Maxwell Equations INEL 4151 ch 9

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Maxwell EquationsINEL 4151 ch 9

• Sandra Cruz-Pol, Ph. D.
• ECE UPRM
• Mayagüez, PR

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Electricity gt Magnetism

• In 1820 Oersted discovered that a steady current
  produces a magnetic field while teaching a
  physics class.

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Would magnetism would produce electricity?

• Eleven years later, and at the same time, Mike
  Faraday in London and Joe Henry in New York
  discovered that a time-varying magnetic field
  would produce an electric current!

Faradays Law
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Lens Law (-)

• If N1 (1 loop)

• The time change
• can refer to B or S

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Two cases of

• B changes

• S (area) changes

Stokes theorem
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3 cases

• Stationary loop in time-varying B field
• Moving loop in static B field
• Moving loop in time-varying B field

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PE 9.1 If B
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PE 9.1
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Example 9.3 Circuit with 10-3 m2 cross-section,
radius 10cm, i1(t)3 sin wt A, 50 Hz, N1200
turns, N2 100 , m500m0, find Vemf on both coils

• Find reluctance and use Faradays Law

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Example PE 9.3 A magnetic core of uniform
cross-section 4 cm2 is connected to a 120V, 60Hz
generator. Calculate the induced emf V2 in the
secondary coil.N1 500, N2300

• Use Faradays Law

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Some terms

• E electric field intensity V/m
• D electric field density C/m2
• H magnetic field intensity, A/m
• B magnetic field density, Teslas

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Take a look at a capacitor

• dielectric is an electrical insulator
• d.c., C open circuits
• Whats the current in an open circuit?

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Maxwells Eqs.

• the equation of continuity
• Maxwell added the term to Amperes Law so
  that it not only works for static conditions but
  also for time-varying situations.
• This added term is called the displacement
  current density, Jd, while JJc is the conduction
  current.

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• For static fields we had
• But the divergence of the curl of ANY vector ?0
• The continuity of the current requires
• We need to define
• Now when we take the divergence of curl
• Therefore

This doesnt work for time-varying fields!
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Maxwell put them together

• If we didnt have the displacement current, Jd,
  then Amperes Law

I
L
At low frequencies JgtgtJd, but at radio
frequencies both terms are comparable in
magnitude.
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• For dynamic fields we had
• Therefore
• Substituting in curl of H
• This is Maxwells equation for Amperes Law

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Maxwell Equations in General Form
Differential form Integral Form
Gausss Law for E field.
Gausss Law for H field. Nonexistence of monopole
Faradays Law
Amperes Circuit Law
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Electromagnetics

• This is the principle of motors, hydro-electric
  generators and transformers operation.

This is what Oersted discovered accidentally
Mention some examples of em waves
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Phasors complex s

• Working with harmonic fields is easier, but
  requires knowledge of phasor, lets review
• complex numbers and
• phasors

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Maxwell Equations for Harmonic fields
Differential form
Gausss Law for E field.
Gausss Law for H field. No monopole
Faradays Law
Amperes Circuit Law
(substituting and
)
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Earth Magnetic Field Declination from 1590 to
1990
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News

• Jan 2011
• The Earth's magnetic north pole is slowly heading
  toward Russia, according to scientists, but one
  of the places being affected by this is Tampa
  International Airport.Airport officials closed
  its main runway this week until Jan. 13 to adjust
  the taxiway signs accounting for the magnetic
  pole shift, Tampa Bay Online reports.