Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments

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Fundamentals of ElectromagneticsA Two-Week,
8-Day, Intensive Course for Training Faculty in
Electrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments

• by
• Nannapaneni Narayana Rao
• Edward C. Jordan Professor Emeritus
• of Electrical and Computer Engineering
• University of Illinois at Urbana-Champaign, USA
• Distinguished Amrita Professor of Engineering
• Amrita Vishwa Vidyapeetham, India
• Amrita Viswa Vidya Peetham, Coimbatore
• August 11, 12, 13, 14, 18, 19, 20, and 21, 2008

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• Module 3
• Maxwells Equations
• In Differential Form
• Faradays law and Amperes Circuital Law
• Gauss Laws and the Continuity Equation
• Curl and Divergence

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Instructional Objectives

• Determine if a given time-varying
  electric/magnetic field satisfies Maxwells curl
  equations, and if so find the corresponding
  magnetic/electric field, and any required
  condition, if the field is incompletely specified
• Find the electric/magnetic field due to
  one-dimensional static charge/current
  distribution using Maxwells divergence/curl
  equation for the electric/magnetic field
• 10. Establish the physical realizability of a
  static electric field by using Maxwells curl
  equation for the static case, and of a magnetic
  field by using the Maxwells divergence equation
  for the magnetic field

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• Faradays Law and
• Ampères Circuital Law
• (FEME, Secs. 3.1, 3.2 EEE6E, Sec. 3.1)

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• Maxwells Equations in Differential Form
• Why differential form?
• Because for integral forms to be useful, an a
  priori knowledge of the behavior of the field to
  be computed is necessary.
• The problem is similar to the following
• There is no unique solution to this.

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• However, if, e.g., y(x) Cx, then we can find
  y(x), since then
• On the other hand, suppose we have the following
  problem
• Then y(x) 2x C.
• Thus the solution is unique to within a constant.

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• FARADAYS LAW
• First consider the special case
• and apply the integral form to the rectangular
  path shown, in the limit that the rectangle
  shrinks to a point.

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• General Case

Lateral space derivatives of the components of E
Time derivatives of the components of B
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• Combining into a single differential equation,

Differential form of Faradays Law
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• AMPÈRES CIRCUITAL LAW
• Consider the general case first. Then noting
  that
• we obtain from analogy,

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• Thus
• Special case

Differential form of Ampères circuital law
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Ex. For in free space find the value(s) of k
such that E satisfies both of Maxwells curl
equations. Noting that
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• Thus,

Then, noting that we
have from
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Comparing with the original given E, we have
Sinusoidal traveling waves in free space,
propagating in the z directions with
velocity,
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• Gauss Laws and
• the Continuity Equation
• (FEME, Secs. 3.4, 3.5, 3.6 EEE6E, Sec. 3.2)

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• GAUSS LAW FOR THE ELECTRIC FIELD

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Divergence of D r
Ex. Given that Find D everywhere.
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Noting that r r (x) and hence D D(x), we set
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• Thus,
• which also means that D has only an x-component.
  Proceeding further, we have
• where C is the constant of integration.
  Evaluating the integral graphically, we have the
  following

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From symmetry considerations, the fields on the
two sides of the charge distribution must be
equal in magnitude and opposite in direction.
Hence, C r0a
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• GAUSS LAW FOR THE MAGNETIC FIELD
• From analogy
• Solenoidal property of magnetic field lines.
  Provides test for physical realizability of a
  given vector field as a magnetic field.

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• LAW OF CONSERVATION OF CHARGE

Continuity Equation
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• SUMMARY
• (4) is, however, not independent of (1), and (3)
  can be derived from (2) with the aid of (5).

(1)
(2)
(3)
(4)
(5)
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Curl and Divergence (FEME, Secs. 3.3, 3.6
EEE6E, Sec. 3.3)
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Maxwells Equations in Differential Form
Curl
Divergence
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Basic definition of curl
is the maximum value of circulation of
A per unit area in the limit that the area
shrinks to the point.
Direction of is the direction of the
normal vector to the area in the limit that the
area shrinks to the point, and in the right-hand
sense.
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Curl Meter
is a device to probe the field for studying the
curl of the field. It responds to the
circulation of the field.
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3-33
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Basic definition of divergence
is the outward flux of A per unit volume in the
limit that the volume shrinks to the point.
Divergence meter
is a device to probe the field for studying the
divergence of the field. It responds to the
closed surface integral of the vector field.
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Example
At the point (1, 1, 0)
(a)
Divergence zero
(b)
Divergence positive
(c)
Divergence negative
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Two Useful Theorems
Stokes theorem
Divergence theorem
A useful identity
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The End