Electricity and Magnetism Lecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter 25 Sect. 1-5 Kirchhoff s Laws: Y&F Chapter 26 Sect.

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1 Electrcty and Magnetsm Lecture 07 - Physcs Current, esstance, DC Crcuts: Y&F Chapter 5 Sect. -5 Krchhoff s Laws: Y&F Chapter 6 Sect. Crcuts and Currents Electrc Current Current Densty J Drft Speed esstance, esstvty, Conductvty Ohm s Law Power n Electrc Crcuts Examples Krchhoff s ules appled to Crcuts EMF s - Pumpng Charges Work, Energy, and EMF Smple Sngle Loop and Mult-Loop Crcuts Summary Copyrght. Janow - Fall 03

2 Electrc Current: Net charge crossng a surface per unt tme dq dt or dq dt Current s the same across each cross-secton of a wre q(t) + - t (t')dt' Unts: Ampere Coulomb per second Conventon: flow s from + to as f free charges are + 0 x t (f s constant) Charge / current s conserved - charge does not ple up or vansh At any n out juncton krchhoff s ules: (summary) Juncton ule: Σ currents n Σ currents out at any juncton Voltage ule: Σ V s 0 for any closed path Current densty J may vary [J] current/area Energy n a crcut: EMFs provde energy (electro-motve force) esstances dsspate energy as heat Capactances store energy n E feld Inductances store energy n B feld Copyrght. Janow - Fall 03

3 CUENT CONSEVATION EXAMPLE: Fnd the unknown current Name the junctons Name the lnks by the junctons they connect Apply the juncton rule: n out Usng Juncton : A + A 3 A, Usng Juncton 5: A + 3 A 5 A 5, Usng Juncton : + A,3, 5, Usng Juncton 3: 4 A + A 6 A 3,4 Usng Juncton 4: 6 A + A 8 A nto juncton, out of juncton out of juncton 5, nto juncton out of juncton, nto juncton 3 out of juncton 3, nto juncton 4 out of juncton 4, to the rght, 5,,3 Copyrght. Janow - Fall 03

4 Try ths one yourself 7-: What s the current n the wre secton marked? A. A. B. A. C. 5 A. D. 7 A. E. Cannot determne from nformaton gven. A 3 A A A 5 A n out 6 A Copyrght. Janow - Fall 03

5 Current densty J: Current / Unt Area (Vector) A A + - J / A (large) Hgh current densty n ths regon J / A (small) Same current crosses larger or smaller Surfaces, current densty J vares Small current densty n ths regon For unform densty d J A or J /A J nˆ da What makes current flow? area J da J unts: Amperes/m E J σe E feld n sold wre drves current. APPLIED FIELD 0 andom moton flow left flow rght + + APPLIED FIELD NOT ZEO Movng charges collde wth fxed ons and flow wth drft velocty - - Copyrght. Janow - Fall 03

6 Do charges n a current keep acceleratng as they flow? Electrons collde wth ons, mpurtes, etc. causng resstance Move at constant drft speed v D : Thermal motons (random motons) have speed Drft speed s tny compared wth thermal motons. Drft speed n copper s m/s. J n nv densty D E # of of J charge carrers charge σe For E 0: no current, v D 0, J 0, 0 For E not 0 (battery voltage not 0): carrers crossng Unts : unt area J qnv D net charge crossng area A per - v th 6 0 # /volume per unt tme unt tme m/s ( 3 k Boltz + T) Note: for electrons, q & v D are both reversed J stll to left q e.6 x 0-9 C. Copyrght. Janow - Fall 03

7 EXAMPLE: Calculate the current densty J ons for ons n a gas Assume: Doubly charged postve ons Densty n x 0 8 ons/cm 3 Ion drft speed v d 0 5 m/s Fnd J ons the current densty for the ons only (forget J electrons ) J qnv.6 x D 0 J 6.4 A./m coul/on ons/cm 3 m/s cm 3 /m 3 6 Copyrght. Janow - Fall 03

8 Increasng the Current 7-: When you ncrease the current n a wre, what changes and what s constant? A. The densty of charge carrers stays the same, and the drft speed ncreases. B. The drft speed stays the same, and the number of charge carrers ncreases. C. The charge carred by each charge carrer ncreases. D. The current densty decreases. J E J σe J qnv D Copyrght. Janow - Fall 03

9 esstance: Determnes how much current flows through a devce n response to a gven potental dfference. E V L A V unts : Ohm Ω volt/ampere depends on the materal & geometry Note: C Q/ V nverse to Apply voltage to a conductng wre. Very large current so s small. Apply voltage to a poor conductor materal lke carbon Tny current so s very large. V Crcut Dagram esstvty ρ : Property of a materal tself (as s delectrc constant). Does not depend on dmensons The resstance of a devce depends on resstvty ρ and also depends on shape For a gven shape, dfferent materals produce dfferent currents for same V Assume cylndrcal resstors ρl resstance A For nsulators: ρ nfnty ρ resstvt y resstvty unts : A L for Copyrght. Janow - Fall 03 a resstor Ohm - meters Ω.m

10 Calculatng resstance, gven the resstvty resstance resstvty ρl A proportonal to length nversely proportonal to cross secton area EXAMPLE: Fnd for a 0 m long ron wre, mm n dameter ρl A 9. 7 π x x 0 8 (0-3 Ω.m x / ) 0 m m. Ω Fnd the potental dfference across f 0 A. (Amperes) EXAMPLE: V V Fnd resstvty of a wre wth 50 mω, dameter d mm, length L m ρ 3 A 50x0 Ω x0 m x π /.96x0 8 L m Ω.m Use a table to dentfy materal. Not Cu or Al, possbly an alloy Copyrght. Janow - Fall 03

11 esstvty depends on temperature: esstvty depends on temperature: Hgher temperature greater thermal moton more collsons hgher resstance. SOME SAMPLE ESISTIVITY VALUES ρ n 0 o C..7 x 0-8 copper 9.7 x 0-8 ron.5 x 0 +3 pure slcon eference Temperature Smple model of resstvty: α temperature coeffcent Change the temperature from reference T 0 to T Coeffcent α depends on the materal ρ ρ0( + α(t T0 )) temperature coeffcent α Conductvty s the recprocal of resstvty E V L ρl A A Defnton: σ ρ J σ E unts : "mho" ( Ω.m) - V EL JA JρL E/J ρ Copyrght. Janow - Fall 03

12 Current Through a esstor 7-3: What s the current through the resstor n the followng crcut, f V 0 V and 00 Ω? A. 0 ma. B. 5 ma. C. 0. A. D. 00 A. E. 5 A. V Crcut Dagram V Copyrght. Janow - Fall 03

13 Current Through a esstor 7-4: If the current s doubled, whch of the followng mght also have changed? A. The voltage across the resstor doubles. B. The resstance of the resstor doubles. C. The voltage n the wre between the battery and the resstor doubles. D. The voltage across the resstor drops by a factor of. E. The resstance of the resstor drops by a factor of. V Crcut Dagram V Copyrght. Janow - Fall 03

14 esstvty of a esstor ρl A 7-5: Three resstors are made of the same materal, wth szes n mm shown below. ank them n order of total resstance, greatest frst. A. I, II, III. B. I, III, II. C. II, III, I. D. II, I, III. E. III, II, I. I. 4 4 II. 5 Each has square cross-secton III. 6 3 Copyrght. Janow - Fall 03

15 Ohm s Law and Ohmc materals (a specal case) V / but could depend on appled V Defntons of resstance: σ / ρ J/E but ρ could depend on E Defnton of OHMIC conductors and devces: ato of voltage drop to current s constant t does not depend on appled voltage.e., current s proportonal to appled V esstvty does not depend on magntude or drecton of appled voltage Ohmc Materals e.g., metals, carbon, Non-Ohmc Materals e.g., semconductor devces band gap constant slope / varyng slope / OHMIC CONDITION d s CONSTANT dv Copyrght. Janow - Fall 03

16 Power s dsspated n resstve crcuts + - V L O A D a b Apply voltage drop V across load Current flows through load whch dsspates energy An EMF (e.g., a battery) does work, holdng V and current constant by expendng potental energy As charge dq flows from a to b t loses P.E. du - potental s PE / unt charge - charge current x tme Power V dsspaton P du P dt V / V [Watts] resstors only du V dq for any load V dt EXAMPLE: Space heater: Fnd rate of convertng electrcal energy to heat Copyrght. Janow - Fall 03

17 EXAMPLE: EXAMPLE: EXAMPLE: Copyrght. Janow - Fall 03

18 Ohmc and non-ohmc conductors SUPECONDUCTOS: At very low temperatures (~4 K) some conductors lose all resstance. Once you start current flowng, t wll contnue to flow forever, - The current becomes enormous once the appled voltage exceeds a small value. 7-6: The three plots show voltage vs. current (so the slope s ) for three knds of devces. Identfy the devces n order appearng n charts I, II, III? A. esstor, superconductor, dode B. Dode, superconductor, resstor C. esstor, dode, superconductor D. Dode, resstor, superconductor E. Superconductor, resstor, dode V I. II. III. Potental dfference (V) Current (ma) Copyrght. Janow - Fall 03

19 Crcut analyss wth resstances and EMFs GENEAL ANALYSIS METHOD: krchhoff S LAWS or ULES Juncton ule n out Charge conservaton Loop ule CICUIT ELEMENTS: V 0 (closed loop) Energy conservaton PASSIVE: ESISTANCE, CAPACITANCE, INDUCTANCE ACTIVE: EMF s (SOUCES OF POTENTIAL DIFFEENCE AND ENEGY) JUNCTIONS and BANCHES.etc ESISTANCE: POWE: OHM s LAW: V L ρ A ρ dw du P V dt dt P V slope / resstvty (resstor) s ndependent of V or Copyrght. Janow - Fall 03

20 EMFs pump charges to hgher energy EMFs move charges from low to hgh potental (potental energy). EMF s (electromotve force) such as batteres supply energy: mantan constant potental at termnals do work dw Edq on charges (source of the energy s usually chemcal) EMFs are charge pumps Unt: volts (V). Symbol: scrpt E. Types of EMFs: batteres, electrc generators, solar cells, fuel cells, etc. DC versus AC E + - Current flows CW through crcut from + to outsde of EMF from to + nsde EMF E work done unt charge P power dw dq Power suppled by EMF: dw E dq E dt P dt ± E E P emf dw dt Power dsspated by resstor: P V V / Copyrght. Janow - Fall 03

21 Ideal EMF devce eal EMF devce Multple EMFs Zero nternal battery resstance Open swtch: EMF E no current, zero power Closed swtch: EMF E s also appled across load crcut Current & power not zero Open swtch: EMF stll E r nternal EMF resstance n seres, usually small ~ Ω Closed swtch: V E r across load, P ckt V Power dsspated n EMF P emf (E-V) r Assume E B > E A (deal EMF s) Whch way does current flow? Apply krchhoff Laws to fnd out Answer: From E B to E A E B does work, loses energy E A s charged up converts PE to heat Load (motor, other) produces moton and/or heat Copyrght. Janow - Fall 03

22 Generatng Crcut Equatons wth the krchhoff Loop ule The algebrac sum of voltage changes zero around all closed loops through a crcut (ncludng mult-loop) Assume ether current drecton. Expect mnus sgns when choce s wrong. Traverse crcut wth or aganst assumed current drecton Across resstances, voltage drop V - f followng assumed current drecton. Otherwse, voltage change s +. When crossng EMFs from to +, V +E. Otherwse V -E Dot product.e determnes whether power s actually suppled or dsspated EXAMPLE: Sngle loop crcut wth battery (nternal resstance r) Power n External Ckt P V (E r) P E r Follow crcut from a to b to a, same drecton as E r 0 E r + Potental around the crcut crcut dsspaton battery dran battery dsspaton Copyrght. Janow - Fall 03

23 Example: CW or CCW around a sngle-loop crcut Assume current drecton as shown b c Traverse clockwse from a: Vba Vb Va + ε Vcb 0 Vdc Vd Vc V 0 ad d closed SAME ESULT V loop 0 ε ε 0 ε b c d Traverse counterclockwse from a: V V V V da cd bc ab 0 V 0 V V V + ε V ε closed loop c a d b ε 0 ε Copyrght. Janow - Fall 03

24 Equvalent resstance for resstors n seres Juncton ule: The current through all of the resstances n seres (a sngle branch) s dentcal: 3 Loop ule: The sum of the potental dfferences around a closed loop equals zero: ε 3 0 The equvalent crcut replaces the seres resstors wth a sngle equvalent resstance: ε eq 0 eq same E, same as above ε ε eq The equvalent resstance for a seres combnaton s the sum of the ndvdual resstances and s always greater than any one of them. eq n nverse of seres capactance rule Copyrght. Janow - Fall 03

25 Equvalent resstance for resstors n parallel Loop ule: The potental dfferences across each of the parallel branches are the same. E 0 E 0 E E E E,, 3 not n equatons eq E Juncton ule: The sum of the currents flowng n equals the sum of the currents flowng out. Combne equatons for all the junctons at a & b eq The equvalent crcut replaces the seres resstors wth a sngle equvalent resstance: same E, same as above ε eq 0 3 n ε eq The recprocal of the equvalent resstance for a parallel combnaton s the sum of the ndvdual recprocal resstances and s always smaller than any one of them. nverse of parallel capactance rule + eq Copyrght. Janow - Fall 03

26 esstors n seres and parallel 7-7: Four dentcal resstors are connected as shown n the fgure. Fnd the equvalent resstance between ponts a and c. A. 4. B. 3. C..5. D E.Cannot determne from nformaton gven. ε c a eq n eq n Copyrght. Janow - Fall 03

27 Capactors n seres and parallel 7-8: Four dentcal capactors are connected as shown n fgure. Fnd the equvalent capactance between ponts a and c. A. 4 C. B. 3 C. C..5 C. D. 0.4 C. E. Cannot determne from nformaton gven. ε c a C C C C C eq n C C eq n C Copyrght. Janow - Fall 03

28 EXAMPLE: 0 Ω Fnd, V, V, V 3, P, P, P E 7 V 3 8 Ω 7 Ω + - E 9 V EXAMPLE: Fnd currents and voltage drops eq Copyrght. Janow - Fall 03 +

29 EXAMPLE: MULTIPLE BATTEIES SINGLE LOOP + - E 8 V 0 Ω + - E 3 V 5 Ω A battery (EMF) absorbs power (charges up) when I s opposte to E ± E E P emf Copyrght. Janow - Fall 03

30 EXAMPLE: Fnd the average current densty J n a copper wre whose dameter s mm carryng current of ma. -3 amps -3 m 0 J A π x (.5 x 0 ) 73 Suppose dameter s mm nstead. Fnd J : J J' A' 4 38 amps/m amps/m Current s unchanged Calculate the drft velocty for the mm wre as above? J v d 3 8 Cuvd where ncu # conducton electrons/m 8.49 x 0 en J x0 m / s About 3 m/year!! en x0 x8. 49x0 Cu So why do electrcal sgnals on wres seem to travel at the speed of lght (300,000 km/s)? Calculatng n for copper: One conducton electron per atom n Cu electron / atom 8.49 x x electrons/m 3 gm/cm x gm/mole 6.0 x 0 3 atoms/mole x cm /m Copyrght. Janow - Fall 03