xE x y z t E z t x E e t z xα
ω β−
= = −
r ) )

Some points to be noted:
First point is that the solution of the wave equation is a
vector
Secondly, Fig. 4.5 shows a typical plot of the normalized
propagating electric field at a fixed point in time
propagating electric field at a fixed point in time
Thirdly, its phase φ=ωt-βz is a function of both time,
frequency and propagation distance

So, if we fix the phase and let the wave travel a distance of ∆z
over a period of time ∆t
Mathematically, this can be expressed as
∆ φ = ω ∆ t-β ∆ z = 0 ,
therefore,
therefore,
vp = ∆ z/ ∆ t = ω / β,
which is the phase velocity of the wave
Hence,
vp = ∆ z/ ∆ t = ω / {ω √ (µε) } = 1/√ (µε)

For free space this turns out to be vp = 1/√ (µ0ε0) ≈ 3 × 108
m/s
Note that the phase velocity can be greater than the speed of
light
For instance, hollow metallic pipe in the form of rectangular
For instance, hollow metallic pipe in the form of rectangular
waveguide has phase velocity
2
1
p
c
c
v
f
f
ω
β
= =
 
− 
 

For guided-wave propagation inside rectangular waveguide
fc< f and
correspondingly such guided waves propagate with phase
velocity greater than the speed of light
this doesn’t contradict the Eiensten’s theory of relativity
this doesn’t contradict the Eiensten’s theory of relativity
because
there is no energy or information transfer associated with this
velocity

Note that beside phase velocity, we have another term called
group velocity which is associated with the energy or
information transfer
Inside rectangular waveguide, in fact, the group velocity is
much lower than the speed of light
much lower than the speed of light
This can be seen from the another relation between the phase
and group velocity inside rectangular waveguide vpvg=c2

phase velocity vp = ω / β
group velocity vg = dω / dβ
Fourthly, phase constant is also generally known as wave
number
β = ω / v = 2 f / v = 2 / λ
β = ω / vp = 2 f / vp = 2 / λ
Fifth point is that ratio of the amplitude of the electric field
to the magnetic field is known as intrinsic or wave impedance
of the medium

η = |E| / |H| = ωµ / β = ωµ / {ω √ (µε) = µ / {√ (µε)
= √ (µ/ε)
For free space η = √ (µ0/ε0) ≈ 377
4.4.3 Propagation in spherical coordinates
We can also calculate the propagation of electric fields from
We can also calculate the propagation of electric fields from
an isotropic point source in spherical coordinates
We will assume that the source is isotropic and the electric
field solution is independent of (θ,φ)
Assuming that the electric field is polarized along θ-direction
and it is only a function of r

we can write the vector wave equation as follows:
( ) ( ), , , ,E r t E r tθθ ϕ θ=
r )
2 2 2 2 2
0 0
d dE
E E r r Eγ γ
 
∇ − = ⇒ − = 
r
r r r
For finite fields, the solution is of the form
2 2 2 2 2
0 0
d dE
E E r r E
dr dr
γ γ
 
∇ − = ⇒ − = 
 
( ) ( ) 0
, , rE
E r E r e
r
γ
θθ ϕ θ −
= =
r )

Like before, we can add the time dependence to the phasor
and get the real part of the electric field as follows
Note that there is an additional 1/r dependence term in the
( ){ } ( ){ } ( )0
Re , , , Re , cosrE
E r t E r t e t r
r
α
θθ ϕ θ ω β θ−
= = −
r ) )
Note that there is an additional 1/r dependence term in the
expression for electric field here
this factor is termed as spherical spreading for the isotropic
point source

4.5 Summary
Electrodynamics
Faraday’s law
Maxwell’s
equations
Boundary
conditions
Wave
equations
Differential Phasor
dlBvds
t
B
dlE
CSC
•×+•
∂
∂
−=• ∫∫∫ )(
2 2
0H Hγ∇ − =
r r2 2
0E Eγ∇ − =
r r
0)
~~
(ˆ 21 =−× EEn
form
Fig. 4.6 Electrodynamics in a nutshell
Differential
form
Phasor
form
dvsdD v∫ ∫=• ρ
rr
0∫ =• sdB
rr
sd
t
D
JldH
r
r
rrr
•







∂
∂
+=• ∫∫
sd
t
B
ldE
r
r
rr
•







∂
∂
−=• ∫∫
0=•∇ B
r
t
D
JH
∂
∂
+=×∇
r
rr
t
B
E
∂
∂
−=×∇
r
r
ε
ρv
E =•∇
r
ε
ρv
E =•∇
~
0
~
=•∇ B
DjJH
~~
ω+=×∇
r
BjE
~~
ω−=×∇
21
sJHHn
r
=−× )
~~
(ˆ 21
0)
~~
(ˆ 21 =−• BBn
snn DD ρ=− 21
0)
~~
(ˆ 21 =−• JJn
1 1
2 2
ˆ
J
n
J
σ
σ
 
× = 
 
%
%

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