ENE 325 Electromagnetic Fields and Waves

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ENE 325Electromagnetic Fields and Waves

• Lecture 5 Ampéres law, Scalar and Vector
  Magnetic Potentials, Magnetic Force, Torque, and
  Magnetic Material

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Review (1)

• Amperes circuital law - the integration of
  around any closed path is equal to the net
  current enclosed by that path.
• Curl is employed to find the point form of
  Ampères circuital law.
• Curl of or is the maximum
  circulation of per unit area as the area
  shrinks to zero

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Review (2)

• Magnetic flux density is related to the
  magnetic field intensity in the free space
  by
• Weber/m2 or Tesla (T)
• where ?0 is the free space permeability, given in
  units of henrys per meter, or
• ?0 4??10-7 H/m.
• Magnetic flux ? (units of Webers) passing
  through a surface is found by

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Outline

• Curl and point form of Ampéres law
• Magnetic flux density
• Scalar and vector magnetic potentials
• Magnetic force and torque
• Magnetic material and permeability

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Curl and the point form of Ampéres circuital law
(1)

• Curl is employed to find the point form
  Ampères circuital law, analogous to Divergence
  to find the point form of Gausss law.
• Curl of or is the maximum
  circulation of per unit area as the area
  shrinks to zero.

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Curl and the point form of Ampéres circuital law
(2)

• Curl operator perform a derivative of vector
  and returns a vector quantity. For Cartesian
  coordinates, can be written as

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Physical view of curl

• Field lines indicating divergence A simple
  way to see the
• Field lines indicating curl
  direction of curl using
• right hand rule

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Stokess Theorem

• Stokess Theorem relates a closed line integral
  into a surface integral

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Magnetic flux density, B

• Magnetic flux density is related to the
  magnetic field intensity in the free space
  by
• Magnetic flux ? (units of Webers) passing through
  a surface is found by

Weber/m2 or Tesla (T)
1 Tesla 10,000 Gauss. where ?0 is the free
space permeability, given in units of henrys per
meter, or ?0 4??10-7 H/m.
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Gausss law for magnetic fields
or
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EX1 A solid conductor of circular cross section
is made of a homogeneous nonmagnetic material. If
the radius a 1 mm, the conductor axis lies on
the z axis, and the total current in the
direction is 20 A, find

• a) H? at ? 0.5 mm
• b) B? at ? 0.8 mm
• c) The total magnetic flux per unit length inside
  the conductor

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Maxwells equations for static fields
Integral form Differential form
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The scalar and vector magnetic potentials (1)

• Scalar magnetic potential (Vm) is
  the simple practical concept to determine the
  electric field. Similarly, the scalar magnetic
  potential, Vm, is defined to relate to the
  magnetic field but there is no physical
  interpretation.

Assume
To make the above statement true, J 0.
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The scalar and vector magnetic potentials (2)
From
Laplaces equation This equations solution to
determine the potential field requires that the
potential on the boundaries is known.
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The scalar and vector magnetic potentials (3)

• The difference between V (electric potential)
  and Vm
• (scalar magnetic potential) is that the electric
  potential is a
• function of the positions while there can be many
  Vm values
• for the same position.

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The scalar and vector magnetic potentials (4)
While for the electrostatic case
does not depend on path.
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The scalar and vector magnetic potentials (5)

• Vector magnetic potential (A) is useful to find
  a
• magnetic filed for antenna and waveguide.

From Let assume so and
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The scalar and vector magnetic potentials (6)

• It is simpler to use the vector magnetic
  potential to determine
• the magnetic field. By transforming from
  Bio-savart law,
• we can write

The differential form
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Ex2 Determine the magnetic field from the
infinite length line of current using the vector
magnetic potential
Find
at point P(?, ?, z)
then
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Vector magnetic potential for other current
distributions

• For current sheet
• For current volume

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Magnetic force

• Force on a moving charge
• Force on a differential current element

N
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Hall effect

• Hall effect is the voltage exerted from the
  separation of electrons and positive ions
  influenced by the magnetic force in the
  conductor. This Hall voltage is perpendicular to
  both magnetic field and the charge velocity.

N
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Magnetic force on the current carrying conductor
(1)

• For the current carrying conductor, consider the
  magnetic force on the whole conductor not on the
  charges.

From and then
dQ ?vdv
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Magnetic force on the current carrying conductor
(2)

• From
• we can write
• then
• For a straight conductor in a uniform magnetic
  field
• (still maintains the closed circuit),
• Force between differential current elements
  determine
• the force on the conductor influenced by the
  other nearby.

or F ILBsin?
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Ex3 Determine the force action on circuit 2 by
circuit 1.
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Force and torque on a closed circuit (1)
N?m
where
torque (N?m)
distance from the origin (m)
Force (N)
If the current is uniform,
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Force and torque on a closed circuit (2)

• For a current loop, we can express torque as
• If is constant or uniform, we can express
  torque as
• Define magnetic dipole moment

where m magnetic dipole moment (A?m2).
Therefore, torque can be shown as
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Ex4 To illustrate some force and torque
calculations, consider the rectangular loop
shown. Calculate the total force and torque
contribution on each side. Let the current I flow
in the loop lied in the uniform magnetic field
tesla.
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Ex5 A 2.5 m length conductor is located at z 0,
x 4m and has a uniform current of 12 A in the
direction . Determine in this area if
the force acting on the conductor is 1.2?10-2 N
in the direction .
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The nature of magnetic materials

• Combine our knowledge of the action of a magnetic
  field on a current loop with a simple model of an
  atom and obtain some appreciation of the
  difference in behavior of various types of
  materials in magnetic fields.
• The magnetic properties of the materials depend
  on magnetic moment. Three types of magnetic
  moment are
• 1. The circular orbiting of electrons around the
  positive nucleus results in the current and then
  the magnetic field m IdS.
• 2. Electron spinning around its own axis and
  thus generates a magnetic dipole moment.
• 3. Nuclear spin, this factor provides a
  negligible effect on the overall magnetic
  properties of materials.

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Types of magnetic material (1)

• diamagnetic The small magnetic filed produced by
  the motion of the electrons in their orbits and
  those produced by the electron spin combine to
  produce a net field of zero or we can say the
  permanent magnetic moment m0 0.
• The external field would produce an internal
  magnetic field.
• Some examples of materials that has diamagnetic
  effect are Metallic bismuth, hydrogen, helium,
  the other inert gases, sodium chloride, copper,
  gold silicon, germanium, graphite, and sulfur.

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Types of magnetic material (2)

• paramagnetic The net magnetic moment of each atom
  is not zero but the average over the volume is,
  due to random orientation of the atoms. The
  material shows no magnetic effects in the absence
  of the external field.
• Whenever there is an external field and the
  alignment of magnetic moments acts to increase
  the value of , the material is called
  paramagnetic but if it acts to decrease the
  value of , it is still called diamagnetic.
• For example, Potassium, Oxygen, Tungsten, and
  some rare earth elements.

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Types of magnetic material (3)

• Ferromagnetic each atom has a relatively large
  dipole moment due to uncompensated electron spin
  moments. These moments are forced to line up in
  parallel fashion over region containing a large
  number of atoms, these regions are called
  domains. The domain moments vary in direction
  from domain to domain. The overall effect is
  therefore one of cancellation, and the material
  as a whole has no magnetic moment.
• When the external field is applied, those domains
  which are in the direction of the applied field
  increase their size at the expense of their
  neighbors, and the internal field increases
  greatly over that of the external field alone.
  When the external field is removed, a completely
  random domain alignment is not usually attained,
  and a residual dipole field remains in the
  macroscopic structure.

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Types of magnetic material (4)

• The magnetic state of material is a function of
  its magnetic history or hysteresis. For
  example, Iron, Nickel, and Cobalt.
• Antiferromagnetic The forces between adjacent
  atoms cause the atomic moments to line up in anti
  parallel fashion. The net magnetic moment is
  zero. The antiferromagnetic materials are
  affected slightly by the presence of and external
  magnetic field.
• For example, nickel oxide (NiO), ferrous sulfide
  (FeS), and cobalt chloride (CoCl2).

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Types of magnetic material (5)

• Ferrimagnetic Substances show an antiparallel
  alignment of adjacent atomic moments, but the
  moments are not equal. A large response to an
  external magnetic field therefore occurs.
• For example, the ferrites, the iron oxide
  magnetite (Fe3O4), a nickel-zinc ferrite, and a
  nickel ferrite.

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Types of magnetic material (6)

• Superparamagnetic materials are composed of an
  assembly of ferromagnetic particles in a
  nonferromagnetic matrix. The domain walls cannot
  penetrate the intervening matrix material to the
  adjacent particles.
• For example, the magnetic tape.