EEE 498598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 5
• Electrostatics Dielectric Breakdown,
  Electrostatic Boundary Conditions, Electrostatic
  Potential Energy Conduction Current and Ohms Law

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Lecture 5 Objectives

• To continue our study of electrostatics with
  dielectric breakdown, electrostatic boundary
  conditions and electrostatic potential energy.
• To study steady conduction current and Ohms law.

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Dielectric Breakdown

• If a dielectric material is placed in a very
  strong electric field, electrons can be torn from
  their corresponding nuclei causing large currents
  to flow and damaging the material. This
  phenomenon is called dielectric breakdown.

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Dielectric Breakdown (Contd)

• The value of the electric field at which
  dielectric breakdown occurs is called the
  dielectric strength of the material.
• The dielectric strength of a material is denoted
  by the symbol EBR.

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Dielectric Breakdown (Contd)

• The dielectric strength of a material may vary by
  several orders of magnitude depending on various
  factors including the exact composition of the
  material.
• Usually dielectric breakdown does not permanently
  damage gaseous or liquid dielectrics, but does
  ruin solid dielectrics.

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Dielectric Breakdown (Contd)

• Capacitors typically carry a maximum voltage
  rating. Keeping the terminal voltage below this
  value insures that the field within the capacitor
  never exceeds EBR for the dielectric.
• Usually a safety factor of 10 or so is used in
  calculating the rating.

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Fundamental Laws of Electrostatics in Integral
Form
Conservative field
Gausss law
Constitutive relation
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Fundamental Laws of Electrostatics in
Differential Form
Conservative field
Gausss law
Constitutive relation
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Fundamental Laws of Electrostatics

• The integral forms of the fundamental laws are
  more general because they apply over regions of
  space. The differential forms are only valid at
  a point.
• From the integral forms of the fundamental laws
  both the differential equations governing the
  field within a medium and the boundary conditions
  at the interface between two media can be derived.

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Boundary Conditions

• Within a homogeneous medium, there are no abrupt
  changes in E or D. However, at the interface
  between two different media (having two different
  values of e), it is obvious that one or both of
  these must change abruptly.

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Boundary Conditions (Contd)

• To derive the boundary conditions on the normal
  and tangential field conditions, we shall apply
  the integral form of the two fundamental laws to
  an infinitesimally small region that lies
  partially in one medium and partially in the
  other.

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Boundary Conditions (Contd)

• Consider two semi-infinite media separated by a
  boundary. A surface charge may exist at the
  interface.

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Boundary Conditions (Contd)

• Locally, the boundary will look planar

rs
x x x x x x
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Boundary Condition on Normal Component of D

• Consider an infinitesimal cylinder (pillbox)
  with
• cross-sectional area Ds and height Dh lying
  half in
• medium 1 and half in medium 2

Ds
Dh/2
rs
x x x x x x
Dh/2
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Boundary Condition on Normal Component of D
(Contd)

• Applying Gausss law to the pillbox, we have

0
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Boundary Condition on Normal Component of D
(Contd)

• The boundary condition is
• If there is no surface charge

For non-conducting materials, rs 0 unless an
impressed source is present.
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Boundary Condition on Tangential Component of E

• Consider an infinitesimal path abcd with width
  Dw
• and height Dh lying half in medium 1 and half
  in
• medium 2

Dw
d
a
Dh/2
Dh/2
b
c
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Boundary Condition on Tangential Component of E
(Contd)
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Boundary Condition on Tangential Component of E
(Contd)

• Applying conservative law to the path, we have

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Boundary Condition on Tangential Component of E
(Contd)

• The boundary condition is

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Electrostatic Boundary Conditions - Summary

• At any point on the boundary,
• the components of E1 and E2 tangential to the
  boundary are equal
• the components of D1 and D2 normal to the
  boundary are discontinuous by an amount equal to
  any surface charge existing at that point

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Electrostatic Boundary Conditions - Special Cases

• Special Case 1 the interface between two perfect
  (non-conducting) dielectrics
• Physical principle there can be no free surface
  charge associated with the surface of a perfect
  dielectric.
• In practice unless an impressed surface charge
  is explicitly stated, assume it is zero.

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Electrostatic Boundary Conditions - Special Cases

• Special Case 2 the interface between a conductor
  and a perfect dielectric
• Physical principle there can be no
  electrostatic field inside of a conductor.
• In practice a surface charge always exists at
  the boundary.

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Potential Energy

• When one lifts a bowling ball and places it on a
  table, the work done is stored in the form of
  potential energy. Allowing the ball to drop back
  to the floor releases that energy.
• Bringing two charges together from infinite
  separation against their electrostatic repulsion
  also requires work. Electrostatic energy is
  stored in a configuration of charges, and it is
  released when the charges are allowed to recede
  away from each other.

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Electrostatic Energy in a Discrete Charge
Distribution

• Consider a point charge Q1 in an otherwise empty
  universe.
• The system stores no potential energy since no
  work has been done in creating it.

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• Now bring in from infinity another point charge
  Q2.
• The energy required to bring Q2 into the system is

• V12 is the electrostatic potential due to Q1
• at the location of Q2.

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• Now bring in from infinity another point charge
  Q3.
• The energy required to bring Q3 into the system is

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• The total energy required to assemble the system
  of three charges is

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• Now bring in from infinity a fourth point charge
  Q4.
• The energy required to bring Q4 into the system
  is
• The total energy required to assemble the system
  of four charges is

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• Bring in from infinity an ith point charge Qi
  into a system of i-1 point charges.
• The energy required to bring Qi into the system
  is
• The total energy required to assemble the system
  of N charges is

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Electrostatic Energy in a Discrete Charge
Distribution (Contd)

• Note that

? Physically, the above means that the partial
energy associated with two point charges is equal
no matter in what order the charges are assembled.
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Electrostatic Energy in a Discrete Charge
Distribution (Contd)
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Electrostatic Energy in a Discrete Charge
Distribution (Contd)
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Electrostatic Energy in a Discrete Charge
Distribution (Contd)
? Physically, Vi is the potential at the location
of the ith point charge due to the other
(N-1) charges.
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Electrostatic Energy in a Continuous Charge
Distribution
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Electrostatic Energy in a Continuous Charge
Distribution (Contd)
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Electrostatic Energy in a Continuous Charge
Distribution (Contd)
Divergence theorem and
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Electrostatic Energy in a Continuous Charge
Distribution (Contd)

• Let the volume V be all of space. Then the
  closed surface S is sphere of radius infinity.
  All sources of finite extent look like point
  charges. Hence,

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Electrostatic Energy in a Continuous Charge
Distribution (Contd)
Electrostatic energy density in J/m3.
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Electrostatic Energy in a Continuous Charge
Distribution (Contd)
energy required to set the field up in free space
energy required to polarize the dielectric
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Electrostatic Energy in a Capacitor
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Electrostatic Energy in a Capacitor

• Letting V V12 V2 V1

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Electrostatic Forces The Principle of Virtual
Work

• Electrostatic forces acting on bodies can be
  computed using the principle of virtual work.
• The force on any conductor or dielectric body
  within a system can be obtained by assuming a
  differential displacement of the body and
  computing the resulting change in the
  electrostatic energy of the system.

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Electrostatic Forces The Principle of Virtual
Work (Contd)

• The electrostatic force can be evaluated as the
  gradient of the electrostatic energy of the
  system, provided that the energy is expressed in
  terms of the coordinate location of the body
  being displaced.

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Electrostatic Forces The Principle of Virtual
Work (Contd)

• When using the principle of virtual work, we can
  assume either that the conductors maintain a
  constant charge or that they maintain a constant
  voltage (i.e, they are connected to a battery).

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Electrostatic Forces The Principle of Virtual
Work (Contd)

• For a system of bodies with fixed charges, the
  total electrostatic force acting on the body is
  given by

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Electrostatic Forces The Principle of Virtual
Work (Contd)

• For a system of bodies with fixed potentials, the
  total electrostatic force acting on the body is
  given by

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Force on a Capacitor Plate

• Compute the force on one plate of a charged
  parallel plate capacitor. Neglect fringing of
  the field.

• The force on the
• upper plate can be
• found assuming a
• system of fixed
• charge.

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Force on a Capacitor Plate (Contd)

• The capacitance can be written as a function of
  the location of the upper plate
• The electrostatic energy stored in the capacitor
  may be evaluated as a function of the charge on
  the upper plate and its location

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Force on a Capacitor Plate (Contd)

• The force on the upper plate is given by
• Using Q CV,

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Force on a Capacitor Plate (Contd)

• Compute the force on one plate of a charged
  parallel plate capacitor. Neglect fringing of
  the field.

• The force on the
• upper plate can be
• found assuming a
• system of fixed
• potential.

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Force on a Capacitor Plate (Contd)

• The capacitance can be written as a function of
  the location of the upper plate
• The electrostatic energy stored in the capacitor
  may be written as a function of the voltage
  across the plates and the location of the upper
  plate

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Force on a Capacitor Plate (Contd)

• The force on the upper plate is given by
• Manipulating, we obtain

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Steady Electric Current

• Electrostatics is the study of charges at rest.
• Now, we shall allow the charges to move, but with
  a constant velocity (no time variation).
• steady electric current direct current (DC)

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Conductors and Conductivity

• A conductor is a material in which electrons are
  free to migrate over macroscopic distances within
  the material.
• Metals are good conductors because they have many
  free electrons per unit volume.
• Other materials with a smaller number of free
  electrons per unit volume are also conductors.
• Conductivity is a measure of the ability of the
  material to conduct electricity.

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Semiconductor

• A semiconductor is a material in which electrons
  in the outermost shell are able to migrate over
  macroscopic distances when a modest energy
  barrier is overcome.
• Semiconductors support the flow of both negative
  charges (electrons) and positive charges (holes).

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Conduction Current

• When subjected to a field, an electron in a
  conductor migrates through the material
  constantly colliding with the lattice and losing
  momentum.
• The net effect is that the electron moves
  (drifts) with an average drift velocity that is
  proportional to the electric field.

electron mobility
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Conduction Current (Contd)

• Consider a conducting wire in which charges
  subject to an electric field are moving with
  drift velocity vd.

electron
E
vd
Ds
cross-section
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Conduction Current (Contd)

• If there are nc free electrons per cubic meter of
  material, then the charge density within the wire
  is
• Consider an infinitesimal volume associated with
  Ds

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Conduction Current (Contd)

• The total charge contained within Dv is
• This charge packet moves through the surface Ds
  with speed
• The amount of time it takes for the charge packet
  to move through Ds is

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Conduction Current (Contd)

• Current is the rate at which charges passes
  through a specified surface area (such as the
  cross-section of a wire).
• The incremental current through Ds is given by

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Current Density

• The component of the current density in the
  direction normal to Ds is
• In general, the current density is given by

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Current Density (Contd)

• The constant of proportionality between the
  electric field and the conduction current density
  is called the conductivity of the material
• Ohms law at a point

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Current Density (Contd)

• The conductivity of the medium is the macroscopic
  quantity which allows us to treat conduction
  current without worrying about the microscopic
  behavior of conductors.
• In semiconductors, we have both holes and
  electrons

hole mobility
hole density
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Current Density (Contd)

• The total current flowing through a
  cross-sectional area S may be found as
• If the current density is uniform throughout the
  cross-section, we have

cross-sectional area
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Current Flow

• Consider a wire of non-uniform cross-section

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Current Flow (Contd)

• To maintain a constant electric field and a
  steady current flow, both E and J must be
  parallel to the conductor boundaries.
• The total current passing through the
  cross-section A1 must be the same as through the
  cross-section A2. So the current density must be
  greater in A2.

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Ohms Law and Resistors

• Consider a conductor of uniform cross-section

• Let the wires and the two
• exposed faces of the
• resistor be perfect
• conductor.

• In a perfect conductor
• J is finite
• s is infinite
• E must be zero.

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Ohms Law and Resistors (Contd)

• To derive Ohms law for resistors from Ohms law
  at a point, we need to relate the circuit
  quantities (V and I) to the field quantities (E
  and J)
• The electric field within the material is given
  by
• The current density in the wire is

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Ohms Law and Resistors (Contd)

• Plugging into J sE, we have
• Define the resistance of the device as
• Thus,

Ohms law for resistors