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## Electromagnetism INEL 4152 CH 9

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### Electromagnetism INEL 4152 CH 9 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayag ez, PR – PowerPoint PPT presentation

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Title: Electromagnetism INEL 4152 CH 9

1
ElectromagnetismINEL 4152 CH 9
• Sandra Cruz-Pol, Ph. D.
• ECE UPRM
• Mayagüez, PR

2
In summary
• Stationary Charges
• Time-varying currents
• Electrostatic fields
• Magnetostatic fields
• Electromagnetic (waves!)

3
Outline
• Faradays Law Origin of emag
• Maxwell Equations explain waves
• Phasors and Time Harmonic fields
• Maxwell eqs for time-harmonic fields

4
• 9.2

5
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
6
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!

7
Electromagnetics was born!
• This is Faradays Law -the principle of motors,
hydro-electric generators and transformers
operation.

Mention some examples of em waves
8
• For N1 and B0

9
Transformer Motional EMF
• 9.3

10
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

11
1. Stationary loop in a time-varying B field
12
2. Time-varying loop area in a static B field
13
3. A t-varying loop area in a t-varying B field
14
Transformer Example
15
Displacement Current, Jd
• 9.4

16
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.
17
Maxwells Equation in Final Form
• 9.4

18
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

19
Maxwell Equations in General Form
Differential form Integral Form
Gausss Law for E field.
Gausss Law for H field. Nonexistence of monopole
Amperes Circuit Law
20
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.

21
Relations B.C.
22
?Time Varying Potentials
• 9.6

23
• Electric Scalar Magnetic Vector potentials
• Related to B as
• To find out what happens for time-varying fields

24
Electric Magnetic potentials
• If we take the divergence of E
• We have
• Taking the curl of add
Amperes
• we get

25
Electric Magnetic potentials
• If we apply this vector identity
• We end up with

26
Electric Magnetic potentials
• We use the Lorentz condition
• To get
• and

Which are both wave equations.
27
?Time Harmonic FieldsPhasors Review
• 9.7

28
Time Harmonic Fields
• Definition is a field that varies periodically
with time.
• Ex. Sinusoid
• Lets review Phasors!

29
Phasors complex s
• Working with harmonic fields is easier, but
requires knowledge of phasor, lets review
• complex numbers and
• phasors

30
COMPLEX NUMBERS
• Given a complex number z
• where

31
Review
• Subtraction,
• Multiplication,
• Division,
• Square Root,
• Complex Conjugate

32
For a Time-varying phase
• Real and imaginary parts are

33
PHASORS
• For a sinusoidal current
• equals the real part of

34
• Time derivative in time is equivalent to
multiplying its phasor by jw
• Time integral is equivalent to dividing by the
same term.

35
How to change back from Phasor to time domain
• The phasor is
• multiplied by the time factor, e jwt,
• and taken the real part.

36
?Time Harmonic Fields
• 9.7

37
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.

38
Maxwell Equations for Harmonic fields (phasors)
Differential form
Gausss Law for E field.
Gausss Law for H field. No monopole
Amperes Circuit Law
(substituting and
)
39
Ex. Given E, find H
• E Eo cos(wt-bz) ax

40
Ex. 9.23
• In free space,
• Find k, Jd and H using phasors and Maxwells eqs.
Recall