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# General Physics II PHY 212

Syracuse

GPA 3.64

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This 101 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 212 at Syracuse University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/225626/phy-212-syracuse-university in Physics 2 at Syracuse University.

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Date Created: 10/21/15

Lecture 31 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Review of Last Lecture 1 Coulomb s Law Electric eld by a point charge 2 Principle of linear superposition eld by a distribution of charges 3 Example eld of a dipole Exmaple eld of an in nitly long charged line Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley d The derivation of electric eld by a charged line in the last lecture A It was clear I understood it completely B I got the main part and with a little help I can repeat the calculation myself C I got the basic idea but still do not understand how the calculation was done D I did not understand anything E I missed the class Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley ProblemSolving Strategy The electric eld of a WN 7 8 continuous distribution of charge Draw the charge distribution and the point P Set up a coordinate system such that the calculation is as simple as possible Divide the charge distrbution into small pieces each carrying charge AQ Write down the eld AE due to AQ at P using Coulomb s Law Usually we write down each component of AE Express all distances and angles in terms of coordinates X y z also express AQ in terms of X y z or r 6 as well as AX Ay etc Now you have AE as function of coordinates and AX Ay etc Summing up AE due to all pieces you get the total eld created by the whole charge distribution Tranform the sum into integral AX gt dX and 2 gt Integral Calculate the integral Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley ProblemSolving Strategy The electric field of a continuous distribution of charge A5555 Check that your result is consistent with any limits for which you know What the eld should be Copyright 2008 Pearson Edieauon In publishing as Pearson AddsomWesley A gewgu Ohm ed 1 V J x 39 W 46 mm quot gtlt P W fog7ff M 4 9 X X 37 M Ag KAQC LL 34 r Y w m s 6amp4 Z quot7 A W 2 A AE E x 394 12 AEX 2 0930 2quot fk A39A g W x 4 xz39ry ff52 m UtaM 14 Tim owlxoLeM E Z A EX 7quot z 39 68 yl3 FIGURE 2114 The electric field of an infinite line of charge The eld points straight away from theline at all points 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 lt lt lt 474 4 gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt p p p gt gt gt In nite line of charge The eld strength decreases with distance Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Surface Charge Density The surface charge density of a twodimensional distribution of charge across a surface of area A is defined as b Charge Q on a surface of area A The surface charge density is 1 QA Q AreaA 39 39 The charge in a small Surface charge dens1ty w1th m M is Ag 2 quotAA units Crnz is the amount of charge per square meter Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley A piece of plastic is uniformly charged with surface charge density 71 The plastic is then broken into a large piece With surface charge density 772 and a small piece with surface charge density 773 Rank in order from largest to smallest the surface charge densities 171 to 173 A 7722773gt771 B 771 gt772gt773 C 771 gt7722773 D 773 gt772gt771 E 771 27722773 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley A piece of plastic is uniformly charged with surface charge density 171 The plastic is then broken into a large piece with surface charge density 172 and a small piece with surface charge density 173 Rank in order from largest to smallest the surface charge densities 171 to 173 A 7722773gt771 B 771gt772gt773 C mgt712713 D 713gt712gtm E quot10203 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley A Disk of Charge FIGURE 2117 Calculating the on axis field of a charged disk The charge of the ring is AQI Field due to ring i Ei Ring 139 with radius rt and area AAi If we unroll the ring it looks as shown below Area AAI 27riAr Ar vi 2117 I Copynght 2008 Pearson Education Inc publishing as Pearson AddisonWesley M VJ CL W M933 M3 AG 7 i 2 f it El ff yly N958 2 45 AE givaCQA d2 gd QueUL meud 60 122 Z CLKIQ PampV39h a dish int mull n n AM WW 2 sf L ng quotS Q M m w 7 I g A M a 0 A C L M 39 Vadius V 39 f7 w FELLA 14 0 a NORM Yquot 10 CWv WA 1312 5 392 7 iquot 394 W e 14 a 25 92 Y1 Z oz EVA W31 mun ma 7 dt sL A 1 2315 1 Srngfdr 2 60 z YL3 O A Disk of Charge The onaXis electric eld of a charged disk of radius R centered on the origin with axis parallel to Z and surface charge density 11 Q7rR2 is z z2 R2 NOTE This expression is only valid for z gt 0 The eld for Z lt 0 has the same magnitude but points in the opposite direction 7 Ea l dlhk4 2EO Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley A Plane of Charge The electric field of an infinite plane of charge with surface charge density 17 is 7 Eplane i constant 260 For a positively charged plane with 17 gt 0 the electric field points away om the plane on both sides of the plane For a negatively charged plane with 17 lt 0 the electric field points towards the plane on both sides of the plane 39 Iquot quotum Mu Pearson Addi onWesley FIGURE 2118 Two views of the electric field of a plane of charge Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Rank in order T a T from largest to T smallest the electric eld 1 strengths Ea t0 Ee at these ve points 1 1 near a plane of charge A EagtEc gtEb gtEe gtEd B EaEb Ec EdEe C EagtEbEc gtEdEe D Eb Ec EdEegtEa E E6 gtEdgtEc gtEb gtEa Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Rank in order T from largest to T smallest the electric field 1 l strengths Ea to Ee at these five points near a plane of ch a C Copyright 7 2004 Peanon Education inc publishing it Addison Wesley A EagtEcgtEb gtEegtEd VB EaEbEcEdEe C EagtEbEC gtEd Ee D Eb EC Ed EegtEa E E6 gtEd gtEc gtEb gtEa T 1 1 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley The ParallelPlate Capacitor The figure shows two electrodes one with charge Q and the other Q FIGURE 2720 A parallel plate capacitor placed facetoface a A distance d apart AreaA o This arrangement of two electrodes charged equally but oppositely is called a 0 Q parallelplate capacitor Capacitors play important roles in many electric circuits Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Copyright 2008 HGU E 272 The eectr eld of capacfon a Idoal capacitor i TIiS i aectrod The fie is mustang poin ia rm that pogi ivo to t a aegaiva alactroda Pearson Education 1110 publishing as Pearson AddisonWesley t S i Edcoeiv paz Z 0 790 N0 FIGURE 2722 The electric eld of a ca pa cito r 3 Ideal capacitor gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt This is an edge View of the electrodes The eld is constant pointing from the positive to the negative electrode Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Lecture 142 Electromagnetic waves Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Maxwell s Equations Eda Q Gauss s law 50 3 39 da 0 Gauss s law for magnetism a dCIDm Eds Faraday s law dz 6 a 6171e B ds Molthmugh 0114 Ampere Maxwell law F q E q 17 X I Lorentz Force Law Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Maxwell Equations Differential Forms 7 Co VBCI 8B E V X B 0 0506 Electric Gauss Law Magnetic Gauss Law Faraday s Law AmpereMaxwell s Law Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Electromagnetic Waves Maxwell using his equations of the electromagnetic field was the first to understand that light is an oscillation of the electromagnetic field Maxwell was able to predict that 0 Electromagnetic waves can exist at any frequency not just at the frequencies of visible light This prediction was the harbinger of radio waves 0 All electromagnetic waves travel in a vacuum with the same speed a speed that we now call the speed of light V 1 em 300 X 108 ms c V EOMO Copyright 2008 Pearson Education 1nc publishing as Pearson AddisonWesley FIGURE 3519 A sinusoidal electromag netic wave l A sinusoidal wave with frequencyfzmd wavelength A travels with wave speed vw Ex Ez 07 Ey E0 Sin 27T lt ft 3 A 39 Wavelength A i i i J E BagiByiO7 BZiBOSIH27TltX ft E A wavelength f frequency C f is the speed of light a AI l 2 E and B are pelpendicular to each other and to A 39 lhe direction of 3 E and E are in phase travel The elds That is they have have amplitudes matching crests EU and B troughs and zeros Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Properties of Electromagnetic Waves Any electromagnetic wave must satisfy four basic conditions 1 The elds E and B and are perpendicular to the direction of propagation vemThus an electromagnetic wave is a transverse wave 2 E and B are perpendicular to each other in a manner such that E X B is in the direction of vem 3 The wave travels in vacuum at speed vem c 4 E CB at any point on the wave Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Energy Density and Energy Flow 1 l u uE uB 60E2 32 2 2M0 1 l B E7 c xeouo l gt m uE 560E27 u 60E2 Energy density The energy flow of an electromagnetic wave is described by the Poynting vector defined as gt 1 a gt E E X B M0 The magnitude of the Poynting vector is Energy transferred by the wave per S M0 CEOE cu un1t area and per un1t time The intensity of an electromagnetic wave whose electric field amplitude is E0 is P 1 06 E02 70E02 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley 2 2 6 E019 L Fgt 1 SJBX u A VojlwvL 39 1 0E 1 2 60 E0 L L Z SMG quotf39CODG 2 Radiation Pressure Momentum transferred to an object that absorbs EM waves Energy sbsorbed Ap C The radiation pressure on an object that absorbs all the light is E P 1 prad A C C where I is the intensity of the light wave The subscript on pmd is important in this context to distinguish the radiation pressure from the momentum p Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Science Fiction Solar Sailing CcnynghKQ 2003 Pearson Edunslmm Vnz publishing as Pearson Addlsunrwasley Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley FIGURE 3527 The plane of polarization is the plane in which the electric field vector OSCillates Light re ected by at surfaces are usually a Vertical polarization horizontally polarized V Sun glasses are designed to ltered out Plane of polarization waves With this polarization E Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley FIGURE 3523 A polarizing filter The polymers are parallel to each other 39 Polaroid The electric eld gnly the coinponent of of unpolarized light E perpendicular to the oscillates randomly polymer molecules in all directions is transmitted Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Malu y The incident light is polarized at angle with respect to the 4 polarizer s axis 3 0 39 19 j 39139 E0 Polarizer axis Only the component of E in the direction of the axis is transmitted mrvaa 2 Pearson Edward irct awashing 3 Pearson msm iesiw Suppose a polarized light wave of intensity IO approaches a polarizing lter 6 is the angle between the incident plane of polarization and the polarizer axis The transmitted intensity is given by Malus s Law Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Malu s s Law If the light incident on a polarizing lter is unpolarized the transmitted intensity is lt1gtlt05 7 1w Imm ed 2 510 incident light unpolarized In other words a polarizing lter passes 50 of unpolarized light and blocks 50 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley W0 Polarizers a Unpolarized light 2 10 Polarizer Analyzer chyrizghz is 2038 Pearso E acation incquot pumath as Pearson AddisonaWesiey W 3 m 39 was its E I if la SEE Hi ii V H w m r r M satw naps193 V 6 0 6 45 Copyright 2608 Pearson Educaxian 1m gub shing as Parson AddisomWesSey Copyright AUUO rearson nuucauon 1110 puousmng as rcarson Auulson weswy Electromagnetic Spectrum THE ELECTRO MAGNETIC SPECTRUM Wavelength l39rIicIrong Radio Microwave infrared Visible Uitraviolet XRay Gamma Ray I I I I I I I IV I I h I I I l 105 103 105 106 108 10quot 10quot Frequency Hz 104 103 10393 i015 10 10395 102 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Lecture 152 AC Circuits Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley AC Sources and Phasors FIGURE 36 An oscillating emf can be represented as a graph or as a phasor diagram 3 The emf oscillates 5 Peak S Sums wt z 39 0 emf I It V T V H 50 The oscillation period is T lf 27710 Copyright 2008 Pearson Education 1nc publishing as Pearson AddisonWesley AC Sources and Phasors FIGURE 36 An oscillating emf can be represented as a graph or as a phasor The phasor ro FateS CCW diagram tracrng out a Circle b 575 50 C08 cut The length of quot39Tl pll LISOI rotates the phusor is SUM E ccw ul angular r frequency m In one period the phase angle I chan es b I The phase 9 y Erquot mg1 is of 27 wT27r gt T w quot 391 The tip of The instantaneous emf NOt jUSt for EMF SI thc pliusor goes 5 value Eucoswl is the current Charge on Cruoscmating once around the projection of the phasor circle in time T onto the horizontal axis quantltles generally Copylight 2008 Pearson Education Inc publishing as PeaIson AddisonWesley AC Resistor Circuits In an AC resistor circuit Ohm s law applies to both the instantaneous and peak currents and voltages FIGURE 363 Instantaneous current iR through a resistor The instantaneous current in the resistor The instantaneous resistor voltage is UR iRR The potential decreases in the direction of the current Copyright e 2008 Pearson Education Inc publishing as Pearson AddisonWesley AC Circuits With single resistor The resistor voltage vR is FIGURE 364 An AC resistor circuit giVel l This is the currcnt direction when 8 gt 0 A half cycle later it will be in the opposite direction vR VRcoswt AV source 7 where VR is the peak or maximum voltage The current through the resistor is vR VRcoswt R R iR Reoswt where R VRR is the peak current Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley An AC resistor circuit phasor diagram curren l39 pkqsor V V365 VKC S not 1 m it In St 39 it u 39m PMSE Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley AC Capacitor Circuits FIGURE 361 An AC capacitor circuit rchoff s Loop Law Instantaneous voltage across single capacitor is the same as driving EMF 21015 V0 cos wt V0 50 a The instantaneous CLliTcnt to and from the capacitor ltTgt The instantaneous capacitor What is the current voltage is t c qC The potential 39 decreases from to S 50005 cat l C In Capacitor Circuits We can relate voltage across capacitor to stored charge 9075 022005 2 CV0 cos wt d t i005 3 wCVC sinwt 7139 2075 wCVC cos cut I The current in a single capacitor circuit driven by an alternating EMF leads the applied EMF by g radians Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley A Capacitor Phasor Diagram V V as art wsor39 lt39 c We V K39 L la ccf quot 1 C03 we 439 g wt I N CVg Copyiight 2008 Pearson Education Inc publishing as Pearson AddisonWesley Capacitive Reactance The capacitive reactance Xc is defined as The units of reactance like those of resistance are ohms Reactance relates the peak voltage VC and current 1c V0 0 wCVC 01 V0 2 0X0 X0 BEWARE Reactance differs from resistance in that it does not relate the instantaneous capacitor voltage and current because they are out of phase vc 76 iCXC Reactance eq s tell you the length of the capacitor current phasor Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Capacitive reactance decreases with increasing frequency Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley An Inductor AC circuit FIGURE 3614 Using an inductor in an AC Circuit Kirchoff s Loop Law a The instantaneous CUI I CHI Instantaneous voltage across single IhFOUgh If 1de capacitor is the same as driving EMF iL gt vLt VL cos wt L VL80 Tlle Instnntallfous lilductor What is the current voltage IS vL 7 LdIdf b S 9000sz Q Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley RC Filter Circuits RC filter circuits R AAAA VVVV J C Vc VC SOXCvRZ Vcagoasweo A lowpass lter transmits low frequencies and blocks high frequencies VVVV Va VR EORVRZ x VR gtansw gtOO A highpass lter transmits high frequencies and blocks low frequencies Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Inductor Circuits diL t vLt VL coswt L dt ftdz39 z39 t z39 0 VL ftdtCoswtl VL Sinwt 0 L L L L 0 wL V V iLt w Z sinwt w Z COS wt The current in a single inductor circuit driven by an alternating EMF lags the applied EMF by Z radians 2 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Phasor Diagram for Inductor Circuits VJ VL C 03 it VL LIE IL C GS wt voH uac VMsor 1quot t IL 12 x L cu r f n w phar 11 1 5 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Inductive Reactance V 7r 7r 2Lt cos wt IL cos wt The inductive reactance XL is de ned as X L E wL Reactance again relates the peak voltage VL and peak current IL I E or V I X L XL L L L NOTE Inductive reactance differs from resistance in that it does not relate the instantaneous inductor voltage and current because they are out of phase That is vL g iLXL Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Inductive Reactance FIGURE 3615 The Inductive reactance as a function of frequency XL The reactance increases with increasing frequency Inductive reactance XL wL Pear nnF duca nn an 39 39 Addi quotWesley AC RLC circuits FIGURE 3617 A series RLC circuit Cupynghua 2m Fearsun Educauun Inc pubhshmg as Fearsun Admsuanesley The Series RLC Circuit The impedance Z of a series RLC circuit is de ned as Impedance like resistance and reactance is measured in ohms The circuit s peak current is related to the source emf and the circuit impedance by 5 Z I Z is at a minimum making I a maximum when XL XC at the circuit s resonance frequency 10 1 VLC Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Driving RLC circuit at resonance FIGURE 3619 A graph of the current I versus emf frequency for a series RLC circuit 1 a The maximum 39 current is EaR R 8 l R 25 O R 50 Q I I I I U 0 m0 2wo Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Power in AC Circuits The rootmeansquare current ms is related to the peak current IR by 1R rms 2 Similarly the rootmeansquare voltage and emf are VR 50 Vrms Erms W The average power supplied by the emf is PSOUI CB Irmsgrms Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Important Concepts Basic circuit elements Resistance Element iand v reactance I and V Power Resistor In phase R is fixed V R Vrmsllms Capacitor ileads v by 90 XC lwC V IXC Inductor i lags v by 900 XL wL V XL 0 For many purposes especially calculating power the rootmeansquare rms quantities Vrms VVE 1m m5 SW SQVi are equivalent to the corresponding DC quantities Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Applications Series RLC circuits I SOZ where Z is the impedance 86 z VR2 XL XC2 L VR R vL XL vC IXC C T When a m0 l V LC the resonance frequency the current in the circuit is a maximum 1max SOR In general the current i lags behind S by the phase angle tan 39XL XCR The power supplied by the emf is Psource Imm 39mscos d where cos d is called the power factor The power lost in the resistor is PR 2 1 me 1ms2R Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley What are covered in the second exam 1 Gauss Law using it to calculate electric eld 2 Electric potential energy and potential relation with electric eld 3 Charges and dipoles in electric eld 4 Field lines and equipotential surface 5 Capacitors and battery circuit of capacitors capacitance energy and equivalent capacitance 6 Current current density drift velocity Resistance and Ohm s Law 8 Kirchhoff s loop rule and junction rule Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Lecture 81 Current and Resistance Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley The Electron Current FIGURE 315 The electron current The sea of electrons ows through a wire at the drift speed vd much like a fluid owing through a pipe Electrons Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Electron Current as Function of Field A cross section Wire at time t t I d ot the wne The sea ofelectrons is moving to the right with drift speed d There are nc electrons per cubic meter of wire Copyrighl o 2005 Pearson Education inc publishing as Pearson AddvsonWesiey N D neAVd Wire at time I At Cross section area A 21d 4 w Ar The sea ol eiectrons has moved forward distance A39 l d Al The shaded volume is V A Ax BET Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley quotWm xxx 7 39 s 39 Irvquot9 d A 7 L t Conductiomelec tron U I 39 density in metals I Electron i 4 I W a quot n 5 i Metal density 111 3 Aluminum 60 X 71028 1 7 A a gopper 85 X 10quot e g8 60 K L A 00 Iron Md 3 M Gold 59 X 1025 Q K to K0 Silvsr 58 X 1028 c LN X WSW5 Copyright 2008 Pearson Education 1110 publishing as Pearson AddisonWesley 3 I r M 1 WA U4 7 9sz x wk v0 alkxo M lily Current If Q is the total amount of charge that has moved past a point in a wire we define the current I in the wire to be the rate of charge ow dQ E current is the rare at which charge ows The SI unit for current is the coulomb per second which is called the ampere l ampere lA l Cs The conventional current I and the electron current ie are related by A cross section quot of the wire V o o I V tie 0 2 id There are nC electrons The sea ot electrons Wire at time t AI Wire at lime I Cross section area A per cubic meter of wire is moving to the right with drift sneed r The sea ol39eledtrons has moved N ward distance Ar lquot AI Q 6 e shaded volume is V Z A Ax ere 12 Ar Ar Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley I l 39 l 83949 2 1 6X O 9C K 1 x 0 61 quotSf iqA Copyright 2008 Pearson Education 1110 publishing as Pearson AddisonWesley FIGURE 5115 The current I is opposite the direction of motion of the electrons But be careful electron is negatively charged So the direction of current is 39 the opposite of the ow of electrons This is an awkward result ofthe convention of the sign of charges Unfortunately we will have to live with it The nu l39cnl In In the dimnun llml positive hinges would quotway is in ihe lilecrinn of E The electron in 39 IS quotI ma things cal opposite to E mid 1 qu ahm Tn The Current Density in a Wire The current density J in a Wire is the current per square meter of cross section N J current density Z neevd The current density has units of Arnz Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Reminder Kirchhoff s Loop Law For any path that starts and ends at the same point AVloop 0 Stated in words the sum of all the potential differences encountered While moving around a loop is zero The origin of this law is that electrostatic force is conservative The work done by the electric force is independent of the path Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Kirchhoff s Junction Law Law of conservation of current The current is the same at all points in a current carrying wire Really conservation of charge and only applies to stationary current For a junction the law of conservation of current requires that E I in 2 I out Kirchhoff s junction law The current owing in must equal to the current owing out Otherwise charge accumulate at the junction 9 nonstationary current Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley j Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWeslay Conductors in and out of equilibrium Conductors in Equilibrium Vanishing electric field potential constant Electron do not move on average drift velocity vanishes Conductors out of equilibrium Electrons drift Current ow inside conductors Electric potential not constant over the space Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley jquot I Q39Vl 39Vol 3 elnezu E 1 r 397 Vol 5 L 39 gt 0 M MARC m Lliesistivity is de ned as the inverse of conductivity and tells us how reluctantly the electrOns move in response to an electric eld Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley TABLE 312 Resistivity and conductivity of conducting materials Resistivity Conductivity Material 9 m 0 1 mquot1 Aluminum 28 x 10 8 35 x 107 Copper 17 x 10 8 60 x 107 Gold 24 X 10 8 41 X 107 Iron 97 x 10 8 10 X 107 Silver 16 X 10 8 62 X 107 Tungsten 56 X 10 8 18 X 107 Nichronie 15 X 10396 67 X 105 Carbon 35 X 10 5 29 X 104 Nickel chromium alloy used for heating wires Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley QUESTION Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley AmS LCUA FIGURE 31139 The current I is related to the potential di rence AV A petenxtiel difference Creates an electric eld inside conductor and causes charges to ow thrgugh i 39 L Equipotential surfaces are perpendicular e the electric field M l AVZIK j M e Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley H Ohm s Law FIGURE 3119 The current I is related to the potential difference AV The potential difference creates an electric eld inside the conductor and causes charges to flow thrpugh il L Area A Equipetential surfaces are perpendicular to the electric eld I 39 R S law Copyright 2008 Pealson Educalion Inc publishing as Pealson AddisonWesley Resistance and Ohm s Law The resistance of a long thin conductor of length L and crosssectional areaA is M R A The SI unit of resistance is the ohm 1 ohm 1 2 1 VA The current through a conductor is determined by the potential difference A V along its length AV I Y Ohm s law Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Ohm s Law Ohm s law is limited to those materials Whose resistance R remains constant or very nearly so during use The materials to which Ohm s law applies are called ohmic The current through an ohmic material is directly proportional to the potential difference Doubling the potential difference doubles the current Metal and other conductors are ohmic deVices Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley FIGURE 3122 Current versus potentiaI difference graphs for ohmic and nonohmic materials a Ohmic material The current is directly proportional to the 39 potential difference The resistance is slope AV b Nonohmic materials This curve is not linear and doesn t have a wellde ned slope v39 Diode Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Lecture 61 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Review of Last Lecture 0 Electric Potential Energy and Energy conversation 0 Point charge in uniform eld 0 Dipole in uniform eld The Electric Potential Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Rank in order from largest to smallest the potential energies Ua to Ucl of these four pairs of charges Each symbol represents the same amount of charge to 0 o5o 03931quot 3 b C d Agtq Bgtm CQgt Dgtgtgt Egtmgt Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley Aproton is 100V 0V 100V released from rest at point B where the A B C potential is 0 V Afterward the proton Cupyrighl E 3004 Pearson Educuliun Inc publishing Lb Addison N39eslcy A moves toward A with a steady speed B moves toward A with an increasing speed C moves toward C with a steady speed D moves toward C with an increasing speed E remains at rest at B Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley I Rank in order from largest to 339 smallest the c potentials Va to Ve l l d 1e at the points a to e A VdVegtVCgtVaVb B VbVCVegtVaVd C VaVbVCVdVe D VaVbgtVCgtVdVe E VaVbVdVegtVC Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley The Electric Potential We de ne the electric potential Vor for brevity just the otential as p H Uqsources 1 Charge q is used as a probe test charge to determine the electric potential but the value of Vis independent of q The electric potential like the electric eld is a property of the source charges The unit of electric potential is the joule per coulomb which is called the volt V lVOltlVElJlC Cupyngme 2m Fearsun Educauun Inc publishing as Fearsun Addisuanesley The Electric Potential of a Point Charge Let C be the source charge and let a second charge 6139 a distance r away probe the electric potential of q The potential energy of the two point charges is 1 6161 39 47760 I By definition the electric potential of charge 6 is UL39 l if g electric potential of a point charge q 47760 139 The potential extends through all of space showing the in uence of charge 6 but it weakens with distance as lr This expression for Vassumes that we have chosen V O to be at r 00 Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley FIGURE 2927 Four graphical representations of the electric potential of a poimt Charge C m our map Potential graph Equipolenliu surl39uces Elevutiun graph Copyn39ght 2008 Pearson Education Inc publishing as Pearson AddisonWesley Rank in order from largest to smallest the 3 potential differences AV AV13 and AV23 between points 1 and If 7392 2 points 1 and 3 and points2 and 3 Copyrighl o 2004 Pearson Eiluculiuu lnc publishing as Addison Wesley AVB gt AV12 gt AV23 AVB AV23 gt AV12 AVB gt AV23 gt AV12 AV12 gt AVB AV23 AV23 gt AV12 gt AVB Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley macw The Electric Potential of Many Charges The electric potential Vat a point in space is the sum of the potentials due to each charge 1 qi V 7 47760 r where r1 is the distance from charge qt to the point in space where the potential is being calculated In other words the electric potential like the electric eld obeys the principle of linear superposition Cupyngm 2m Parson Edumhun in publishing as 12mm Addisuanesley Potential by a dipole a contour map 1 Elevation graph I V I I I l I v I I I I 33 I l 00 i I M I I 5v te quot I 39 I V quot I quot 9 13 i I 39 I Equipotential surfaces Cupyngln zoos Psarsuu Emmanun m2 uumshmg as Pearson Addisavvrwesley Copyright 2008 Pearson Education Inc publishing as Pea l l The Electric Potential of a Charged Sphere In practice you are more likely to work with a charged sphere of radius R and total charge Q than with a point charge Outside a uniformly charged sphere the electric potential is identical to that of a point charge Q at the center That is 1 Q sphere of charge r 2 R 4776 1 Or in a more useful form the potential outside a sphere that is charged to potential V0 is R V 7V0 sphere charged to potential V0 ProblemSolving Strategy The electric potential of a continuous distribution of charge 1 Draw the charge distribution and the point P Set up a coordinate system such that the calculation is as simple as possible Divide the charge distrbution into small pieces each carrying charge AQ Write down the eld dV due to dQ at P using Coulomb s Law UIbUJN Express all distances and angles in terms of coordinates X y z also express dQ in terms of X y z or r 6 as well as dX dy etc Now you have dQ as function of coordinates and dX dy etc 6 Summing up dV due to all pieces you get the total eld created by the whole charge distribution 7 Tranform the sum into integral 8 Calculate the integral Copyright 2008 Pearson Education Inc publishing as Pearson AddisonWesley

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#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

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#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

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Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.