# 329fa08notes - ECE 329 Lecture Notes Fall08 Erhan Kudeki 1...

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ECE 329 Lecture Notes — Fall08, Erhan Kudeki 1 Vector felds and Forces Fundamental building blocks of matter, electrons and protons , are elec- trically charged and exert attractive/repulsive forces upon one another (depending on algebraic signs of the charges involved) — in general, all electric and magnetic phenomena observed in nature can be attributed to the existence and motions of electric charge carriers . In classical descriptions, individual charge carriers (e.g., electrons and pro- tons) are modeled as point particles having: – mass m and charge q , conserved and quantized as integer multiples of proton charge e 1 . 6 × 10 - 19 C (coulombs) in all reference frames, well de±ned locations r =( x,y,z ) , and trajectories 1 r = r ( t ) obeying Newton’s 2nd Law of motion , m ¨ r = F , where ¨ r d 2 r dt 2 denotes a particle acceleration and F the force applied on the particle. Electric and magnetic components of force F on a point charge q are pro- portional to q and expressed in terms of an electric Feld E and magnetic Feld B produced by distant charge carriers — speci±cally Lorentz force F = q ( E + v × B ) , which is known as Lorentz force , where v ˙ r = d r dt is the velocity vector of charge q , – electric Feld E is by defnition the vector force per unit charge when q is stationary (i.e., v =0 ), Maxwell’s eqns in diferential Form (used starting in Chpt 3): ∇· E = ρ ± o B ∇× E = - B ∂t B = μ o J + μ o ± o E with, in MKSA units, μ o 4 π × 10 - 7 H m , and ± o 1 μ o c 2 , where c 3 × 10 8 m/s is the speed oF light in Free space. (In Gaussian-cgs units B c is used in place oF B , while ± o = 1 4 π and μ o = 1 ± o c 2 = 4 π c 2 .) – magnetic Feld B describes by defnition an additional force per unit charge which is experienced when q is in motion with a ±nite v within the frame of reference where E is measured (typically called the “lab frame”). E and B are related to and produced by distant charge carriers and their motions as described by Maxwell’s equations (see margin), which e²ectively reduce, in static cases (i.e., non-time varying cases), to Coulomb ’s and Biot-Savart laws (to be reviewed shortly). In the MKSA system where q is measured in coulombs (C) units, E is in Nt/C=V/m and B in V.s/m 2 =Wb/m 2 =T, where Nt, V, Wb, and T are abbreviations for newtons, volts, webers, and teslas, respectively. Particle positions r , velocities ˙ r , and accelerations ¨ r , as well as forces F and ±elds E and B are described in terms of 3D vectors . In Cartesian coordinates such vectors and vector functions are expressed in terms of mutually orthogonal unit vectors ˆ x , ˆ y , and ˆ z as in r = x ˆ x + y ˆ y + z ˆ z ( ) and E = E x ˆ x + E y ˆ y + E z ˆ z ( E x ,E y z ) etc., where 1 Quantum models (as opposed to classical) describe the likelihood of detection of charge carriers at locations r (as opposed to their speciFc trajectories), depending on applied force Felds F and related potentials — quantum approach remedies the shortcomings of classical models based on Newton’s dynamics at atomic and smaller scales.

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329fa08notes - ECE 329 Lecture Notes Fall08 Erhan Kudeki 1...

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