ELECTRODYNAMIC FIELDS

KONGUNADU COLLEGE OF ENGINEERING AND TECHNOLOGY
DEPARTMENT OF ELECTRICALAND ELECTRONICS ENGINEERING
1Prepared by Mr.K.Karthik-AP/EEE

 Integral form  Differential form




V
ev
S
C
dvqsdD
ldE 0
evqD
E

 0
ED 

 Integral form  Differential form
0



S
SC
sdB
sdJldH
0

B
JH
HB 

 In the static case (no time variation), the
electric field (specified by E and D) and
the magnetic field (specified by B and H)
are described by separate and
independent sets of equations.
 In a conducting medium, both
electrostatic and magnetostatic fields
can exist, and are coupled through the
Ohm’s law (J = sE). Such a situation is
called electromagnetostatic.

 In an electromagnetostatic field, the
electric field is completely determined
by the stationary charges present in
the system, and the magnetic field is
completely determined by the current.
 The magnetic field does not enter into
the calculation of the electric field, nor
does the electric field enter into the
calculation of the magnetic field.

 Electric charges attract/repel each
other as described by Coulomb’s law.
 Current-carrying wires attract/repel
each other as described by Ampere’s law
of force.
 Magnetic fields that change with time
induce electromotive force as
described by Faraday’s law.

battery
switch
toroidal iron
core
compass
primary
coil
secondary
coil

 Upon closing the switch, current
begins to flow in the primary coil.
 A momentary deflection of the compass
needle indicates a brief surge of current
flowing in the secondary coil.
 The compass needle quickly settles back
to zero.
 Upon opening the switch, another
brief deflection of the compass needle is
observed.

 “The electromotive force induced
around a closed loop C is equal to the
time rate of decrease of the magnetic
flux linking the loop.”
C
S
dt
d
Vind



 
S
sdB
• S is any surface
bounded by C
 
C
ind ldEV
 
SC
sdB
dt
d
ldE
integral form
of Faraday’s
law








SS
SC
sd
t
B
sdB
dt
d
sdEldE
Stokes’s theorem
assuming a stationary surface S

 Since the above must hold for any S, we have
t
B
E



differential form
of Faraday’s law
(assuming a
stationary frame
of reference)

 Faraday’s law states that a changing magnetic
field induces an electric field.
 The induced electric field is non-conservative.

 “The sense of the emf induced by the
time-varying magnetic flux is such that
any current it produces tends to set up a
magnetic field that opposes the change
in the original magnetic field.”
 Lenz’s law is a consequence of
conservation of energy.
 Lenz’s law explains the minus sign in
Faraday’s law.

 “The electromotive force induced
around a closed loop C is equal to
the time rate of decrease of the
magnetic flux linking the loop.”
 For a coil of N tightly wound turns
dt
d
Vind


dt
d
NVind



 
S
sdB
• S is any surface
bounded by C
 
C
ind ldEV
C
S

 Faraday’s law applies to situations where
◦ (1) the B-field is a function of time
◦ (2) ds is a function of time
◦ (3) B and ds are functions of time

 The induced emf around a circuit can be
separated into two terms:
◦ (1) due to the time-rate of change of the B-field
(transformer emf)
◦ (2) due to the motion of the circuit (motional emf)

 






CS
S
ind
ldBvsd
t
B
sdB
dt
d
V
transformer emf
motional emf

 Consider a conducting bar moving
with velocity v in a magnetostatic field:
B
v
2
1
+
-
• The magnetic force on an
electron in the conducting
bar is given by
BveFm 

 Electrons are pulled
toward end 2. End 2
becomes negatively
charged and end 1
becomes + charged.
 An electrostatic
force of attraction is
established
between the two
ends of the bar.
B
v
2
1
+
-

 The electrostatic force on an electron due
to the induced electrostatic field is given
by
 The migration of electrons stops
(equilibrium is established) when
EeFe 
BvEFF me 

 A motional (or “flux cutting”) emf is produced
given by
  ldBvVind  
1
2

 Electrostatics:
 EE 0
scalar electric potential

 Electrodynamics:
 



















t
A
E
t
A
E
A
tt
B
E
0
AB 

 Electrodynamics:
t
A
E



scalar
electric
potential
vector
magnetic
potential
• both of these
potentials are now
functions of time.

 The differential form of Ampere’s law in the
static case is
 The continuity equation is
JH 
0



t
q
J ev

 In the time-varying case, Ampere’s law in the
above form is inconsistent with the continuity
equation
  0 HJ

 To resolve this inconsistency, Maxwell
modified Ampere’s law to read
t
D
JH c



conduction
current density
displacement
current density

 The new form of Ampere’s law is consistent
with the continuity equation as well as with
the differential form of Gauss’s law
    0


 HD
t
J c
qev

Derivation of Displacement Current
q EA I
dq
dt
d EA
dt
   0 0
( )
For a capacitor, and .
I
d
dt
E
 0
( )
Now, the electric flux is given by EA, so: ,
where this current , not being associated with charges, is
called the “Displacement current”, Id.
Hence:
and: B ds I I
B ds I d
dt
d
E


  
   

  
0
0 0 0
( )

I
d
dtd
E
  0 0


Derivation of Displacement Current
q EA I
dq
dt
d EA
dt
   0 0
( )
For a capacitor, and .
I
d
dt
E
 0
( )
Now, the electric flux is given by EA, so: ,
where this current, not being associated with charges, is
called the “Displacement Current”, Id.
Hence:
and: B dl I I
B dl I d
dt
d
E


  
   

  
0
0 0 0
( )

I
d
dtd
E
  0 0


 Ampere’s law can be written as
dc JJH 
where
)(A/mdensitycurrentntdisplaceme 2




t
D
J d

 Consider a parallel-plate capacitor with
plates of area A separated by a dielectric
of permittivity  and thickness d and
connected to an ac generator:
tVtv cos)( 0
+
-
z = 0
z = d

icA
id
z

 The displacement current is given by
c
d
S
dd
i
dt
dv
CtCV
tV
d
A
AJsdJi

 



sin
sin
0
0
conduction
current in
wire

 Consider a conducting medium
characterized by conductivity s and
permittivity .
 The conduction current density is
given by
 The displacement current density is
given by
EJ c s
t
E
J d


 

10
0
10
2
10
4
10
6
10
8
10
10
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Frequency (Hz)
Humid Soil (r
= 30, s = 10-2
S/m)

s
good
conductor
good insulator

 We have been examining a variety of
electrical and magnetic phenomena
 James Clerk Maxwell summarized all of
electricity and magnetism in just four
equations
 Remarkably, the equations predict the
existence of electromagnetic waves

As you can see we need some vector calculus to use these equations. That isn’t
going to happen in this class, but I wanted you to see the equations anyway.

 Start with Ampere’s Law

B||l  0I
Earlier, we just went on a closed path
enclosing surface 1. But according to
Ampere’s Law, we could have
considered surface 2. The current
enclosed is the same as for surface 1.
We can say that the current flowing
into any volume must equal that
coming out.

 Suppose we have a charged capacitor and
it begins to discharge
Surface 1 works but
surface 2 has no current
passing through the
surface yet there is a
magnetic field inside the
surface.

Same problem here. Surface
1 works, but no current
passes through surface two
which encloses a magnetic
field. What is happening???

Maxwell solved this problem
by realizing that....
B EInside the capacitor there must
be an induced magnetic field...
How?.

B E
x
x x x x
x x x x x
x x
A changing
electric field
induces a
magnetic field
Inside the capacitor there must
How?. Inside the capacitor there is a changing E 
E
B

B E
x
x x x x
x x x x x
x x
A changing
electric field
induces a
magnetic field
where Id is called the
displacement current
B dl
d
dt
IE
d     0 0 0

E
B

B E
B dl I d
dt
E
     0 0 0

x
x x x x
x x x x x
x x
A changing
electric field
induces a
magnetic field
where Id is called the
displacement current
Therefore, Maxwell’s revision
of Ampere’s Law becomes....
B dl
d
dt
IE
d     0 0 0

E
B

1
2
E dA
q
  
0
B dA   0
..monopole..
?
...there’s no
magnetic monopole....!!
Gauss’s Laws

4
E dl d
dt
B
    
B dl I   0
3
.. if you change a
magnetic field you
induce an electric
field.........
.......is the reverse
true..?
Faraday’s Law
Ampere’s Law

...lets take a look at charge flowing into a
capacitor...
E
...when we derived Ampere’s Law
we assumed constant current...
.. if the loop encloses one
plate of the capacitor..there is a
problem … I = 0
B
Side view:(Surface
is now like a bag:)
EB
B dl I   0

Consider these equations in a vacuum.....
......no mass, no charges. no currents.....
B dl
d
dt
E
    0 0

E dl
d
dt
B
   

E dA
q
  
0
B dA   0
B dl I d
dt
E
     0 0 0

E dA   0
E dl
d
dt
B
   

B dA   0

 So, a magnetic field will be produced in
space if there is a changing electric field
 But, this magnetic field is changing since
the electric field is changing
 A changing magnetic field produces an
electric field that is also changing
 We have a self-perpetuating system

Close switch and current flows briefly.
Sets up electric field. Current flow
sets up magnetic field as little circles
around the wires. Fields not
instantaneous, but form in time.
Energy is stored in fields and cannot
move infinitely fast.

Picture a shows first half cycle. When current reverses in picture b, the fields reverse.
See the first disturbance moving outward. These are the electromagnetic waves.

Notice that the electric and magnetic
fields are at right angles to one another!
They are also perpendicular to the
direction of motion of the wave.

 Now that we have shown how the waves
are formed from oscillating charges, we
need to see if we can predict how fast the
move
 We move far away from the source so that
the wave fronts are essentially flat
 Just like dropping a rock in a pond and
looking at the waves a few hundred feet
away from the impact point

This picture defines the coordinate system we will use in our discussion. Wave
propagates along the x-axis. The electric field varies in the y-direction and the
magnetic field in the z-direction.

We are going to apply Faraday’s Law to the imaginary moving rectangle abcd. Compute
the magnetic flux change
emf 
B
t

BA
t

By0vt
t
 By0v

 We can say the emf around the loop is the
sum of the individual emfs going along
each straight line segment in the loop
 We look at the work done in moving a test
charge around the loop
 emf = W/q = Fd/q = Ed
 emf = Ey0 = By0v
 E = Bv

Now we are going to look at the change in electric flux. Set a new imaginary rectangle
and play the same game as before.
B||l  00
E
t
  00
Ez0vt
t
 00Ez0v

Bz0  00Ez0v
B  00Ev
E  Bv
B  00(Bv)v
1 00v2
v 
1
00
v 
1
8.851012
 4 107
v  3108

The Electromagnetic Spectrum

...lets take a look at charge flowing into a
capacitor...
...when we derived Ampere’s Law
we assumed constant current...
EB
B dl I   0

B dl
d
dt
E
    0 0
 E dl
d
dt
B
   

Faraday’s law: dB/dt electric field
Maxwell’s modification of Ampere’s law
dE/dt magnetic field
These two equations can be solved simultaneously.
The result is:
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) ˆz
ˆj

B dl
d
dt
E
    0 0

E dl
d
dt
B
   

B
E
dE
dt

dB
dt


B dl
d
dt
E
    0 0

E dl
d
dt
B
   

B
E
Special case..PLANE WAVES...
satisfy the wave equation
   A tsin( )Maxwell’s Solution
v
 
 
 

2
2 2
2
2
1
x t

 
E E x t j B B x t ky z ( , ) ( , )
dE
dt

dB
dt


x
Ey
Bz
ˆj
c

Static wave
F(x) = FP sin (kx + )
k = 2  
k = wavenumber
 = wavelength
F(x)
x

Moving wave
F(x, t) = FP sin (kx - t )
 = 2  f
 = angular frequency
f = frequency
v =  / k
F(x)
x

v

x
v
Moving wave
F(x, t) = FP sin (kx - t )
What happens at x = 0 as a function of time?
F(0, t) = FP sin (-t)
F
For x = 0 and t = 0  F(0, 0) = FP sin (0)
For x = 0 and t = t  F (0, t) = FP sin (0 – t) = FP sin (– t)
This is equivalent to: kx = - t  x = - (/k) t
F(x=0) at time t is the same as F[x=-(/k)t] at time 0
The wave moves to the right with speed /k

x
Ey
Bz
Notes: Waves are in Phase,
but fields oriented at 900.
k=2.
Speed of wave is c=/k (= f
At all times E=cB.
c m s  1 3 100 0
8/ / 
ˆj
c

ELECTRODYNAMIC FIELDS

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