Intro Vector

04/28/23 EEL3472 - Fields I, Spring 2009, Module 1 1

FAMUFAMU--FSUFSU College of EngineeringCollege of Engineering

EEL 3472 EEL 3472 Electromagnetic Fields IElectromagnetic Fields I

Spring 2009Spring 2009

Instructor: Michael P. FrankSlide Module 1:

Course Introduction


FAMU-FSU College of Engineering

Outline of LectureOutline of Lecture Course Introduction: Some Perspective

Importance of Electromagnetic Field Theory Some History of the Subject Outline of Topics

Review Syllabus, Schedule, etc. Course Objectives Grading Policies Tentative Schedule

Begin course material: Vector algebra



Subject of this CourseSubject 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 What do we mean by a FieldField?? 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 Electric field E (or D)

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 Present-Day Status of Electromagnetic Field TheoryElectromagnetic 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 TheorySome Applications of EMF Theory EM theory provides the basis for the design

and analysis of the following structures: Capacitors Electromagnets, Inductors Transmission Lines (e.g., coaxial cables) Transformers (for AC power conversion) Antennas (to transmit/receive EM signals) Waveguides (e.g., optic fibers) Cavities (e.g., microwave ovens)



Major Historical Developments in Major Historical Developments in Electromagnetic Field TheoryElectromagnetic Field Theory

Laplace, Gauss, Poisson, Stokes – Vector analysis

Coulomb – Described electrostatic force Faraday – Electromagnetic induction 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



Maxwell: 1831-1879 Lorentz: 1853-1928 Poincare: 1854-1912 Hertz: 1857-1894 Marconi: 1874-1937 Einstein: 1879-1955

Coulomb: 1736-1806 Laplace: 1749-1827 Ampère: 1775-1836 Gauss: 1777-1855 Poisson: 1781-1840 Ohm: 1789-1854 Faraday: 1791-1867 Stokes: 1819-1903

Some Important Figures in the Some Important Figures in the History of Electromagnetic TheoryHistory of Electromagnetic Theory



Outline of Topics to be CoveredOutline of Topics to be Covered1. Course Introduction2. Vector Algebra3. Vector Calculus4. Electrostatics5. Electric fields in media6. Boundary-value probs.7. [Exam I]8. Electric current9. Magnetostatics

10. (SPRING BREAK) 11. Magnetic force/torque12. Induction13. Maxwell’s equations14. [Exam II]15. Electromagnetic waves16. Review17. [Final Exam Week]


FAMUFAMU--FSUFSU College of EngineeringCollege of Engineering

TOPIC #1TOPIC #1

Vector Algebra and Vector Algebra and Coordinate SystemsCoordinate Systems



Vector AlgebraVector Algebra Basic Vector Concepts & Notation Vector Operations

Dot Product Cross Product

Orthogonal Coordinate systems Cartesian Circular-cylindrical Spherical



Vector NotationsVector Notations Definition: A vector is a geometric entity that has

both a magnitude and a direction. For our purposes:

Magnitudes (scalars) are identified with real numbers (+, −, or 0). Magnitudes are typically quantified in terms of physical units

e.g. Newtons of force, Volts/meter of electric field strength 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: or A A



Some Vector-Valued Quantities in Some Vector-Valued Quantities in Electromagnetic Field TheoryElectromagnetic Field Theory

Electric field vectors: E – Electric field strength (or intensity), dimensioned in

voltage/distance = force/charge. D – Electric flux density, dimensioned in charge/area.

Magnetic field vectors: B – Magnetic flux density, dimensioned in “magnetic charge”/area. H – Magnetic field intensity, dimensioned in force/“magnetic charge”

Others: A – The vector potential

From which electric and magnetic fields can be derived F – Force on a particle J – Current density R – Position or displacement vector And more: velocity, acceleration, angular velocity, torque, etc.



Vector MagnitudeVector 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

A

|A|



Basic Vector OperationsBasic Vector Operations Vectors can be added together:

Example: A = B + C Vectors can be multiplied by scalars:

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

BC

B B

A



Unit VectorsUnit Vectors A unit vector has magnitude 1.

Convention used in Sadiku, Schaum books: 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.

a

A AaA



Component FormComponent 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).

Einsteinsummationnotation

ˆ ˆ ˆ, ,i j k



Magnitudes in Terms of ComponentsMagnitudes 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.

2 2 2x y zA A A A A

Ax

Ay Az

A



Vector Magnitude – Practice ProblemVector Magnitude – Practice Problem What is the distance

from the origin to the point (3, 4, 0)? Use Pythagorean

Theorem What is |rP|, the

magnitude of the vector r(3,4,5) =3ax + 4ay + 5az?

(3,4,0)



Addition in Terms of ComponentsAddition 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



Basic Algebraic LawsBasic Algebraic Laws Vector addition and vector multiplication by scalars

obey commutative/associative/distributive laws: Commutative laws:

Vector addition is commutative: A + B = B + A 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” OperatorThe “Dot Product” Operator Takes two vectors and produces a scalar:

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.



Geometrical Interpretations Geometrical Interpretations of the Dot Productof the Dot Product

A•B is B = |B| times the magnitude of A’s projection onto (i.e., component along) B’s direction. Or vice-versa.

Or, area of parallelogramformed by rotating oneof the vectors 90° towards the other one. But, value gets negated if

we don’t overshoot.

A

B

A cos θ = A•B/B

θ

A

B

AreaA•BA′

AB



Cross ProductCross Product The cross product of vectors A,B is the vector

AB = (AB sin θ)an

Where AB are magnitudes, θ is the interior angle, and an is the normal (perpendicular) unit vector to A and B, by the “right-hand rule”

Magnitude is area of parallelogram formedby A and B.

A

B

Area |AB|



[ ]

( ) ( ) ( )x y z z y y z x x z z x y y x

x y z

x y z

x y z

i j k

A B A B A B A B A B A B

A A AB B B

A B

A B a a a

a a a

a

Some Properties of Cross ProductSome Properties of Cross Product The cross product is anticommutative:

A B = −(B A) It can be written in terms of components as a

matrix determinant:

Remember this rule:Forwards order (xyz): positive.Backwards order (zyx): negative.

x y z

x y z

x y z

A A AB B B

a a a

Modern notation. i,j,k{x,y,z}; [ ] = Anti-symmetrized sum over permutations of x,y,z



Some Other Concepts Not CoveredSome Other Concepts Not Covered See Sadiku text for information about…

The Scalar Triple Product A•(BC) The Vector Triple Product A(BC)

Intro Vector

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