Electromagnetism INEL 4152 CH 9

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ElectromagnetismINEL 4152 CH 9

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

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In summary

• Stationary Charges
• Steady currents
• Time-varying currents

• Electrostatic fields
• Magnetostatic fields
• Electromagnetic (waves!)

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Outline

• Faradays Law Origin of emag
• Maxwell Equations explain waves
• Phasors and Time Harmonic fields
• Maxwell eqs for time-harmonic fields

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Faradays Law

• 9.2

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Electricity gt Magnetism

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

This is what Oersted discovered accidentally
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Would magnetism would produce electricity?

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

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Electromagnetics was born!

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

Mention some examples of em waves
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Faradays Law

• For N1 and B0

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Transformer Motional EMF

• 9.3

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Three ways B can vary by having

1. A stationary loop in a t-varying B field
2. A t-varying loop area in a static B field
3. A t-varying loop area in a t-varying B field

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1. Stationary loop in a time-varying B field
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2. Time-varying loop area in a static B field
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3. A t-varying loop area in a t-varying B field
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Transformer Example
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Displacement Current, Jd

• 9.4

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Maxwell noticed something was missing

• And added Jd, the displacement current

I
L
At low frequencies JgtgtJd, but at radio
frequencies both terms are comparable in
magnitude.
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Maxwells Equation in Final Form

• 9.4

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Summary of Terms

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

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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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Maxwells Eqs.

• Also 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, while J is the conduction
  current.

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Relations B.C.
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?Time Varying Potentials

• 9.6

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We had defined

• Electric Scalar Magnetic Vector potentials
• Related to B as
• To find out what happens for time-varying fields
• Substitute into Faradays law

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Electric Magnetic potentials

• If we take the divergence of E
• We have
• Taking the curl of add
  Amperes
• we get

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Electric Magnetic potentials

• If we apply this vector identity
• We end up with

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Electric Magnetic potentials

• We use the Lorentz condition
• To get
• and

Which are both wave equations.
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?Time Harmonic FieldsPhasors Review

• 9.7

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Time Harmonic Fields

• Definition is a field that varies periodically
  with time.
• Ex. Sinusoid
• Lets review Phasors!

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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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COMPLEX NUMBERS

• Given a complex number z
• where

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Review

• Addition,
• Subtraction,
• Multiplication,
• Division,
• Square Root,
• Complex Conjugate

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For a Time-varying phase

• Real and imaginary parts are

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PHASORS

• For a sinusoidal current
• equals the real part of

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Advantages of phasors

• Time derivative in time is equivalent to
  multiplying its phasor by jw
• Time integral is equivalent to dividing by the
  same term.

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How to change back from Phasor to time domain

• The phasor is
• multiplied by the time factor, e jwt,
• and taken the real part.

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?Time Harmonic Fields

• 9.7

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Time-Harmonic fields (sines and cosines)

• The wave equation can be derived from Maxwell
  equations, indicating that the changes in the
  fields behave as a wave, called an
  electromagnetic wave or field.
• Since any periodic wave can be represented as a
  sum of sines and cosines (using Fourier), then we
  can deal only with harmonic fields to simplify
  the equations.

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Maxwell Equations for Harmonic fields (phasors)
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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Ex. Given E, find H

• E Eo cos(wt-bz) ax

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Ex. 9.23

• In free space,
• Find k, Jd and H using phasors and Maxwells eqs.
  Recall