EEE 498598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 7 Magnetostatics Amperes Law Of
  Force Magnetic Flux Density Lorentz Force
  Biot-savart Law Applications Of Amperes Law In
  Integral Form Vector Magnetic Potential
  Magnetic Dipole Magnetic Flux

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Lecture 7 Objectives

• To begin our study of magnetostatics with
  Amperes law of force magnetic flux density
  Lorentz force Biot-Savart law applications of
  Amperes law in integral form vector magnetic
  potential magnetic dipole and magnetic flux.

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Overview of Electromagnetics
Maxwells equations
Fundamental laws of classical electromagnetics
Geometric Optics
Electro-statics
Magneto-statics
Electro-magnetic waves
Special cases
Statics
Transmission Line Theory
Input from other disciplines
Kirchoffs Laws
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Magnetostatics

• Magnetostatics is the branch of electromagnetics
  dealing with the effects of electric charges in
  steady motion (i.e, steady current or DC).
• The fundamental law of magnetostatics is Amperes
  law of force.
• Amperes law of force is analogous to Coulombs
  law in electrostatics.

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Magnetostatics (Contd)

• In magnetostatics, the magnetic field is produced
  by steady currents. The magnetostatic field does
  not allow for
• inductive coupling between circuits
• coupling between electric and magnetic fields

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Amperes Law of Force

• Amperes law of force is the law of action
  between current carrying circuits.
• Amperes law of force gives the magnetic force
  between two current carrying circuits in an
  otherwise empty universe.
• Amperes law of force involves complete circuits
  since current must flow in closed loops.

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Amperes Law of Force (Contd)

• Experimental facts
• Two parallel wires carrying current in the same
  direction attract.
• Two parallel wires carrying current in the
  opposite directions repel.

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Amperes Law of Force (Contd)

• Experimental facts
• A short current-carrying wire oriented
  perpendicular to a long current-carrying wire
  experiences no force.

F12 0
?
I2
I1
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Amperes Law of Force (Contd)

• Experimental facts
• The magnitude of the force is inversely
  proportional to the distance squared.
• The magnitude of the force is proportional to the
  product of the currents carried by the two wires.

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Amperes Law of Force (Contd)

• The direction of the force established by the
  experimental facts can be mathematically
  represented by

unit vector in direction of current I1
unit vector in direction of current I2
unit vector in direction of force on I2 due to I1
unit vector in direction of I2 from I1
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Amperes Law of Force (Contd)

• The force acting on a current element I2 dl2 by a
  current element I1 dl1 is given by

Permeability of free space m0 4p ? 10-7 F/m
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Amperes Law of Force (Contd)

• The total force acting on a circuit C2 having a
  current I2 by a circuit C1 having current I1 is
  given by

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Amperes Law of Force (Contd)

• The force on C1 due to C2 is equal in magnitude
  but opposite in direction to the force on C2 due
  to C1.

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Magnetic Flux Density

• Amperes force law describes an action at a
  distance analogous to Coulombs law.
• In Coulombs law, it was useful to introduce the
  concept of an electric field to describe the
  interaction between the charges.
• In Amperes law, we can define an appropriate
  field that may be regarded as the means by which
  currents exert force on each other.

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Magnetic Flux Density (Contd)

• The magnetic flux density can be introduced by
  writing

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Magnetic Flux Density (Contd)

• where

the magnetic flux density at the location of dl2
due to the current I1 in C1
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Magnetic Flux Density (Contd)

• Suppose that an infinitesimal current element Idl
  is immersed in a region of magnetic flux density
  B. The current element experiences a force dF
  given by

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Magnetic Flux Density (Contd)

• The total force exerted on a circuit C carrying
  current I that is immersed in a magnetic flux
  density B is given by

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Force on a Moving Charge

• A moving point charge placed in a magnetic field
  experiences a force given by

The force experienced by the point charge is in
the direction into the paper.
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Lorentz Force

• If a point charge is moving in a region where
  both electric and magnetic fields exist, then it
  experiences a total force given by
• The Lorentz force equation is useful for
  determining the equation of motion for electrons
  in electromagnetic deflection systems such as
  CRTs.

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The Biot-Savart Law

• The Biot-Savart law gives us the B-field arising
  at a specified point P from a given current
  distribution.
• It is a fundamental law of magnetostatics.

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The Biot-Savart Law (Contd)

• The contribution to the B-field at a point P from
  a differential current element Idl is given by

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The Biot-Savart Law (Contd)
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The Biot-Savart Law (Contd)

• The total magnetic flux at the point P due to the
  entire circuit C is given by

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Types of Current Distributions

• Line current density (current) - occurs for
  infinitesimally thin filamentary bodies (i.e.,
  wires of negligible diameter).
• Surface current density (current per unit width)
  - occurs when body is perfectly conducting.
• Volume current density (current per unit cross
  sectional area) - most general.

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The Biot-Savart Law (Contd)

• For a surface distribution of current, the B-S
  law becomes
• For a volume distribution of current, the B-S law
  becomes

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Amperes Circuital Law in Integral Form

• Amperes Circuital Law in integral form states
  that the circulation of the magnetic flux
  density in free space is proportional to the
  total current through the surface bounding the
  path over which the circulation is computed.

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Amperes Circuital Law in Integral Form (Contd)
By convention, dS is taken to be in the direction
defined by the right-hand rule applied to dl.
Since volume current density is the most
general, we can write Iencl in this way.
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Amperes Law and Gausss Law

• Just as Gausss law follows from Coulombs law,
  so Amperes circuital law follows from Amperes
  force law.
• Just as Gausss law can be used to derive the
  electrostatic field from symmetric charge
  distributions, so Amperes law can be used to
  derive the magnetostatic field from symmetric
  current distributions.

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Applications of Amperes Law

• Amperes law in integral form is an integral
  equation for the unknown magnetic flux density
  resulting from a given current distribution.

known
unknown
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Applications of Amperes Law (Contd)

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

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Applications of Amperes Law (Contd)

• Closed form solution to Amperes law relies on
  our ability to construct a suitable family of
  Amperian paths.
• An Amperian path is a closed contour to which the
  magnetic flux density is tangential and over
  which equal to a constant value.

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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law

• Consider an infinite line current along the
  z-axis carrying current in the z-direction

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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law (Contd)

• (1) Assume from symmetry and the right-hand rule
  the form of the field
• (2) Construct a family of Amperian paths

circles of radius r where
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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law (Contd)

• (3) Evaluate the total current passing through
  the surface bounded by the Amperian path

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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law (Contd)
Amperian path
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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law (Contd)

• (4) For each Amperian path, evaluate the integral

length of Amperian path.
magnitude of B on Amperian path.
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Magnetic Flux Density of an Infinite Line Current
Using Amperes Law (Contd)

• (5) Solve for B on each Amperian path

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Applying Stokess Theorem to Amperes Law
? Because the above must hold for any
surface S, we must have
Differential form of Amperes Law
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Amperes Law in Differential Form

• Amperes law in differential form implies that
  the B-field is conservative outside of regions
  where current is flowing.

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Fundamental Postulates of Magnetostatics

• Amperes law in differential form
• No isolated magnetic charges

B is solenoidal
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Vector Magnetic Potential

• Vector identity the divergence of the curl of
  any vector field is identically zero.
• Corollary If the divergence of a vector field
  is identically zero, then that vector field can
  be written as the curl of some vector potential
  field.

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Vector Magnetic Potential (Contd)

• Since the magnetic flux density is solenoidal, it
  can be written as the curl of a vector field
  called the vector magnetic potential.

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Vector Magnetic Potential (Contd)

• The general form of the B-S law is
• Note that

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Vector Magnetic Potential (Contd)

• Furthermore, note that the del operator operates
  only on the unprimed coordinates so that

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Vector Magnetic Potential (Contd)

• Hence, we have

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Vector Magnetic Potential (Contd)

• For a surface distribution of current, the vector
  magnetic potential is given by
• For a line current, the vector magnetic potential
  is given by

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Vector Magnetic Potential (Contd)

• In some cases, it is easier to evaluate the
  vector magnetic potential and then use B
  ?? A, rather than to use the B-S law to directly
  find B.
• In some ways, the vector magnetic potential A is
  analogous to the scalar electric potential V.

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Vector Magnetic Potential (Contd)

• In classical physics, the vector magnetic
  potential is viewed as an auxiliary function with
  no physical meaning.
• However, there are phenomena in quantum mechanics
  that suggest that the vector magnetic potential
  is a real (i.e., measurable) field.

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Magnetic Dipole

• A magnetic dipole comprises a small current
  carrying loop.
• The point charge (charge monopole) is the
  simplest source of electrostatic field. The
  magnetic dipole is the simplest source of
  magnetostatic field. There is no such thing as a
  magnetic monopole (at least as far as classical
  physics is concerned).

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Magnetic Dipole (Contd)

• The magnetic dipole is analogous to the electric
  dipole.
• Just as the electric dipole is useful in helping
  us to understand the behavior of dielectric
  materials, so the magnetic dipole is useful in
  helping us to understand the behavior of magnetic
  materials.

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Magnetic Dipole (Contd)

• Consider a small circular loop of radius b
  carrying a steady current I. Assume that the
  wire radius has a negligible cross-section.

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Magnetic Dipole (Contd)

• The vector magnetic potential is evaluated for R
  gtgt b as

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Magnetic Dipole (Contd)

• The magnetic flux density is evaluated for R gtgt
  b as

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Magnetic Dipole (Contd)

• Recall electric dipole
• The electric field due to the electric charge
  dipole and the magnetic field due to the magnetic
  dipole are dual quantities.

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Magnetic Dipole Moment

• The magnetic dipole moment can be defined as

Magnitude of the dipole moment is the product of
the current and the area of the loop.
Direction of the dipole moment is determined by
the direction of current using the right-hand
rule.
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Magnetic Dipole Moment (Contd)

• We can write the vector magnetic potential in
  terms of the magnetic dipole moment as
• We can write the B field in terms of the magnetic
  dipole moment as

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Divergence of B-Field

• The B-field is solenoidal, i.e. the divergence of
  the B-field is identically equal to zero
• Physically, this means that magnetic charges
  (monopoles) do not exist.
• A magnetic charge can be viewed as an isolated
  magnetic pole.

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Divergence of B-Field (Contd)

• No matter how small the magnetic is divided, it
  always has a north pole and a south pole.
• The elementary source of magnetic field is a
  magnetic dipole.

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Magnetic Flux

• The magnetic flux crossing an open surface S is
  given by

Wb
Wb/m2
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Magnetic Flux (Contd)

• From the divergence theorem, we have
• Hence, the net magnetic flux leaving any closed
  surface is zero. This is another manifestation
  of the fact that there are no magnetic charges.

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Magnetic Flux and Vector Magnetic Potential

• The magnetic flux across an open surface may be
  evaluated in terms of the vector magnetic
  potential using Stokess theorem