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EEL 3472 Electromagnetic Fields I Spring 2009

EEL 3472 Electromagnetic Fields I Spring 2009. Instructor: Michael P. Frank Slide Module 1: Course Introduction. Outline of Lecture. Course Introduction: Some Perspective Importance of Electromagnetic Field Theory Some History of the Subject Outline of Topics

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EEL 3472 Electromagnetic Fields I Spring 2009

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  1. EEL 3472 Electromagnetic Fields ISpring 2009 Instructor:Michael P. Frank Slide Module 1:Course Introduction EEL3472 - Fields I, Spring 2009, Module 1

  2. Outline 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 EEL3472 - Fields I, Spring 2009, Module 1

  3. 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. EEL3472 - Fields I, Spring 2009, Module 1

  4. 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 • 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. EEL3472 - Fields I, Spring 2009, Module 1

  5. 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. EEL3472 - Fields I, Spring 2009, Module 1

  6. Some 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) EEL3472 - Fields I, Spring 2009, Module 1

  7. Major Historical Developments in Electromagnetic 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 EEL3472 - Fields I, Spring 2009, Module 1

  8. 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 History of Electromagnetic Theory EEL3472 - Fields I, Spring 2009, Module 1

  9. Course Introduction Vector Algebra Vector Calculus Electrostatics Electric fields in media Boundary-value probs. [Exam I] Electric current Magnetostatics (SPRING BREAK) Magnetic force/torque Induction Maxwell’s equations [Exam II] Electromagnetic waves Review [Final Exam Week] Outline of Topics to be Covered EEL3472 - Fields I, Spring 2009, Module 1

  10. TOPIC #1 Vector Algebra and Coordinate Systems EEL3472 - Fields I, Spring 2009, Module 1

  11. Vector Algebra • Basic Vector Concepts & Notation • Vector Operations • Dot Product • Cross Product • Orthogonal Coordinate systems • Cartesian • Circular-cylindrical • Spherical EEL3472 - Fields I, Spring 2009, Module 1

  12. Vector 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: EEL3472 - Fields I, Spring 2009, Module 1

  13. Some Vector-Valued Quantities in Electromagnetic 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. EEL3472 - Fields I, Spring 2009, Module 1

  14. 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 EEL3472 - Fields I, Spring 2009, Module 1

  15. Basic 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 vector0. • For any A, 0A = 0. C B A B B A EEL3472 - Fields I, Spring 2009, Module 1

  16. Unit 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. EEL3472 - Fields I, Spring 2009, Module 1

  17. 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 componentAxax (a vector) has the numerical coefficientAx (a scalar). Einsteinsummationnotation EEL3472 - Fields I, Spring 2009, Module 1

  18. 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. A Ay Az Ax EEL3472 - Fields I, Spring 2009, Module 1

  19. Vector 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) EEL3472 - Fields I, Spring 2009, Module 1

  20. 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 EEL3472 - Fields I, Spring 2009, Module 1

  21. Basic 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) EEL3472 - Fields I, Spring 2009, Module 1

  22. The “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. EEL3472 - Fields I, Spring 2009, Module 1

  23. Geometrical Interpretations of 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 θ B AB A cos θ = A•B/B A B AreaA•B A′ EEL3472 - Fields I, Spring 2009, Module 1

  24. Cross Product • The cross product of vectors A,B is the vectorAB = (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 Area |AB| B EEL3472 - Fields I, Spring 2009, Module 1

  25. Some 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. Modern notation. i,j,k{x,y,z}; [ ] = Anti-symmetrized sum over permutations of x,y,z EEL3472 - Fields I, Spring 2009, Module 1

  26. Some Other Concepts Not Covered • See Sadiku text for information about… • The Scalar Triple Product A•(BC) • The Vector Triple Product A(BC) EEL3472 - Fields I, Spring 2009, Module 1

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