EEL3472-Mod1-Intro Vectors

Reversible Computing & Truly Adiabatic CircuitsFAMU-FSU College of Engineering
Outline of Topics
EEL3472 - Fields I, Spring 2009, Module 1
FAMU-FSU College of Engineering
Subject of this Course
This course introduces the theoretical study of classical electromagnetic fields.
With an introduction to a variety of basic applications, used in examples throughout.
Electromagnetism (EM): The physical phenomena of electricity and magnetism.
Found by Faraday, Ampere, Maxwell in 1800’s to be fundamentally interlinked with each other.
Two aspects of a single underlying fundamental force.
What do we mean by a Field?
Definition: A (physical) field is a physical quantity that varies as a function of position (and possibly also of time) throughout some region of space (or spacetime).
Examples of fields we’ll encounter in this course:
Charge density ρ (greek “rho” not latin “p”)
Electric potential V
These represent the same physical entity in different ways
Magnetic field B (or H) - Likewise
The electric/magnetic field quantities are vectors.
We’ll begin this course with a review of vector math.
Present-Day Status of
Electromagnetic Field Theory
EM field theory is the classical theory from which emerged quantum electrodynamics (QED).
QED is the modern quantum field theory describing how electric charges interact via the electromagnetic force.
Includes quantum phenomena: photons, entangled light, etc.
QED has been confirmed to extremely high precision.
At least about 11 decimal places so far
Classical EM theory does not describe nature precisely, but is only an approximation to QED.
However, it is sufficiently precise for a wide variety of large-scale electrical engineering applications (next slide)
For most of these, quantum effects are not very important.
Some Applications of EMF Theory
EM theory provides the basis for the design and analysis of the following structures:
Capacitors
Waveguides (e.g., optic fibers)
Cavities (e.g., microwave ovens)
Laplace, Gauss, Poisson, Stokes – Vector analysis
Coulomb – Described electrostatic force
Maxwell, 1864 – Derived unified theory of electromagnetism, predicted EM waves
Hertz, Marconi – Verified EM waves exist
Lorentz, Poincaré, Einstein – Deduced relativistic mechanics from EM theory
History of Electromagnetic Theory
Chart1
Coulomb
Coulomb
Coulomb
Laplace
Laplace
Laplace
Ampere
Ampere
Ampere
Gauss
Gauss
Gauss
Poisson
Poisson
Poisson
Ohm
Ohm
Ohm
Faraday
Faraday
Faraday
Stokes
Stokes
Stokes
Maxwell
Maxwell
Maxwell
Lorentz
Lorentz
Lorentz
Poincare
Poincare
Poincare
Hertz
Hertz
Hertz
Marconi
Marconi
Marconi
Einstein
Einstein
Einstein
1806
1736
1736
1827
1749
1749
1836
1775
1775
1855
1777
1777
1840
1781
1781
1854
1789
1789
1867
1791
1791
1903
1819
1819
1879
1831
1831
1928
1853
1853
1912
1854
1854
1894
1857
1857
1937
1874
1874
1955
1879
1879
Sheet1
Coulomb
1806
1736
1736
Laplace
1827
1749
1749
Ampere
1836
1775
1775
Gauss
1855
1777
1777
Poisson
1840
1781
1781
Ohm
1854
1789
1789
Faraday
1867
1791
1791
Stokes
1903
1819
1819
Maxwell
1879
1831
1831
Lorentz
1928
1853
1853
Poincare
1912
1854
1854
Hertz
1894
1857
1857
Marconi
1937
1874
1874
Einstein
1955
1879
1879
Sheet1
Sheet2
Sheet3
Course Introduction
Vector Algebra
Vector Calculus
Vector Notations
Definition: A vector is a geometric entity that has both a magnitude and a direction.
For our purposes:
Directions are in three-dimensional space.
A vector can be represented by a list of numbers.
Assumes a particular coordinate system.
A notational convention:
In typed documents, we write vectors in bold: A, B, C
When handwriting, put an arrow over symbol:
Electric field vectors:
D – Electric flux density, dimensioned in charge/area.
Magnetic field vectors:
Others:
F – Force on a particle
J – Current density
And more: velocity, acceleration, angular velocity, torque, etc.
Vector Magnitude
The magnitude of a vector A is a scalar quantity A.
The operation of taking the magnitude of a vector is denoted by enclosing the vector in vertical bars “|”.
The magnitude can be thought of as the length of the vector (if it is pictured as an arrow).
Note: This differs from the “length”
of a “vector” in computer programming:
That means, number of elements in a list.
A
|A|
Example: A = B + C
E.g.: A = B + B = 2B
They can also be divided by scalars (≠0).
Any vector, when multiplied by the scalar 0, becomes the unique null vector 0.
For any A, 0A = 0.
A
B
C
B
B
A
Unit vector is written in bold lowercase: a.
Another common convention:
Unit vector has an angled “hat” over it:
The unit vector aA in the direction of a given (nonzero) vector A can be found by dividing A by its magnitude: so A = |A|aA.
Component Form
Any vector A can be written as a sum of 3 compenent vectors, i.e., as a linear combination of unit vectors along any 3 linearly independent axes.
This is called a component representation of the vector.
E.g., use unit vectors ax, ay, az in the direction of the
x, y, z axes in a Cartesian coordinate system.
Another common notation: i, j, k or
More about coordinate systems later.
That is, for any vector A, we can write:
A = Ax ax + Ay ay + Az az = ∑i{x,y,z} Aiai = Aiai
I’ll say the component Axax (a vector)
has the numerical coefficient Ax (a scalar).
Einstein
summation
notation
Magnitudes in Terms of Components
To find the magnitude of a vector in terms of the magnitudes of its components, we can use a 3D extension of the Pythagorean Theorem.
Note: This formula assumes we are using an orthogonal coordinate system, i.e. using unit vectors that are at right angles to each other.
Ax
Ay
Az
A
Vector Magnitude – Practice Problem
What is the distance from the origin to the point (3, 4, 0)?
Use Pythagorean Theorem
3ax + 4ay + 5az?
Addition in Terms of Components
To add or subtract two vectors, just add or subtract their corresponding components
Assuming both component representations are using the same unit vectors (coordinate system)
If not, you have to first convert one of the vector representations into the coordinate system of the other!
We’ll cover how to do this shortly.
A + B = (Axax + Ayay + Azaz) + (Bxax + Byay + Bzaz)
= (Ax + Bx)ax + (Ay + By)ay + (Az + Bz)az
Commutative laws:
Scalar multiplication is commutative: sA = As.
Associative laws:
Vector addition is associative: A + (B + C) = (A + B) + C
Scalar multiplication is associative: s(tA) = (st)A.
Scalar multiplication is distributive, in two ways:
s(A + B) = sA + sB (across vector addition)
(s + t)A = sA + tA (across scalar addition)
The “Dot Product” Operator
A•B = s = AB cos θ
Where A=|A| etc. and θ is the angle between A and B.
Definition in terms of orthogonal coefficients:
A•B = AxBx + AyBy + AzBz
Some algebraic properties of dot product:
Distributes over vector addition: A•(B+C)=A•B+A•C
Commutes with scalar multiplication: s(A•B) = A•(sB)
Sometimes, the dot product is referred to as an “inner product” operator.
As opposed to cross product “outer product.”
We’ll explain this later.
of A’s projection onto (i.e.,
component along) B’s direction.
Or vice-versa.
the other one.
we don’t overshoot.
θ
A
B
Area
AB = (AB sin θ)an
Where AB are magnitudes, θ is the
interior angle, and an is the normal
(perpendicular) unit vector to
Magnitude is area of
A B = −(B A)
It can be written in terms of components as a matrix determinant:
Remember this rule:
Modern notation. i,j,k{x,y,z}; [ ] = Anti-symmetrized sum over permutations of x,y,z
See Sadiku text for information about…
The Scalar Triple Product A•(BC)
The Vector Triple Product A(BC)
[]
()()()
1700
1750
1800
1850
1900
1950
2000
Coulomb
LaplaceAmpere
Gauss
Poisson
Ohm
Faraday
Stokes
Maxwell
Lorentz
Hertz
MarconiEinstein
1700
1750
1800
1850
1900
1950
2000
Coulomb
LaplaceAmpere
Gauss
Poisson
Ohm
Faraday
Stokes
Maxwell
Lorentz
Poincare
Hertz
MarconiEinstein

EEL3472-Mod1-Intro Vectors

Documents

Transcript of EEL3472-Mod1-Intro Vectors