EEE 498598 Overview of Electrical Engineering

1
EEE 498/598Overview of Electrical Engineering

• Lecture 4
• Electrostatics Electrostatic Shielding
  Poissons and Laplaces Equations Capacitance
  Dielectric Materials and Permittivity

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Lecture 4 Objectives

• To continue our study of electrostatics with
  electrostatic shielding Poissons and Laplaces
  equations capacitance and dielectric materials
  and permittivity.

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Ungrounded Spherical Metallic Shell

• Consider a point charge at the center of a
  spherical metallic shell

Electrically neutral
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Ungrounded Spherical Metallic Shell (Contd)

• The applied electric field is given by

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Ungrounded Spherical Metallic Shell (Contd)

• The total electric field can be obtained using
  Gausss law together with our knowledge of how
  fields behave in a conductor.

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Ungrounded Spherical Metallic Shell (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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Ungrounded Spherical Metallic Shell (Contd)

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

1)
2)
3)
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Ungrounded Spherical Metallic Shell (Contd)
(3) Evaluate the total charge within the volume
enclosed by each Gaussian surface
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Ungrounded Spherical Metallic Shell (Contd)
Gaussian surfaces for which
Gaussian surfaces for which
Gaussian surfaces for which
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Ungrounded Spherical Metallic Shell (Contd)

• For
• For

Shell is electrically neutral The net charge
carried by shell is zero.
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Ungrounded Spherical Metallic Shell (Contd)

• For
• since the electric field is zero inside
  conductor.
• A surface charge must exist on the inner surface
  and be given by

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Ungrounded Spherical Metallic Shell (Contd)

• Since the conducting shell is initially neutral,
  a surface charge must also exist on the outer
  surface and be given by

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Ungrounded Spherical Metallic Shell (Contd)

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

surface area of Gaussian surface.
magnitude of D on Gaussian surface.
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Ungrounded Spherical Metallic Shell (Contd)

• (5) Solve for D on each Gaussian surface

(6) Evaluate E as
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Ungrounded Spherical Metallic Shell (Contd)
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Ungrounded Spherical Metallic Shell (Contd)

• The induced field is given by

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Ungrounded Spherical Metallic Shell (Contd)
E
total electric field
Eapp
r
a
b
Eind
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Ungrounded Spherical Metallic Shell (Contd)

• The electrostatic potential is obtained by taking
  the line integral of E. To do this correctly, we
  must start at infinity (the reference point or
  ground) and move in back toward the point
  charge.
• For r gt b

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Ungrounded Spherical Metallic Shell (Contd)

• Since the conductor is an equipotential body (and
  potential is a continuous function), we have for

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Ungrounded Spherical Metallic Shell (Contd)

• For

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Ungrounded Spherical Metallic Shell (Contd)
V
No metallic shell
r
a
b
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Grounded Spherical Metallic Shell

• When the conducting sphere is grounded, we can
  consider it and ground to be one huge conducting
  body at ground (zero) potential.
• Electrons migrate from the ground, so that the
  conducting sphere now has an excess charge
  exactly equal to -Q. This charge appears in the
  form of a surface charge density on the inner
  surface of the sphere.

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Grounded Spherical Metallic Shell

• There is no longer a surface charge on the outer
  surface of the sphere.
• The total field outside the sphere is zero.
• The electrostatic potential of the sphere is zero.

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Grounded Spherical Metallic Shell (Contd)
E
total electric field
Eapp
R
a
b
Eind
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Grounded Spherical Metallic Shell (Contd)
V
Grounded metallic shell acts as a shield.
R
a
b
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The Need for Poissons and Laplaces Equations

• So far, we have studied two approaches for
  finding the electric field and electrostatic
  potential due to a given charge distribution.

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The Need for Poissons and Laplaces Equations
(Contd)

• Method 1 given the position of all the charges,
  find the electric field and electrostatic
  potential using
• (A)

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The Need for Poissons and Laplaces Equations
(Contd)

• (B)

? Method 1 is valid only for charges in free
space.
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The Need for Poissons and Laplaces Equations
(Contd)

• Method 2 Find the electric field and
  electrostatic potential using

Gausss Law
? Method 2 works only for symmetric charge
distributions, but we can have materials other
than free space present.
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The Need for Poissons and Laplaces Equations
(Contd)

• Consider the following problem

? What are E and V in the region?
Conducting bodies
Neither Method 1 nor Method 2 can be used!
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The Need for Poissons and Laplaces Equations
(Contd)

• Poissons equation is a differential equation for
  the electrostatic potential V. Poissons
  equation and the boundary conditions applicable
  to the particular geometry form a boundary-value
  problem that can be solved either analytically
  for some geometries or numerically for any
  geometry.
• After the electrostatic potential is evaluated,
  the electric field is obtained using

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Derivation of Poissons Equation

• For now, we shall assume the only materials
  present are free space and conductors on which
  the electrostatic potential is specified.
  However, Poissons equation can be generalized
  for other materials (dielectric and magnetic as
  well).

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Derivation of Poissons Equation (Contd)
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Derivation of Poissons Equation (Contd)
Poissons equation
? ?2 is the Laplacian operator. The Laplacian of
a scalar function is a scalar function equal to
the divergence of the gradient of the original
scalar function.
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Laplacian Operator in Cartesian, Cylindrical, and
Spherical Coordinates
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Laplaces Equation

• Laplaces equation is the homogeneous form of
  Poissons equation.
• We use Laplaces equation to solve problems where
  potentials are specified on conducting bodies,
  but no charge exists in the free space region.

Laplaces equation
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Uniqueness Theorem

• A solution to Poissons or Laplaces equation
  that satisfies the given boundary conditions is
  the unique (i.e., the one and only correct)
  solution to the problem.

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Potential Between Coaxial Cylinders Using
Laplaces Equation

• Two conducting coaxial cylinders exist such that

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Potential Between Coaxial Cylinders Using
Laplaces Equation (Contd)

• Assume from symmetry that

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Potential Between Coaxial Cylinders Using
Laplaces Equation (Contd)

• Two successive integrations yield
• The two constants are obtained from the two BCs

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Potential Between Coaxial Cylinders Using
Laplaces Equation (Contd)

• Solving for C1 and C2, we obtain
• The potential is

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Potential Between Coaxial Cylinders Using
Laplaces Equation (Contd)

• The electric field between the plates is given
  by
• The surface charge densities on the inner and
  outer conductors are given by

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Capacitance of a Two Conductor System

• The capacitance of a two conductor system is the
  ratio of the total charge on one of the
  conductors to the potential difference between
  that conductor and the other conductor.

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Capacitance of a Two Conductor System

• Capacitance is a positive quantity measured in
  units of Farads.
• Capacitance is a measure of the ability of a
  conductor configuration to store charge.

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Capacitance of a Two Conductor System

• The capacitance of an isolated conductor can be
  considered to be equal to the capacitance of a
  two conductor system where the second conductor
  is an infinite distance away from the first and
  at ground potential.

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Capacitors

• A capacitor is an electrical device consisting of
  two conductors separated by free space or another
  conducting medium.
• To evaluate the capacitance of a two conductor
  system, we must find either the charge on each
  conductor in terms of an assumed potential
  difference between the conductors, or the
  potential difference between the conductors for
  an assumed charge on the conductors.

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Capacitors (Contd)

• The former method is the more general but
  requires solution of Laplaces equation.
• The latter method is useful in cases where the
  symmetry of the problem allows us to use Gausss
  law to find the electric field from a given
  charge distribution.

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Parallel-Plate Capacitor

• Determine an approximate expression for the
  capacitance of a parallel-plate capacitor by
  neglecting fringing.

Conductor 2
d
A
Conductor 1
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Parallel-Plate Capacitor (Contd)

• Neglecting fringing means to assume that the
  field that exists in the real problem is the same
  as for the infinite problem.

z
V V12
z d
V 0
z 0
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Parallel-Plate Capacitor (Contd)

• Determine the potential between the plates by
  solving Laplaces equation.

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Parallel-Plate Capacitor (Contd)
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Parallel-Plate Capacitor (Contd)

• Evaluate the electric field between the plates

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Parallel-Plate Capacitor (Contd)

• Evaluate the surface charge on conductor 2

• Evaluate the total charge on conductor 2

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Parallel-Plate Capacitor (Contd)

• Evaluate the capacitance

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Dielectric Materials

• A dielectric (insulator) is a medium which
  possess no (or very few) free electrons to
  provide currents due to an impressed electric
  field.
• Although there is no macroscopic migration of
  charge when a dielectric is placed in an electric
  field, microscopic displacements (on the order of
  the size of atoms or molecules) of charge occur
  resulting in the appearance of induced electric
  dipoles.

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Dielectric Materials (Contd)

• A dielectric is said to be polarized when induced
  electric dipoles are present.
• Although all substances are polarizable to some
  extent, the effects of polarization become
  important only for insulating materials.
• The presence of induced electric dipoles within
  the dielectric causes the electric field both
  inside and outside the material to be modified.

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Polarizability

• Polarizability is a measure of the ability of a
  material to become polarized in the presence of
  an applied electric field.
• Polarization occurs in both polar and nonpolar
  materials.

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Electronic Polarizability
electron cloud

• In the absence of an applied electric field, the
  positively charged nucleus is surrounded by a
  spherical electron cloud with equal and opposite
  charge.
• Outside the atom, the electric field is zero.

nucleus
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Electronic Polarizability (Contd)
Eapp

• In the presence of an applied electric field, the
  electron cloud is distorted such that it is
  displaced in a direction (w.r.t. the nucleus)
  opposite to that of the applied electric field.

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Electronic Polarizability (Contd)

• The net effect is that each atom becomes a small
  charge dipole which affects the total electric
  field both inside and outside the material.

dipole moment (C-m)
polarizability (F-m2)
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Ionic Polarizability
negative ion
positive ion

• In the absence of an applied electric field, the
  ionic molecules are randomly oriented such that
  the net dipole moment within any small volume is
  zero.

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Ionic Polarizability (Contd)
Eapp

• In the presence of an applied electric field, the
  dipoles tend to align themselves with the applied
  electric field.

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Ionic Polarizability (Contd)

• The net effect is that each ionic molecule is a
  small charge dipole which aligns with the applied
  electric field and influences the total electric
  field both inside and outside the material.

dipole moment (C-m)
polarizability (F-m2)
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Orientational Polarizability

• In the absence of an applied electric field, the
  polar molecules are randomly oriented such that
  the net dipole moment within any small volume is
  zero.

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Orientational Polarizability (Contd)
Eapp

• In the presence of an applied electric field, the
  dipoles tend to align themselves with the applied
  electric field.

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Orientational Polarizability (Contd)

• The net effect is that each polar molecule is a
  small charge dipole which aligns with the applied
  electric field and influences the total electric
  field both inside and outside the material.

dipole moment (C-m)
polarizability (F-m2)
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Polarization Per Unit Volume

• The total polarization of a given material may
  arise from a combination of electronic, ionic,
  and orientational polarizability.
• The polarization per unit volume is given by

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Polarization Per Unit Volume (Contd)

• P is the polarization per unit volume. (C/m2)
• N is the number of dipoles per unit volume. (m-3)
• p is the average dipole moment of the dipoles in
  the medium. (C-m)
• aT is the average polarizability of the dipoles
  in the medium. (F-m2)

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Polarization Per Unit Volume (Contd)

• Eloc is the total electric field that actually
  exists at each dipole location.
• For gases Eloc E where E is the total
  macroscopic field.
• For solids

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Polarization Per Unit Volume (Contd)

• From the macroscopic point of view, it suffices
  to use

electron susceptibility (dimensionless)
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Dielectric Materials

• The effect of an applied electric field on a
  dielectric material is to create a net dipole
  moment per unit volume P.
• The dipole moment distribution sets up induced
  secondary fields

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Volume and Surface Bound Charge Densities

• A volume distribution of dipoles may be
  represented as an equivalent volume (qevb) and
  surface (qesb) distribution of bound charge.
• These charge distributions are related to the
  dipole moment distribution

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Gausss Law in Dielectrics

• Gausss law in differential form in free space
• Gausss law in differential form in dielectric

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Displacement Flux Density

• Hence, the displacement flux density vector is
  given by

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General Forms of Gausss Law

• Gausss law in differential form
• Gausss law in integral form

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Permittivity Concept

• Assuming that
• we have
• The parameter e is the electric permittivity or
  the dielectric constant of the material.

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Permittivity Concept (Contd)

• The concepts of polarizability and dipole moment
  distribution are introduced to relate microscopic
  phenomena to the macroscopic fields.
• The introduction of permittivity eliminates the
  need for us to explicitly consider microscopic
  effects.
• Knowing the permittivity of a dielectric tells us
  all we need to know from the point of view of
  macroscopic electromagnetics.

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Permittivity Concept (Contd)

• For the most part in macroscopic
  electromagnetics, we specify the permittivity of
  the material and if necessary calculate the
  dipole moment distribution within the medium by
  using

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Relative Permittivity

• The relative permittivity of a dielectric is the
  ratio of the permittivity of the dielectric to
  the permittivity of free space