Last Time. Equilibrium vs. Steady State in a Circuit What is "used up" in a circuit? Kirchhoff's Current Node Law E-field inside a wire

Transcription

1 Last Time Equilibrium vs. Steady State in a Circuit What is "used up" in a circuit? Kirchhoff's Current Node Law E-field inside a wire 1

2 Electric Field Inside the Wire Constant current in the wire Constant E in the wire. I I I Conventional Current I I I I I Drift Velocity controlled by E Mobility (u) set by the material. Constant current requires constant E 2

3 Today Transient response when connecting a circuit How long until steady state is reached? Introduction to Resistors Energy conservation in a circuit Kirchhoff's Voltage Loop Law Batteries 3

4 Direction of Electric Field in a Wire E must be parallel to the wire E is the same along the wire Does current fill the wire? Is E uniform across the wire? DV ABCDA B = -ò E A C D A 1 dl - ò E3 dl - ò E2 dl - ò E3 dl = B C D V AB 0 V CD 0 E 1 = E 2 4 0

5 Electric Field in a Wire E What charges make the electric field in the wires? Bulb filament and wires are metals there cannot be excess charges in the interior Are excess charges on the battery? ASSUME: E due to dipole field of battery. This cannot be the source of the E which drives current. E E 5

6 Field due to the Battery Surface charge arranges itself in such a way as to produce a pattern of electric field that follows the direction of the wire and has such a magnitude that current is the same along the wire. 6

7 Field due to Battery E Smooth transition from + surface charge to to provide constant E. The amount of surface charge is proportional to the voltage. 7

8 Connecting a Circuit What happens just before and just after a circuit is connected? Before the circuit is connected: No current flows System is in equilibrium: How is E = 0 maintained when there are charges here? There must be surface charges on the wire to prevent current from flowing before we connect the circuit. 8

9 Connecting a Circuit What happens just before and just after a circuit is connected? Before the circuit is connected: No current flows System is in equilibrium: Think about the gap... E due only to gap faces 9

10 Connecting a Circuit What happens just before and just after a circuit is connected? Before the circuit is connected: No current flows System is in equilibrium: Think about the gap... E due to everything else cancels E gap 10

11 Connecting a Circuit What happens just before and just after a circuit is connected? Before the circuit is connected: E due to everything else cancels E gap Now close the gap... The gap face charge 0, and so does 11E gap

12 Connecting a Circuit What happens just before and just after a circuit is connected? Just after the circuit is connected: There is a disturbance in the previous (equilibrium) E-field. Now the region next to the disturbance updates its E-field, and the next region... How fast does this disturbance propagate? At the drift speed of the electrons? At the speed of light? 12

13 iclicker Reality Physics! Drift speed of electrons Speed of light Flip Light Switch On. How long until electrons from the switch reach the light bulb? L = 5 m A) About 1 nanosecond B) About 1 microsecond C) About 1 minute D) About 1 day 13

14 iclicker Reality Physics! Drift speed of electrons Speed of light Flip Light Switch On. How long until information about the change in E-field reaches the light bulb? L = 5 m A) About 16 nanoseconds B) About 16 microseconds C) About 16 minutes D) About 16 days 14

15 Reality Physics! Drift speed of electrons Speed of light Flip Light Switch On. How long until information about the change in E-field reaches the light bulb? L = 5 m 1 day for electrons to travel from light switch to bulb. 16 nanoseconds for the change in E-field to travel from light switch to bulb. Because there are sooooo many electrons in the wire, they don't have to move far to create a large current. 15

16 Connecting a Circuit What happens just before and just after a circuit is connected? Just after the circuit is connected: There is a disturbance in the previous (equilibrium) E-field. Now the region next to the disturbance updates its E-field, and the next region... The disturbance travels at the speed of light, and within a few nanoseconds, steady state is established. 16

17 Electric field in Thin and Thick wires After steady state is reached: i = i i thin i thick thin thick = = na na thin thick ue thin ue thick E = thin A A thick thin 17 E thick

18 A Wide Resistor low mobility i = nav = i i thin thick = = nau nau naue thin E thick thin E thick E = thin u u 18 thick thin E thick

19 Energy in a Circuit DV wire = EL DV battery =? Energy conservation (the Kirchhoff loop rule [2 nd law]): DV 1 + DV 2 + DV 3 + = 0 along any closed path in a circuit DV= DU/q energy per unit charge 19

20 General Use of the Loop Rule DV 1 + DV 2 + DV 3 + DV 4 = 0 (V B -V A )+ (V C -V B )+ (V F -V C )+ (V A -V F )=0 20

21 Two Batteries in Series emf - EL = 0 emf E = L i = naue = nau Why light bulb is brighter with two batteries? emf L æ P 1batt = elnau emf è ç L Two batteries in series can drive more current: Potential difference across two batteries in series is 2emf doubles electric field everywhere in the circuit doubles drift speed doubles current. ö ø 2 Work per second: P = (q / T )EL = ieel P = nauele 2 2 emf - EL = 0 E = 2emf L i = nau 2emf L æ P 2batt = elnau 2emf è ç L P21 2batt = 4 P 1batt 2 ö ø

22 Potential Difference Across the Battery F C E C FC s DVbatt = ECs = = e F e NC s Coulomb force on each e non-coulomb force on each e 1. a=f NC /m F 2. F C =ee C E = C e 3. F C =F NC Energy input per unit charge emf electromotive force C The function of a battery is to produce and maintain a charge separation. The emf is measured in Volts, but it is not a potential difference! The emf is the energy input per unit charge. chemical, nuclear, gravitational 22

23 iclicker Questions i = naue Nichrome wire (resistive) When wire length double, the current will be A. Double B. Halved C.unchanged 23

24 Twice the Length Nichrome wire (resistive) Quantitative measurement of current with a compass DV i = naue = nau L Current is halved when increasing the length of the wire by a factor of 2. 24

25 iclicker Question Nichrome wire (resistive) When wire cross sectional area is doubled, the current will be A. Double B. Halved C.unchanged 25

26 Doubling the Cross-Sectional Area Nichrome wire If A doubles, the current doubles. 26

27 iclicker Questions When plug in two batteries instead of one, the current will be A. Double B. Halved C.unchanged 27

28 Two Batteries in Series emf - EL = 0 emf E = L i = naue = nau Why light bulb is brighter with two batteries? emf L æ P 1batt = elnau emf è ç L Two batteries in series can drive more current: Potential difference across two batteries in series is 2emf doubles electric field everywhere in the circuit doubles drift speed doubles current. ö ø 2 Work per second: P = W / T = qel / T = ( q / T ) EL = ieel P = nauele 2 2 emf - EL = 0 E = 2emf L i = nau 2emf L æ P 2batt = elnau 2emf è ç L P28 2batt = 4 P 1batt 2 ö ø

29 How Do the Currents Know How to Divide? 29

30 Let s be Quantitative Capacitors, Resistors and Batteries 30

31 Capacitor: Charging and Discharging Charging Discharging 31

32 Capacitor: Construction and Symbols Similar to a large parallel plate capacitor D s There is no connecting path through a capacitor 32

33 Capacitor: Discharge Electron Current Electric Field 33

34 Capacitor: Charging 34

35 Capacitor: Charging Why does current ultimately stop flowing in the circuit? Ultimately, the fringe field of the capacitor and the field due to charges on the wire are such that E=0 inside the wire. At this point, i=0. 35

36 The Effect of Different Light Bulbs Thin filament Thick filament Which light bulb will glow longer? Why? 1) Round is brighter capacitor gets charged more? 2) Long bulb glows longer capacitor gets charged more? After current stops, Voltage across capacitor = Voltage across battery no matter which bulb is used. Capacitor charged by same amount in both cases. 36

37 Effect of the Capacitor Disk Size Use two different capacitors in the same circuit In the first moment, which capacitor will cause the bulb to produce more light? Which capacitor will make the light bulb glow longer? Fringe field: E 1 Q / A 2 0 s R 37

38 Effect of the Capacitor Disk Separation In the first moment, which capacitor will cause the bulb to produce more light? Which capacitor will make the light bulb glow longer? Fringe field: E 1 Q / A 2 0 s R 38

39 Effect of Insulator in Capacitor Insulator In the first moment, which capacitor will cause the bulb to produce more light? Which capacitor will make the light bulb glow longer? Fringe field: Q / A s E1-2 R 0 Edipoles 39

40 The capacitors shown are initially uncharged. When connected to identical circuits, after 0.01 s of charging: A) The fringe field of each capacitor is the same. B) The fringe field of the smaller capacitor is greater. C) The fringe field of the larger capacitor is greater. R 1 R 2 s E fringe = Q / A ( s s 2 0 R ) 40

41 Parallel Capacitors Initial moment: brighter? Will it glow longer? Fringe field: E 1 Q / A 2 0 s R Capacitors in parallel effectively increase A 41

42 An Isolated Light Bulb Will it glow at all? How do electrons flow through the bulb? 42

43 Capacitor in a Circuit I Charging Bulb Brightness time time Energy conservation Do 19.X.7! E cap time 43

44 The Current Node Rule in a Capacitor Circuit I 1 = I 2 + I 3 Charge conservation: i I i = 0 in steady state I i > 0 for incoming I i < 0 for outgoing Capacitor transients: not a steady state! Cannot use Kirchhoff rule for a part of a capacitor (area 1 or 2) But can use for capacitor as a whole (area 3) 44

45 Capacitance Electric field in a capacitor: E = Q / A 0 -Q +Q DV f = -ò E dl DV = Es i E DV = Q / A 0 s Q A = 0 s DV In general: Q ~ DV Definition of capacitance: Q = C DV Capacitance Capacitance of a parallelplate capacitor: C = 0A 45 s s

46 Exercise A capacitor is formed by two rectangular plates 50 cm by 30 cm, and the gap between the plates is 0.25 mm. What is its capacitance? C = 0A s C = 0A s 0.15 m 2 = C 2 /N m m = F = 5.4 nf 46

47 A Capacitor With an Insulator Between the Plates No insulator: Q / A E = 0 With insulator: Q / A E = K 0 D DV = Es DV Q / A = s A Q = 0 s 0 DV Q DV = Es DV = K A s = 0 Q / A s K 0 DV s C = 0A s A K s 0 C = 47