Quick review: phasor notation and complex numbers

1
Quick review phasor notation and complex numbers

• complex numbers
• complex conjugate
• magnitude

• Eulers formula

• Eulers formula for a complex number

2
The complex plane

• the complex number Z x jy can be plotted in
  two-dimensions

• the complex number Z acts like a vector in
  2-space
• if Z is a function of a variable w (say, for
  instance, frequency), then the vector is
  different for different values of w
• this is formally called a vector field
• well look at this more later

Z 1 j2
q
3
The complex plane

• unit circle

• Z 1
• q 0
• Z j
• q p/2
• Z -1
• q p
• Z -j
• q 3p/2

4
Quick review powers of complex numbers

• easiest to use phasor form, i.e., Eulers formula

• examples
• square root of -1 (this should be j!)
• square root of j (i.e., the fourth root of -1)

5
Location in three dimensional space

• Cartesian coordinates are easy
• three unit vectors, all orthogonal to one another

• infinitesimal surface area
• dx dy
• dy dz
• dx dz
• infinitesimal volume element
• dx dy dz

6
Other coordinate systems cylindrical coordinates

• cylindrical coordinates
• essentially polar coordinates in x-y plane

• infinitesimal surface area

• dz dr

• dz (r d?)

• dr (r d?)

• infinitesimal volume element
• dz dr (r d?)

7
Spherical coordinates

• use spheres to define position
• radial distance r
• plus two angular positions

8
spherical coordinates

• differential area
• dr (r dq)

• dr ((r sinq) d?)

• (r dq) ((r sinq) d?)

• differential volume element
• dr (r dq) ((r sinq) d?)

r
9
operations defined in this space

• scalar operations
• result is a simple number, without a direction
• dot product
• vector operations
• result has both direction and magnitude, i.e., it
  is still a vector
• vector addition / subtraction
• cross product

10
Dot product

• dot product scalar result
• projection of one vector onto another
• properties
• commutative
• distributive

• since we need three unit vectors to specify a
  point in three dimensions, A and B can be written
  as

• now use distributive and commutative properties
  to multiply this out

• if a1, a2 and a3 are normalized and orthogonal
  then

11
Dot products

• the dot product of a vector with a unit vector
  gives you the component of the vector in the
  direction of the unit vector
• we can use this to convert from one coordinate
  system to another
• example convert from rectangular coordinates to
  cylindrical

12
Dot products

• from rectangular coordinates to cylindrical

13
A vector coordinate example

• given in cylindrical coords

• note the this vector does start at the origin

• lets check the results using the matrix version

14
Another vector coordinate example

• given in cylindrical coords

• note the this vector doesnt start at the
  origin!

• lets check the results using the matrix version

15
Dot products

• convert from rectangular coordinates to spherical

• what are q and ? in terms of x, y, and x?

16
Cross products

• cross product vector result
• direction set by right hand rule

• cross product is not commutative
• A x B - B x A

• in rectangular coordinates

run cross product applet http//www.phy.syr.edu/c
ourses/java-suite/crosspro.html
17
Fields

• fancy name for a function whose argument is a
  vector
• roughly, a field is something that is a function
  of several variables
• for us, the variables will usually be position
• also need to identify the dimension of the
  result of the function
• scalar field result is a simple number
• example the temperature (a scalar quantity) in
  this room as a function of position in the room
• vector field result is a vector
• example the air velocity (a vector quantity) in
  this room as a function of position in the room

18
Web resources

• cross product applet http//www.phy.syr.edu/cours
  es/java-suite/crosspro.html
• vector field analyzer http//math.la.asu.edu/kaws
  ki/vfa2/
• a better version of the vector field analyzer
  http//www.physics.orst.edu/tevian/microscope/

19
applets showing some vector fields

• 2-d view http//www.physics.orst.edu/tevian/micr
  oscope/
• 3-d view http//www.falstad.com/vector/
• fields available http//www.falstad.com/vector/fu
  nctions.html
• 1/r single line electric field around an
  infinitely long line of charge. It is inversely
  proportional to the distance from the line.
• 1/r double lines field around two infinitely
  long conductors. The distance between them is
  adjustable.
• 1/r2 single field associated with gravity and
  electrostatic attraction gravitational field
  around a planet and the electric field around a
  single point charge.
• This is a two-dimensional cross section of a
  three-dimensional field.
• In three dimensions, the divergence of this field
  is zero except at the origin in this cross
  section, the divergence is positive everywhere
  (except at the origin, where it is negative).
• 1/r2 double field associated with gravity and
  electrostatic attraction. gravitational field
  around two planets and the electric field around
  two negative point charges are similar to this
  field.

20
Forces between charges

• lets start with the fundamental interaction
  between point charges
• 1600 Gilbert publishes first major
  classification of electric and non-electric
  materials
• Charles Augustin Coulomb (1736-1806)
• 1761-1781 served as a initially a military
  engineer
• 1785 1791 wrote seven important treatises on
  electricity and magnetism, submitted to the
  Académie des Sciences (France)
• developed a theory of attraction and repulsion
  between bodies of the opposite and same
  electrical charge
• demonstrated an inverse square law for such
  forces
• examined perfect conductors and dielectrics
• suggested that there was no perfect dielectric,
  proposing that every substance has a limit above
  which it will conduct electricity
• put forward the case for action at a distance
  between electrical charges in a similar way as
  Newton's theory of gravitation was based on
  action at a distance between masses