Download 1_3_Maxwell

Electricity and Magnetism
INEL 4151
Sandra Cruz-Pol, Ph. D.
ECE UPRM
Mayagüez, PR
Electricity => Magnetism

In 1820, Prof. Oersted discovered that a
steady current produces a magnetic field
while teaching a physics class.
http://micro.magnet.fsu.edu/electromag/java/faraday/index.html
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 timevarying magnetic field
produces an electric
current!
Vemf
d
 N
dt

L E  dl   t s B  dS
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
http://ece.uprm.edu/~pol/cursos
Some terms

E = electric field intensity [V/m]
 D = electric field density
 H = magnetic field intensity, [A/m]
 B = magnetic field density, [Teslas]
D  E
B  H
9
10
 o  8.85 10 12 F / m 
36
 o  4 10 7 H / m
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
D
 H  J 
t
Gauss’s Law for E
field.
Gauss’s Law for H
field. Nonexistence
of monopole

Faraday’s Law
L E  dl   t s B  dS
D 

H

dl

J


L
s  t   dS
Ampere’s Circuit
Law
Moving loop in static B field
When a conducting loop is moving inside a magnet (static B
field), there’s a force on the charges.
http://www.walter-fendt.de/ph14e/electricmotor.htm
http://micro.magnet.fsu.edu/electromag/java/generator/dc.html

 
F  Qu  B
 

F  Il B
Encarta®
Who was NikolaTesla?
 Find
out what inventions he made
 His relation to Thomas Edison
 Why is he not well know?
Vector Analysis Review:
 What
is a vector?
 How to add them, multiply, etc,?
 Coordinate systems

Cartesian, cylindrical, spherical
 Vector
Calculus review
Vector

A vector
has magnitude and direction.

A  â A A
where â A is unit vector.


Ax â x  Ay â y  Az â z
A A

â A   
Ax2  Ay2  Az2
A A

In Cartesian coordinates (x,y,z):

A  Ax â x  Ay â y  Az â z
Vector operations
Commutative
Associative
Distributive
   
AB  BA
   
A B  BA

  
 
A  (B  C)  (A  B)  C


 


k A  B  kA  kB
      
C A  B  CA  CB


 
kA  Ak


k (lA)  (kl)A
Example
Given vectors A=ax+3az and B=5ax+2ay-6az
 (a) |A+B|
 (b) 5A-B
 (c) the component of A along y
 (d) a unit vector parallel to 3A+B
Answers: (a) 7 (b) (0,-2,21) (c) 0 (d) ± (0.9117,.2279,0.3419)
Vector Multiplications

 
Dot product A  B  AB cos  AB
Note that:
 
A  B  Ax Bx  Ay By  Az Bz

Cross product
  2
A  A  A  A2
 
A  B  AB sin  ABaˆn
aˆ x
 
A  B  Ax
aˆ y
Ay
aˆ z
Az
Bx
By
Bz
Also…

Multiplying 3 vectors:
Scalar:
Vector:

  
  
  
A  (B  C)  B  (C  A)  C  (A  B)
  
     
A  (B  C)  B(A  C)  C(A  B)
Projection of vector A along B:


A B  A  âB
Coordinates Systems
 Cartesian
(x,y,z)
 Cylindrical (,f,z)
 Spherical (r,,f)
Cylindrical
coordinates

A  A â   Af â f  Az â z
 Ax  cos f
 A    sin f
 y 
 Az   0
 A   cos f
 A    sin f
 f 
 Az   0
 x y
2
x   cos f
2
f  tan
1
y   sin f
y
x
 sin f
cos f
0
sin f
cos f
0
0  A 
0  Af 
1  Az 
0  Ax 
0  Ay 
1  Az 
Spherical
coordinates

A  Ar â r  A â  Af â f
 Ax  sin  cos f
 A    sin  sin f
 y 
 Az   cos 
 Ar   sin  cos f
  
 A    cos  cos f
 Af    sin f
 
cos  cos f
cos  sin f
 sin 
sin  sin f
cos  sin f
cos f
2
2
x

y
2
2
2
1
1 y
r x y z
  tan
f  tan
z
x
x  r sin  cos f y  r sin  sin f z  r cos 
 sin f   Ar 
 
cos f   A 
0   Af 
cos    Ax 
 sin    Ay 
0   Az 
Vector calculus review
Del (gradient)



  aˆ x  aˆ y  aˆ z
x
y
z
Divergence
 Ax Ay Az
A 


x
y
z
Curl
aˆ x


A 
x
Ax
Laplacian
(del2 )
2
2
2

V

V

V
2
V 2  2  2
x
y
z
aˆ y

y
Ay
aˆ z

z
Az
Theorems

Divergence
 

A

d
S



A
dv


S

Stokes’

Laplacian
Scalar:
Vector:

 


 A  dl     A  dS
L

v
S
2
2
2

V

V

V
2
V 2  2  2
x
y
z

 2 A   2 Ax aˆ x   2 Ay aˆ y   2 Az aˆ z