EEE 498/598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 3
• Electrostatics Electrostatic Potential Charge
  Dipole Visualization of Electric Fields
  Potentials Gausss Law and Applications
  Conductors and Conduction Current

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Lecture 3 Objectives

• To continue our study of electrostatics with
  electrostatic potential charge dipole
  visualization of electric fields and potentials
  Gausss law and applications conductors and
  conduction current.

3
Electrostatic Potential of a Point Charge at the
Origin
spherically symmetric
4
Electrostatic Potential Resulting from Multiple
Point Charges
Q2
P(R,q,f)
Q1
O
No longer spherically symmetric!
5
Electrostatic Potential Resulting from Continuous
Charge Distributions
? line charge
? surface charge
? volume charge
6
Charge Dipole

• An electric charge dipole consists of a pair of
  equal and opposite point charges separated by a
  small distance (i.e., much smaller than the
  distance at which we observe the resulting field).

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Dipole Moment

• Dipole moment p is a measure of the strength
• of the dipole and indicates its direction

p is in the direction from the negative point
charge to the positive point charge
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Electrostatic Potential Due to Charge Dipole
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Electrostatic Potential Due to Charge Dipole
(Contd)
cylindrical symmetry
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Electrostatic Potential Due to Charge Dipole
(Contd)
P
q
d/2
d/2
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Electrostatic Potential Due to Charge Dipole in
the Far-Field

• assume Rgtgtd

• zeroth order approximation

not good enough!
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Electrostatic Potential Due to Charge Dipole in
the Far-Field (Contd)

• first order approximation from geometry

q
d/2
d/2
lines approximately parallel
13
Electrostatic Potential Due to Charge Dipole in
the Far-Field (Contd)

• Taylor series approximation

14
Electrostatic Potential Due to Charge Dipole in
the Far-Field (Contd)
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Electrostatic Potential Due to Charge Dipole in
the Far-Field (Contd)

• In terms of the dipole moment

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Electric Field of Charge Dipole in the Far-Field
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Visualization of Electric Fields

• An electric field (like any vector field) can be
  visualized using flux lines (also called
  streamlines or lines of force).
• A flux line is drawn such that it is everywhere
  tangent to the electric field.
• A quiver plot is a plot of the field lines
  constructed by making a grid of points. An arrow
  whose tail is connected to the point indicates
  the direction and magnitude of the field at that
  point.

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Visualization of Electric Potentials

• The scalar electric potential can be visualized
  using equipotential surfaces.
• An equipotential surface is a surface over which
  V is a constant.
• Because the electric field is the negative of the
  gradient of the electric scalar potential, the
  electric field lines are everywhere normal to the
  equipotential surfaces and point in the direction
  of decreasing potential.

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Visualization of Electric Fields

• Flux lines are suggestive of the flow of some
  fluid emanating from positive charges (source)
  and terminating at negative charges (sink).
• Although electric field lines do NOT represent
  fluid flow, it is useful to think of them as
  describing the flux of something that, like fluid
  flow, is conserved.

20
Faradays Experiment
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Faradays Experiment (Contd)

• Two concentric conducting spheres are separated
  by an insulating material.
• The inner sphere is charged to Q. The outer
  sphere is initially uncharged.
• The outer sphere is grounded momentarily.
• The charge on the outer sphere is found to be -Q.

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Faradays Experiment (Contd)

• Faraday concluded there was a displacement from
  the charge on the inner sphere through the inner
  sphere through the insulator to the outer sphere.
• The electric displacement (or electric flux) is
  equal in magnitude to the charge that produces
  it, independent of the insulating material and
  the size of the spheres.

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Electric Displacement (Electric Flux)
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Electric (Displacement) Flux Density

• The density of electric displacement is the
  electric (displacement) flux density, D.
• In free space the relationship between flux
  density and electric field is

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Electric (Displacement) Flux Density (Contd)

• The electric (displacement) flux density for a
  point charge centered at the origin is

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Gausss Law

• Gausss law states that the net electric flux
  emanating from a close surface S is equal to the
  total charge contained within the volume V
  bounded by that surface.

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Gausss Law (Contd)
By convention, ds is taken to be outward from the
volume V.
Since volume charge density is the most general,
we can always write Qencl in this way.
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Applications of Gausss Law

• Gausss law is an integral equation for the
  unknown electric flux density resulting from a
  given charge distribution.

known
unknown
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Applications of Gausss Law (Contd)

• In general, solutions to integral equations must
  be obtained using numerical techniques.
• However, for certain symmetric charge
  distributions closed form solutions to Gausss
  law can be obtained.

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Applications of Gausss Law (Contd)

• Closed form solution to Gausss law relies on our
  ability to construct a suitable family of
  Gaussian surfaces.
• A Gaussian surface is a surface to which the
  electric flux density is normal and over which
  equal to a constant value.

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Electric Flux Density of a Point Charge Using
Gausss Law

• Consider a point charge at the origin

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Electric Flux Density of a Point Charge Using
Gausss Law (Contd)

• (1) Assume from symmetry the form of the field
• (2) Construct a family of Gaussian surfaces

spherical symmetry
spheres of radius r where
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Electric Flux Density of a Point Charge Using
Gausss Law (Contd)

• (3) Evaluate the total charge within the volume
  enclosed by each Gaussian surface

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Electric Flux Density of a Point Charge Using
Gausss Law (Contd)
Gaussian surface
R
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Electric Flux Density of a Point Charge Using
Gausss Law (Contd)

• (4) For each Gaussian surface, evaluate the
  integral

surface area of Gaussian surface.
magnitude of D on Gaussian surface.
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Electric Flux Density of a Point Charge Using
Gausss Law (Contd)

• (5) Solve for D on each Gaussian surface

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law

• Consider a spherical shell of uniform charge
  density

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• (1) Assume from symmetry the form of the field
• (2) Construct a family of Gaussian surfaces

spheres of radius r where
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• Here, we shall need to treat separately 3
  sub-families of Gaussian surfaces

1)
2)
3)
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)
Gaussian surfaces for which
Gaussian surfaces for which
Gaussian surfaces for which
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• (3) Evaluate the total charge within the volume
  enclosed by each Gaussian surface

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• For
• For

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• For

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• (4) For each Gaussian surface, evaluate the
  integral

surface area of Gaussian surface.
magnitude of D on Gaussian surface.
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• (5) Solve for D on each Gaussian surface

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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)

• Notice that for r gt b

Total charge contained in spherical shell
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Electric Flux Density of a Spherical Shell of
Charge Using Gausss Law (Contd)
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Electric Flux Density of an Infinite Line Charge
Using Gausss Law

• Consider a infinite line charge carrying charge
  per
• unit length of qel

z
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Electric Flux Density of an Infinite Line Charge
Using Gausss Law (Contd)

• (1) Assume from symmetry the form of the field
• (2) Construct a family of Gaussian surfaces

cylinders of radius r where
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Electric Flux Density of an Infinite Line Charge
Using Gausss Law (Contd)

• (3) Evaluate the total charge within the volume
  enclosed by each Gaussian surface

cylinder is infinitely long!
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Electric Flux Density of an Infinite Line Charge
Using Gausss Law (Contd)

• (4) For each Gaussian surface, evaluate the
  integral

surface area of Gaussian surface.
magnitude of D on Gaussian surface.
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Electric Flux Density of an Infinite Line Charge
Using Gausss Law (Contd)

• (5) Solve for D on each Gaussian surface

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Gausss Law in Integral Form
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Recall the Divergence Theorem

• Also called Gausss theorem or Greens theorem.
• Holds for any volume and corresponding closed
  surface.

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Applying Divergence Theorem to Gausss Law
? Because the above must hold for any volume
V, we must have
Differential form of Gausss Law
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Fields in Materials

• Materials contain charged particles that respond
  to applied electric and magnetic fields.
• Materials are classified according to the nature
  of their response to the applied fields.

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Classification of Materials

• Conductors
• Semiconductors
• Dielectrics
• Magnetic materials

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Conductors

• A conductor is a material in which electrons in
  the outermost shell of the electron migrate
  easily from atom to atom.
• Metallic materials are in general good conductors.

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Conduction Current

• In an otherwise empty universe, a constant
  electric field would cause an electron to move
  with constant acceleration.

e 1.602 ? 10-19 C
magnitude of electron charge
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Conduction Current (Contd)

• In a conductor, electrons are constantly
  colliding with each other and with the fixed
  nuclei, and losing momentum.
• The net macroscopic effect is that the electrons
  move with a (constant) drift velocity vd which is
  proportional to the electric field.

Electron mobility
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Conductor in an Electrostatic Field

• To have an electrostatic field, all charges must
  have reached their equilibrium positions (i.e.,
  they are stationary).
• Under such static conditions, there must be zero
  electric field within the conductor. (Otherwise
  charges would continue to flow.)

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Conductor in an Electrostatic Field (Contd)

• If the electric field in which the conductor is
  immersed suddenly changes, charge flows
  temporarily until equilibrium is once again
  reached with the electric field inside the
  conductor becoming zero.
• In a metallic conductor, the establishment of
  equilibrium takes place in about 10-19 s - an
  extraordinarily short amount of time indeed.

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Conductor in an Electrostatic Field (Contd)

• There are two important consequences to the fact
  that the electrostatic field inside a metallic
  conductor is zero
• The conductor is an equipotential body.
• The charge on a conductor must reside entirely on
  its surface.
• A corollary of the above is that the electric
  field just outside the conductor must be normal
  to its surface.

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Conductor in an Electrostatic Field (Contd)
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Macroscopic versus Microscopic Fields

• In our study of electromagnetics, we use
  Maxwells equations which are written in terms of
  macroscopic quantities.
• The lower limit of the classical domain is about
  10-8 m 100 angstroms. For smaller dimensions,
  quantum mechanics is needed.

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Boundary Conditions on the Electric Field at the
Surface of a Metallic Conductor
-
-
-
-
-
E 0





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Induced Charges on Conductors

• The BCs given above imply that if a conductor is
  placed in an externally applied electric field,
  then
• the field distribution is distorted so that the
  electric field lines are normal to the conductor
  surface
• a surface charge is induced on the conductor to
  support the electric field

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Applied and Induced Electric Fields

• The applied electric field (Eapp) is the field
  that exists in the absence of the metallic
  conductor (obstacle).
• The induced electric field (Eind) is the field
  that arises from the induced surface charges.
• The total field is the sum of the applied and
  induced electric fields.