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### L10_CurrentDensityOhm_post

Course: PHY 2049, Spring 2012
School: University of Florida
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Word Count: 1291

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10 Current Lecture Density Ohms Law in Differential Form Sections: 5.1, 5.2, 5.3 Homework: See homework file LECTURE 10 slide 1 Electric Direct Current Review DC is the flow of charge under Coulomb (electrostatic) forces in conductors Georg Simon Ohm was the 1st to observe and explain the lack of charge acceleration in metals electrons move with uniform averaged speed (drift velocity) the electrostatic force...

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10 Current Lecture Density Ohms Law in Differential Form Sections: 5.1, 5.2, 5.3 Homework: See homework file LECTURE 10 slide 1 Electric Direct Current Review DC is the flow of charge under Coulomb (electrostatic) forces in conductors Georg Simon Ohm was the 1st to observe and explain the lack of charge acceleration in metals electrons move with uniform averaged speed (drift velocity) the electrostatic force is provided by external sources: battery, charged capacitor LECTURE 10 slide 2 Current Density the current flowing through the cross-section s of a conductor is the amount of transferred charge Q per unit time I ( s ) Q , A=C s 1 t Q v V v sL v s vt , C I ( s ) V L n I ( s ) J n s where J n v v, A/m2 L vt s e e e e e e e e Q dQ I dt Q v v s, A t J current density, normal component the current density is a vector I V J n J an LECTURE 10 J v v, A/m2 slide 3 Current and Current Density the current I is the flux of the current density J I I ( s ) J n s J s dI J ds I J ds, A S Two cylindrical wires are connected in series. Current I = 10 A flows through the junction. The radii of the wires are: r1 = 1 mm, r2 = 2 mm. Find the current densities J1 and J2 in the two wires. LECTURE 10 slide 4 Charge Mobility charge velocity in a conductor depends on the charge mobility ve eE, v h hE, vi i E, m/s metals support electron current drift electron velocity in metals: vd = eE semiconductors support both electron and hole currents most electrolytes support both electron and ion currents in general plasmas support both electron and ion currents mobility may in general depend on E (nonlinear conductors) LECTURE 10 slide 5 Specific Conductivity 1 J e v e p v p ( e e p p ) E, p h, i note: e < 0 specific conductivity depends on the free-charge density and its mobility e e p p , S/m=( m)1 charge density depends on the number of charge carriers per unit volume (number density), e.g., e = eNe e 1.6022 1019 , C semi ( Ne e N h p )e in pure semiconductors Ne = Nh metal Ne ee LECTURE 10 slide 6 Specific Conductivity 2 typical carrier number densities, mobilities, conductivities (low frequency, below THz) e h Ne (m3) Nh (m3) (S/m) pure Ge 0.39 0.19 2.4x1019 2.4x1019 2.2 pure Si 0.14 0.05 1.4x1016 1.4x1016 4.4x10 4 Cu 0.0032 1.13x1029 5.8x107 Al 0.0015 1.46x1029 3.5x107 Ag 0.005 7.74x1028 Au 6.2x107 4.5 107 S/m Homework: What is the drift velocity of electrons in a Cu wire of length 10 cm if the voltage applied to both ends of the wire is 1 V. (Ans.: 3.2 cm/s. Wire may melt if too thin!) LECTURE 10 slide 7 Ohms Law in Point (Differential) Form J E Ohms law in circuits I GV V / R, A assume uniform current distribution in the cross-section of the conductor between points A and B V VAB E L AB E , l | L AB | I J s l use Ohms law in point form to arrive at Ohms law for resistors s s 1l I Js sE V G , R l l s G R 1 conductance/resistance of a conductor of length l, constant cross-section s, and constant current density distribution in s LECTURE 10 slide 8 General Expression for Resistance B B V R I A E dl , s J ds A E dl , R s E ds in homogeneous medium B A E dl , R E ds s s E ds , S G R 1 B A E dl 1 LECTURE 10 slide 9 DC Resistance per Unit Length L twin-lead line 1 L R 2 , A 1 R 2 , /m A I A I A coaxial line 1 L 1 L R , 2 2 b2 ) a (c 11 1 R 2 2 2 , /m a c b LECTURE 10 I cb I a slide 10 Homework: Resistance per Unit Length Find the resistance per unit length of a coaxial cable whose inner wire is of radius = a 0.5 mm and whose shield has inner radius b = 4 mm and outer radius c = 4.5 mm. Both the inner wire and the shield are made of copper (Cu = 5.8x107 S/m). ANS: 21 m/m LECTURE 10 slide 11 Conservation of Charge/Continuity of Current 1 s consider the current flowing through a closed surface I J ds [v] total positive flux corresponds to an outflow of charge (charge inside volume decreases) continuity of current (conservation dQencl of charge) in integral form I J ds NOTE THE NEGATIVE SIGN! dt s[ v ] in circuits we assume that no charge accumulates at nodes I J ds J ds J ds J ds 0 s[ v ] s1 s2 I1 s3 I2 Kirchhoffs current law follows from conservation of charge In 0 n I3 s3 I3 I1 LECTURE 10 s1 s2 s slide 12 I2 Conservation of Charge/Continuity of Current 2 apply Gauss (divergence) theorem to conservation of charge law I J ds Jdv s[ v ] v v J t dQinside d v dv dt dt v continuity of current (conservation of charge) in point form the equation of charge relaxation v v hm 1 v ( E) v E , also ( E) E t t D J hm v v 0 t (t ) 0e t 0e t / , / charge relaxation constant LECTURE 10 slide 13 Charge Relaxation consider an isolated conductor into which some charge Q0 is injected initially Coulomb forces push the charge carriers apart until they redistribute and settle on the surface the process continues until no free charge is left inside the conductor the time for this to happen is about 3 where / this is also the time required to discharge a charged capacitor through a shorting conductor LECTURE 10 slide 14 Charge Relaxation Illustrated 1 3s 0 1 exp(-t/) 0.8 1/e curve tangent at t = 0, intersects time axis at t = 0.6 d 0e t / dt 0.4 t 0 0 0.2 0 0 = 35 10 time (s) 15 20 Example: Calculate the time required to restore charge neutrality in Cu where = 0 and = 5.8x107 S/m. 3 8.8542 1012 T 3 3 0 / 4.6 1019 , s 5.8 107 LECTURE 10 slide 15 Joules Law in Differential Form consider sufficiently small volume v = sL where the E-field and the charge density v are constant since charge is moving with uniform drift velocity ud, the E-field does work on the charge (this work is converted into heat) We QE L we v, J F power is work done per unit time We w v QE L P pv e QE u d , W t t t power density QE u d p v E u d E J , W/m3 v Joules law in differential form: dissipated power per unit volume p E J (E E) | E |2 , W/m3 LECTURE 10 slide 16 Joules Law in Integral Form power dissipated in conductors P pdv E Jdv | E |2 dv, W v v v Joules law in circuit theory assume that in a piece of conductor, E does not depend on the cross-section, only J (or ) does, while J (or ) does not depend on the length assume that E and J are collinear P E J dsdL EJdLds p v dv SL P EdL Jds V I , W L S LECTURE 10 slide 17 You have learned: what current density is and how it relates to the total current that drift velocity of charge in conductors is proportional to the strength of E and the coefficient of proportionality is the mobility what specific conductivity is and how it relates J to E through Ohms law in differential (point) form how to compute the resistance/conductance of conducting bodies that charge is preserved and the rate of change of the charge density determines the divergence of the current density (continuity of current in point form) what charge relaxation is and how it depends on the permittivity and conductivity of the material how to find from the E field the dissipated power using Joules law LECTURE 10 slide 18
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University of Florida - PHY - 2049
Homework: Repeat for an infinite uniform line charge along the x-axis at z = 3, y = 0 (above ground plane).
University of Florida - PHY - 2049
Lecture 11Perfect Conductors, BoundaryConditions, Method of ImagesSections: 5.4, 5.5Homework: See homework fileLECTURE 11slide1Perfect Conductors 1 metals such as Cu, Ag, Al are closely approximated by theconcept of a perfect electric conductor
University of Florida - PHY - 2049
There are no hand-writtenexamples for Lecture 12LECTURE 12slide1
University of Florida - PHY - 2049
Lecture 12Dielectrics: Dipole, PolarizationSections: 4.7, 6.1 (in 8th ed.: 4.7, 5.7)Homework: See homework fileLECTURE 12slide1Electric Dipole and its Dipole Momentelectric dipole: two point charges of equal charge but oppositepolarity in close p
University of Florida - PHY - 2049
University of Florida - PHY - 2049
Lecture 13Boundary Conditions atDielectric InterfacesSections: 6.2 (in 8th ed.: 5.8)Homework: See homework fileLECTURE 13slide1BCs for the Tangential Field Components 1 we consider interfaces between two perfect ( = 0) dielectric regions use con
University of Florida - PHY - 2049
University of Florida - PHY - 2049
Lecture 14CapacitanceSections: 6.3, 6.4, 6.5 (8th ed.: 6.1, 6.2, 6.3, 6.4)Homework: See homework fileLECTURE 14slide1Definition of Capacitancecapacitance is a measure of the ability of the physical structure toaccumulate electrical free charge un
University of Florida - PHY - 2049
University of Florida - PHY - 2049
Lecture 15Magnetostatic Field Forcesand the Biot-Savart LawSections: 8.1, 9.1, 9.2Homework: See homework fileLECTURE 15slide1Magnetic Forces Review 1Ampres force law (motor equation)LFm IL(a I B), Nmagnetic fluxdensity vectorIBFmforce doe
University of Florida - EEL - 3111
238MICROWAVE AND RF DESIGN: A SYSTEMS APPROACH4.14SummaryIn this chapter a classical treatment of transmission lines was presented.Transmission lines are distributed elements and form the basis of microwavecircuits. A distinguishing feature is they
University of Florida - EEL - 3111
280MICROWAVE AND RF DESIGN: A SYSTEMS APPROACHSlow-Wave ModeThe third possible mode of propagation is the slow-wave mode [76, 77], forwhich the sectional equivalent circuit model of Figure 5-21(d) is applicable.This mode occurs when f is not so large
University of Florida - EEL - 3111
366MICROWAVE AND RF DESIGN: A SYSTEMS APPROACHmade in design, partly because of necessary simplications that mustbe made in modeling, but also because many of the material propertiesrequired in a detailed design can only be approximate. An example is
University of Florida - EEL - 3111
408MICROWAVE AND RF DESIGN: A SYSTEMS APPROACH7.16SummaryMany passive microwave elements exploit particular physical phenomena.Many make use of the characteristics of transmission lines. Each year newvariants of microwave elements are developed and
University of Florida - EEL - 3111
454MICROWAVE AND RF DESIGN: A SYSTEMS APPROACH8.10Exercises1. Repeat the analysis in Example 8.1 on Page419 for the case B = 1/L.2. Develop a two-element matching network forthe source/load conguration shown in theFigure below. The matching networ
University of Florida - EEL - 3111
COUPLED LINES AND APPLICATIONS9.13501SummaryCoupling from one transmission line to a nearby neighbor may often beundesirable. However, the effect can be exploited to realize a coupler thatdoes not have a lumped-element equivalent. This is one exampl
University of Florida - EEL - 3111
FILTERS591DRAIN BIASOUTPUTVGC FBSOURCE BIASFigure 10-90An active resonator circuit. After Karacaoglu and Robertson [167].the gate-source voltage. This induces a negative resistance that is adjustedthrough device selection and feedback capacitanc
University of Florida - EEL - 3111
10Note that Hx is out of phase with Eyand Hy is in phase with Ex. Explain.152224
University of Florida - EEL - 3111
1114161820
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= coshR= -sinhR89
University of Florida - EEL - 3111
23L06/sl. 36 [Exercise 4-6, Steer]8446sl. 17548
University of Florida - EEL - 3111
University of Florida - EEL - 3111
32
University of Florida - EEL - 3111
University of Florida - EEL - 3111
Lecture 1Radio Systems: an OverviewSections: parts from 1.6, 1.7, 1.8, 3.10Acknowledgement: Most diagrams and plots are from M. Steers book Microwaveand RF DesignWhy Do We Need Microwave Engineers?all wireless links use RF and microwave circuits and
University of Florida - EEL - 3111
Lecture 2Maxwells Equations andBoundary Conditions: ReviewAppendix D: all sectionsHomework: From Section 5.10 Exercises 4(a), 5, 6, 8Time-domain Maxwells Equations: Faradays LawFaradays law in integral formV tB E dl t dsCSCFaradays law in dif
University of Florida - EEL - 3111
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University of Florida - EEL - 3111
Lecture 5Plane Wave Reflection andTransmissionNormal Incidence (Review)xIncident wave:Ei ( z ) xE i (0)e 1z i ( z ) y 1 E i (0)e 1zH( 1 , 1 , 1 )EiSiiHy1Reflected wave:Er ( z ) xE r (0)e 1z r ( z ) y 1 E r (0)e 1zHSrErHr(2 , 2 , 2 )
University of Florida - EEL - 3111
Lecture 6Transmission LinesSections: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.10, 4.11Homework: From Section 4.15 Exercises: 1, 3, 4, 5, 7, 9Acknowledgement: Some diagrams are from M. Steers book Microwaveand RF DesignTransmission Lines You Know: Coaxial Lin
University of Florida - EEL - 3111
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University of Florida - EEL - 3111
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University of Florida - EEL - 3111
Lecture 9Scattering ParametersSections: 6.1, 6.2, 6.3, 6.4, 6.5, 6.6Homework: From Section 6.13 Exercises: 1, 2Microwave Networks: Voltages and Currentsthe theory of microwave networks was developed to enable circuitlike analysis methods which are si
University of Florida - EEL - 3111
HP8510Lecture 10Vector Network Analyzers andSignal Flow GraphsSections: 6.7 and 6.11Homework: From Section 6.13 Exercises: 4, 5, 6, 7, 9, 10, 22Acknowledgement: Some diagrams and photos are from M. Steers bookMicrowave and RF DesignVector Network
University of Florida - EEL - 3111
Network AnalyzerError ModelsandCalibration MethodsbyDoug RyttingPage 1This paper is an overview of error models and calibration methodsfor vector network analyzers.1Presentation OutlineNetwork Analyzer Block Diagram and Error ModelSystem Error
University of Florida - EEL - 3111
The Complete Smith ChartBlack Magic Design0.1270(+jX/Z451.00.8552.00.5060.440.1405REACTA75NCECOMPONENT0.450.040.0.30.84.0155.00.2INDUCTIVE20100.250.260.240.270.230.250.240.260.230.27REFLECTION COEFF
University of Florida - EEL - 3111C
ASSIGNMENT #1Radio Systems1. [10 points] The FM (frequency modulation) broadcast band in North America extends from 88MHz to 108 MHz. Standard FM receivers use an IF (intermediate frequency) of 10.7 MHz. What isthe required tuning range of the LO (loc
University of Florida - EEL - 3111C
Assignment 1 SolutionQuestion 1f LO ,min 88 10.7 77.3 MHz ,f LO ,max 108 10.7 97.3 MHz .2Question 2There are more sky wave happen at night for AM signals. Thereceivers can receive signals that they could not receive at day timefrom AM stations loc
University of Florida - EEL - 3111C
ASSIGNMENT #2Maxwells Equations and Boundary Conditions1. [40 points] Answer the questions from L02 Maxwells Equations and Boundary Conditions: slide 4 [4 points] slide 6 (all 3 questions) [6 points] slide 8 [5 points] slide 14 [2 points] slide 15
University of Florida - EEL - 3111C
Assignment 2 SolutionQuestion 1slide 4:magnetostatic Amperes law: H J H Jsince () 0, J 0 twhich is inconsistent with conservation of chargecorrection: H J DtDt 0 J D J tt J twhich is consistent with conservation of charge H 0 J 2Quest
University of Florida - EEL - 3111C
ASSIGNMENT #3TEM Waves1. [40 points] Answer the following questions from Lecture 3 TEM Waves: slide 4 [8 points] slide 7 [11 points] slide 11 [5 points] slide 13 [8 points] slide 25, both questions [8 points]2. [30 points] A uniform plane wave tra
University of Florida - EEL - 3111C
Assignment 3 SolutionQuestion 1slide 4:2Question 1 , contdslide 7:2 ET , x 2 ET , x022xy(1)2 ET , y 2 ET , y022xy(2)ET , x ET , y0xy(3)take derivative of (3) with,x y2 ET , x 2 ET , y02xyx(4)2 ET , x 2 ET , y02xyy(5)
University of Florida - EEL - 3111C
ASSIGNMENT #4Polarization of Plane Waves1. [15 points] The E-field vector of a TEM wave is given byE E0e jkr .The direction of propagation is u k / | k | and is the medium intrinsic impedance. Express thecomplex Poynting vector S of the wave in terms
University of Florida - EEL - 3111C
Assignment 4 SolutionQuestion 111S E H E u E2111u E E E E u211 2 1 u E E E u 02S E H H u H H u H 0= u H H H u H u H2 u H211 2uE22Question 23Question 34Question 45
University of Florida - EEL - 3111C
ASSIGNMENT #5Reflection and Transmission1. [35 points] A plane wave is incident from air (medium 1) upon copper (medium 2), 2 =5.8x107 S/m, at an angle of incidence i = 80. Calculate the complex angle of transmission t (theapproximate answer t 0 is no
University of Florida - EEL - 3111C
Assignment 4 SolutionQuestion 1(1)(2)2Question 1, contdsubstitute: f 1 GHz 2 f0 ;0n1 1; r1 1; 0 8.854 1012 F/m; 2 5.8 107 S/m;i 80 ;into (1) and (2), obtain:t (2.16 105 j 2.16 105 ) rad 2 1 j 0 22 (1 j) 4.78 105 rad/m3Question 1, cont
University of Florida - EEL - 3111C
ASSIGNMENT #6Quasi-TEM Transmission Lines1. Solve the problems in Lecture 6 Transmission Lines, in(a) [2 points] slide 2(b) [10 points] slide 10(c) [2 points] slide 12(d) [2 points] slide 14.2. [24 points] Using the Kirchhoff laws, show that the fo
University of Florida - EEL - 3111C
Assignment 6 SolutionQuestion 1(a)(b)EV1ln(b / a) Ed (*)V1ln(b / a) aVa ln(b / a) V a ln(b / a) EdPmax 2 b0a(*)2E d d2V2a 2 ln(b / a)2 2 d d(*) 2 b0a ln(b / a) Ed 2 d d 0 a 2 ln(b / a)2 2(*) 2 b2Ed a2bln( )a2Questi
University of Florida - EEL - 3111C
ASSIGNMENT #7Voltage and Current Waves in Quasi-TEM Transmission Lines1. [4 points] Solve the problem in Lecture 6 Transmission Lines, slide 36.2. [4 points] Solve the problem in Lecture 6 Transmission Lines, slide 39.3. [4 points] Solve the problem i
University of Florida - EEL - 3111C
Assignment 7 SolutionQuestion 1lgl0.13 0.482 / 2 / 30 2electrical length: l 0.1 30 3 rad/m2Question 2 d E1,2 E0 cos (0 )t (0 ) z d d E0 cos (0t 0 z) t z d AB E0 cos A cos B E0 sin Asin B d d E0 cos(0t 0 z) cos t z E0 sin
University of Florida - EEL - 3111C
ASSIGNMENT #8Impedance Transformation by a Transmission Line1. Answer the following questions in Lecture 7 Terminated Transmission Lines:a. slide 7 [5 points],b. slide 8 [5 points].2. [50 points] Write a MATLAB code which can calculate in a given fre
University of Florida - EEL - 3111C
Assignment 8 SolutionQuestion 1a.I ( z) 1 j zV0 e (1) V0e j z Z0 V0 j z j z ee Z02V0cos z Z0b.I ( z) 1 j zV0 e (1) V0e j z Z0 V0 j z j z j ee Z02V0jsin z Z02Question 23Question 2, contd4Question 35Question 3, contd6
University of Florida - EEL - 3111C
ASSIGNMENT #9Impedance Matching by a Double-stub Tuner1. [2 points] Answer the question in Lecture 8 The Smith Chart and Basic Impedance MatchingConcepts, sl. 9.2. [6 points] Answer the questions in Lecture 8, sl. 11.3. [2 points] Answer the question
University of Florida - EEL - 3111C
ASSIGNMENT #10Scattering Parameters and Preparation for Practice Work1. [34 points] Wilkinson Power DividerBuild the following 3-port circuit in AWR Design Environment and simulate in the frequency rangefrom 1.5 GHz to 2.5 GHz.Fig. 1. Schematic of Wi
University of Florida - EEL - 3111C
AWR DESIGN ENVIRONMENT 9.04 GUIDEThe AWR Design Environment (AWRDE) suite comprises three powerful tools that can beused together to create an integrated system and RF or analog design environment: VisualSystem SimulatorTM (VSS), Microwave Office (MWO)
University of Florida - MAC - 3200
University of Florida - MAC - 3200
University of Florida - MAC - 3200
University of Florida - MAC - 3200
University of Florida - MAC - 3200
University of Florida - MAC - 3200
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University of Florida - MAC - 3200