Dynamic Electromagnetic Fields

1
CHAPTER 3
ELECTROMAGNETIC FIELDS THEORY

• Dynamic Electromagnetic Fields

2
Objectives

• Electromotive force based on faradays law
• Maxwells equation

3
Introduction

• Stationary charges ? electrostatic fields
• Steady current ? magnetostatic fields
• Time-varying currents ? electromagnetic fields
  (waves)

4
Introduction
http//micro.magnet.fsu.edu/primer/java/polarizedl
ight/emwave/index.html
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Faradays Law

• A time varying magnetic field produces an induced
  voltage called electromagnetic force or emf.
• Induced emf, Vemf in any closed circuit is equal
  to the time rate of change of the magnetic flux
  linkage by the circuit
• The negative sign shows that the induced voltage
  acts in such a way as to oppose the flux
  producing it ? Lentzs Law
• Emf sources such as electric generator and
  batteries can convert nonelectrical energy into
  electrical energy

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Faradays Law

• Electrochemical action of the battery results in
  an emf-produced field Ef.
• Due to the accumulation of charge at the battery
  terminals, an electrostatic field Ee also exists.
• Total electric field EEf Ee

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Faradays Law

• Ef is zero outside the battery, Ef and Ee have
  opposite directions in the battery, and the
  direction of Ee inside the battery is opposite to
  that outside it.
• Integrate over a closed loop, we get
• because Ee is conservative

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Faradays Law

• Emf of the battery is the line integral of the
  emf-produced field
• Can also be regarded as as the potential
  difference (Vp-VN) between the batterys
  open-circuit terminals

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Faradays Law

• It is important to note that
• An electrostatic field Ee cannot maintain a
  steady current in a closed circuit since
  IR
• An emf-produced field Ef is nonconservative.
• Except in electrostatics, voltage and potential
  difference are usually not equivalent.

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Connection Between Magnetic and Electric Field

• For a circuit with single turn, N 0,
• Thus Vemf can be written in terms of E and B
• where has been replaced by
  and S is the surface area of the circuit
  bounded by the closed path L
• dl and dS are both in accordance with the
  right-hand rule and Stokes theorem

11
Transformer And Motional Emfs

• Variation of flux with time may be caused
• By having a stationary loop in a time-varying B
  field
• By having a time-varying loop area in a static B
  field
• By having a time-varying loop area in a
  time-varying B field.

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Stationary Loop in Time-varying B Field
(Transformer emf)

• A stationary conducting loop is in a time-varying
  magnetic B field. Thus Vemf is

----------------------- (1)
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Stationary Loop in Time-varying B Field
(Transformer emf)

• emf induced is caused by the time-varying current
  (producing the time-varying B field) in the
  stationary loop.
• It is often referred to as transformer emf in
  power analysis since it is due to transformer
  action.
• By applying Stokes's theorem to the middle term,
  equation (1) becomes

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Stationary Loop in Time-varying B Field
(Transformer emf)

• Thus
• This is one of the Maxwell's equations for
  time-varying fields.
• It shows that the time varying E field is not
  conservative

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Moving Loop in Static B Field (Motional emf)

• When a conducting loop is moving in a static B
  field, an emf is induced in the loop.
• Force on a charge moving with uniform velocity u
  in a magnetic field B is given by

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Moving Loop in Static B Field (Motional emf)

• Motional electric field is defined as
• Consider a conducting loop, moving with uniform
  velocity u as consisting of a large number of
  free electrons, the emf induced in the loop is
• This type of emf is called motional emf or
  flux-cutting emf because it is due to motional
  action.

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Moving Loop in Static B Field (Motional emf)

• For a rod, moving between a pair of rails, B and
  u are perpendicular
• Thus or
• And

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Moving Loop in Static B Field (Motional emf)

• Applying Stokes theorem to previous Vemf In eq
  (4).

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Moving Loop in Static B Field (Motional emf)

• To apply eq (4), the following points should be
  noted
• The integral in eq. (4) is zero along the portion
  of the loop where u 0. Thus dl is taken along
  the portion of the loop that is cutting the field
  where u has nonzero value.
• The direction of the induced current is the same
  as that of Em or u x B - satisfy Lenz's law

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Moving Loop in Time-Varying Field

• A moving conducting loop is in a time-varying
  magnetic field.
• Both transformer emf and motional emf are
  present.
• Total emf is obtained by combining equations (1)
  and (4)
• Total emf can also be found using

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Example
The loop ABCD is inside a uniform magnetic field
B 50 ax mWb/m2. If side DC of the loop cuts the
lines at the frequency of 50 Hz and the loop lies
in the yz-plane at time t 0, find
z
B

• The induced emf at t 1 ms
• The induced current at t 3 ms

B
4cm
C
0.1?
3cm
?
A
y
D
F
x
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Solution

• The induced emf is motional because B is time
  invariant, and the loop is moving
• So make use of the formula in eq. 4
• in this case dl is only along the side DC

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Solution (cont)

• As u and dl are in cylindrical coordinate,
  transform B into cylindrical coordinate using the
  coordinate transformation equation in chapter 1
• In this case,
• where

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Solution (cont)
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Solution (cont)
If we integrate dF, we get
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Solution (cont)
If we integrate dF, we get
At t 0, F p/2, so
At t 1 ms,
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Solution (cont)

• (b) The current induced is
• at t 3 ms,

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Displacement Current

• For static EM field,
• Divergence of curl for any vector is zero
• But the continuity of current requires that
• Thus a new term is added so that

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Displacement Current

• is known as displacement current density
• To define we use the divergence of the curl
  again
• So therefore
• so therefore

Another Maxwells equation for time varying field
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Maxwells Equations In Final Forms

• This is also reffered to as Gausss law for
  magnetic fields
• All the equations agree with Lorentz force
  equation

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Maxwells Equations In Final Forms

• The equation of continuity is implicit in
  Maxwell's equations.
• The concepts of linearity, isotropy, and
  homogeneity of a material medium still apply for
  time-varying fields in a linear, homogeneous,
  and isotropic medium characterized by s, e , and
  µ

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Maxwells Equations In Final Forms

• The boundary conditions
• For a perfect conductor (s 8) in a time varying
  field,
• E 0 , H 0 , J 0 Hence
• For a perfect dielectric (s 0), the equation
  holds except that
• K 0.

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Electromagnetic system
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Maxwells Equations In Final Forms
Is the free magnetic density (similar to ),
which is zero
The principal relationships
(a) compatibility equations
and
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Maxwells Equations In Final Forms
(b) constitutive equations
and
(c) equilibrium equations
and
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Time-varying Potentials

• Recall back for static EM, the potentials are

the electric scalar potential
magnetic vector potential
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Time-varying Potentials

• For time-varying potentials
• Where means that time t in
  ?v(x,y,z,t)
• or J(x,y,z,t) is replace by the retarded time
  t
• Where Rr - r is the distance between the
  source point r and the observation point r.
• And is the velocity of wave propagation

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Time Harmonic Fields

• Time Harmonic Fields varies periodically or
  sinusoidally with time
• Sinusoids are easily expressed in phasors, which
  are more convenient to work with.
• A phasor z is a complex number that can be
  written as
• Or
• Where
• x is the real part of z,
• y is the imaginary part of z

Polar form
Rectangular form
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Time Harmonic Fields

• Magnitude of z, which is r is given by
• And the phase of z, which is F is given by

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Some basic properties of complex numbers
Consider three complex numbers
Addition
Subtraction
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Some basic properties of complex numbers
Multiplication
Division
Square root
Complex conjugate
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