Magnetostatic Field: Ampere Circuital Law

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CHAPTER 3
ELECTROMAGNETIC FIELDS THEORY

• Magnetostatic Field Ampere Circuital Law

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In this chapter you will learn

• Biot-Savats Law
• Ampere Circuital Law
• Magnetic flux density vector
• Magnetic potential vector and magnetic force
• Magnetic circuit
• Faradays Law
• Maxwells Equation

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Analogy between Electric Magnetic Fields
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Amperes Circuit Law

• Similar to Gausss law
• Amperes law states that the line integral of the
  tangential components of H around a closed path
  is the same as the net current Ienc enclosed by
  the path
• Eq 1 is the integral form of Amperes Circuit Law
• Amperes Circuit Law is used when we want to
  determine H when the current distribution is
  symmetrical

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Amperes Circuit Law

• If we apply Stokes theorem to eq 1, we obtain
• But since
• We compare (2) and (3) to obtain
• Third maxwells Equation Amperes law in point
  form

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Amperes Circuit Law

• Magnetostatic field is not conservative. That is

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Applications of Amperes law

• Some symmetrical current distributions
• Infinite line current
• Infinite Sheet of current
• Infinitely long coaxial transmission line

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Infinite Line Current

• Consider an infinitely long filamentary current l
  along the z-axis

To determine H at an observation point P, we
allow a closed path pass through P known as an
Amperian path (analogous to Gaussian surface)
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Infinite Line Current

• Since the amperian path encloses the whole
  current I, according to Amperes law
• Thus

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Infinite Sheet of current

• Consider an infinite current sheet in the z 0
  plane with a uniform current density K Kyay A/m

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Infinite Sheet of current

• Applying Amperes Law, we get
• Regard the infinite sheet as comprising of
  filaments. The resultant dH has only an
  x-component
• Also, H on one side of the sheet is the negative
  of that on the other side.

------------(1)
------------(2)
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Infinite Sheet of current

• Evaluating the Amperes law along the closed path
  we obtain
• Compare eq (1) with eq (3), we get
• Subtitute H0 in eq (2)

------------(3)
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Infinite Sheet of current

• Thus now we can say
• In general, for an infinite sheet of current
  density K A/m,

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Infinitely long coaxial transmission line

• Consider an infinitely long transmission line
  consisting of two concentric cylinders having
  their axes along the z-axis

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Infinitely long coaxial transmission line

• The inner conductor has radius a and carries
  current I while the outer conductor has inner
  radius b and thickness t and carries return
  current -I.
• Since the current distribution is symmetrical, we
  apply Ampere's law for the Amperian path for each
  of the four possible regions
• 0 p a,
• a p b,
• b p b t,
• a n d p b t .

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region 0 p a

• For region 0 p a, we apply Ampere,s law to
  path L1
• Since the current is uniformly distributed over
  the cross section,

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region 0 p a

• Thus

or
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Region a p b

• For region a p b, we use L2

or
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Region b p b t

• For region b p b t, we use L3
• J in this case is the current density (current
  per unit area) of the outer conductor and is
  along az

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Region b p b t

• Thus
• Then we can get H,

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Region p b t

• For region p b t we use L4

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Infinitely long coaxial transmission line

• Putting it all together,

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Infinitely long coaxial transmission line

• Plot of HF against ?.

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Example 2

• A toroid whose dimensions are shown below has N
  turns and carries current I. determine H inside
  and outside the toroid.

2a
?
?0
?0
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Solution

• Apply Amperes circuit law to the Amperian path,
  which is a circle of radius p. Since N wires cut
  through this path each carrying current I, the
  net current enclosed by the path is NI. Hence,
• Where p0 is the mean radius of the toroid. Thus
  the approximate value of H is

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Solution

• Outside the toroid, the current enclosed by the
  Amperean path is
• NI - NI 0
• Hence H 0

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Practice Exercise 7.6

• A toroid of circular cross section whose center
  is at the origin and axis the same as the z-axis
  has 1000 turns with p0 10 cm, a 1cm. if the
  toroid carries a 100-mA current, find H at
• (3cm, -4cm, 0)
• (6cm, 9cm, 0)