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```Electromagnetism
INEL 4151
Sandra Cruz-Pol, Ph. D.
ECE UPRM
Mayagüez, PR
Electricity => Magnetism

In 1820 Oersted discovered that a steady
current produces a magnetic field while
teaching a physics class.
Cruz-Pol, Electromagnetics
UPRM
Would magnetism would produce
electricity?

Eleven years later,
and at the same time,
London and Joe
Henry in New York
discovered that a
time-varying magnetic
field would produce
an electric current!
Vemf
d
 N
dt

L E  dl   t s B  dS
Cruz-Pol, Electromagnetics
UPRM
Electromagnetics was born!

This is the principle of
motors, hydro-electric
generators and
transformers operation.
This is what Oersted discovered
accidentally:
D 

L H  dl  s  J  t   dS
*Mention some examples of em waves
Cruz-Pol, Electromagnetics
UPRM
Cruz-Pol, Electromagnetics UPRM
Electromagnetic Spectrum
Cruz-Pol, Electromagnetics
UPRM
Some terms
E
= electric field intensity [V/m]
 D = electric field density
 H = magnetic field intensity, [A/m]
 B = magnetic field density, [Teslas]
Cruz-Pol, Electromagnetics UPRM
Maxwell Equations
in General Form
Differential form Integral Form
  D  v
 D  dS   v dv
s
 B  0
v
 B  dS  0
s
B
 E  
t

L E  dl   t s B  dS
D
 H  J 
t
D 

H

dl

J


L
s  t   dS
Cruz-Pol, Electromagnetics
UPRM
Gauss’s Law for E
field.
Gauss’s Law for H
field. Nonexistence
of monopole
Ampere’s Circuit
Law
Maxwell’s Eqs.
 v
J  
t
 Also
the equation of continuity
D
 Maxwell added the term  t to Ampere’s
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.
Cruz-Pol, Electromagnetics UPRM
 
 H  J
 For
But the divergence of curl of ANY vector =0


  H  0   J
The continuity of the current requires:
 v
J  
 0 This doesn’t work for time-varying fields!
t
  
to define:   H  J  J d
We need
Now when we take the divergence of curl



  H  0    J    Jd
Therefore:   J    J
d
Cruz-Pol, Electromagnetics, UPRM
 For
 v   D 
  J d    J 
Therefore:
t

t
D
 
t
D
Jd 
t
Substituting in curl of H
  D
 H  J 
t
This is Maxwell’s equation for Ampere’s Law
Cruz-Pol, Electromagnetics, UPRM
Maxwell put them together

current. If we didn’t have it, then:
 H  dl   J  dS  I
L
enc
I
S1
 H  dl   J  dS  0
L
S1
I
L
S2
d
dQ
L H  dl  S J d  dS  dt S D  dS  dt  I
2
2
S2
At low frequencies J>>Jd, but at radio frequencies both
Cruz-Pol, Electromagnetics
terms are comparable in magnitude.
UPRM
Moving loop in static field

When a conducting loop is moving inside a
magnet (static B field, in Teslas), the force on a
charge is

 
F  Qu  B
 

F  Il B
Cruz-Pol, Electromagnetics
UPRM
Encarta®
Phasors & complex #’s
Working with harmonic fields is easier, but
requires knowledge of phasor, let’s review
 complex numbers and
 phasors
Cruz-Pol, Electromagnetics UPRM
Maxwell Equations
for Harmonic fields
Differential form*

 DE
 v  v
Gauss’s Law for E field.
 BH
0

Gauss’s Law for H field.
No monopole
0
  E   jH
 E  
B
t
  H  J  jE
D
 H  J 
t
* (substituting