Chapter 20. Capacitors, Resistors and Batteries
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1 Chapter 20 Capacitors, Resistors and Batteries
2 How is Discharging Possible?! E Positive and negative charges are attracted to each other: how can they leave the plates? Fringe field is not zero! Electrons in the wire near the negative plate feel a force that moves them away from the negative plate. Electrons near the positive plate are attracted towards it.
3 Electron current Capacitor: Discharge
4 Capacitor Discharge The fringe field of the capacitor plus the electric field of the charges on the surface of the wires drive current in a way to REDUCE the charge of the capacitor plates. Recall that we derived an expression for the fringe field by considering the superposition of two charged disks separated by a distance s.
5 Capacitor: Charging Fringe field of a capacitor rises until E=0 in a wire static equilibrium. Fringe field opposes the flow of current!
6 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.
7 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?
8 Analysis When capacitor is fully charged, it has same Q regardless of the bulb. In the initial instant of charging, E = emf/l, is same for both bulbs since bulb filaments are the same length. i = nuae, smaller A in long bulb smaller current smaller fringe field Long bulb takes longer to charge and discharge.
9 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
10 Analysis Capacitor 2 has larger A Smaller fringe field Smaller fringe field takes longer to build up fringe field to oppose current flow When fully charged, Capacitor 2 will hold more charge since Q/A is the same for both capacitors. Bulb stays lit longer for Capacitor 2.
11 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
12 Analysis Capacitor 2 has smaller gap Smaller fringe field Current ceases when fringe field opposes flow of current Therefore bulb with Capacitor 2 will glow longer.
13 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 E 1 "! 2# R 0 Edipoles
14 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
15 An Isolated Light Bulb Will it glow at all? How do electrons flow through the bulb? Why do we show charges near bulb as - on the left and + on the right?
16 Capacitor in a Circuit I Charging Bulb Brightness time time Energy conservation Do 20.X.3! E cap time
17 Capacitance Electric field in a capacitor: E = Q / A! 0 -Q +Q # V f!! = "! E dl!v = Es i E Q / A " A " V = s Q = 0! V! s 0 In general: Q ~!V Definition of capacitance: Q = C! V Capacitance Capacitance of a parallelplate capacitor:! C = 0A s s
18 Capacitance Q = C! V Units: C/V, Farads (F) Michael Faraday ( )
19 Exercise The capacitor in your set is equivalent to a large two-disk capacitor How large would it be? D s=1 mm! 0 C A Cs = A = s! 0 ( 1 F) ( m) A = 9! 10 "12 C 2 /N # m 2 ( ) A = 1.1! 10 8 m 2 D ~ 10 km (6 miles)
20 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 ) = 9 " 10#12 C 2 /N $ m " 10 #4 m = 5.4 " 10 #9 F = 5.4 nf
21 A Capacitor With an Insulator Between the Plates No insulator: Q / A E =! 0 With insulator: Q / A E = K! 0 D!V = Es " V = Q / A s! 0 " A Q = 0! V s!v = Es " V = Q / A s K! 0 K" 0A Q =! V s s! C = 0A s! A K s 0 C =
22 Macroscopic Analysis of Circuits Microscopic treatment: insight into the fundamental physical mechanism of circuit behavior. Not easy to measure directly E, u, Q, v. It is easier to measure conventional current, potential difference macroscopic parameters Need a link between microscopic and macroscopic quantities.
23 Resistance Many elements in a circuit act as resistors: prevent current from rising above a certain value. Goal: find a simple parameter which can predict ΔV and I in such elements. Need to combine the properties of material and geometry.
24 Conductivity Combining the properties of a material Geometry Conventional current: I = q nav = q naue Group the material properties together: I = ( q nu) AE Current density:! J = I A = Different properties of the material ( q nu ) E! =! E! (A/m 2 ) & # Conductivity ( = q nu $! " % A V ' m
25 In copper at room temperature, the mobility of electrons is about (m/s)/(v/m) and the density of electrons is n= m -3. What is σ?! = q nu = ( 1.6 " 10 #19 C) ( 8 " m -3 )( 4.5 " 10 #3 (m/s)/(v/m) ) " = 5.76! 10 7 (A/m 2 )/(V/m) What is the strength of E required to drive a current of 0.3 A through a copper wire which has a cross-section of 1 mm 2? E = I J = =! E A Exercise I E =! A 0.3 A ( 5.76! 10 7 (A/m 2 )/(V/m)) 1! 10 "6 m 2 ( ) = 5.2! 10"3 V/m
26 Exercise The conductivity of tungsten at RT is σ= (A/m 2 )/(V/m) and it decreases 18 times at a temperature of a glowing filament (3000 K). The tungsten filament has a radius of mm. What is E required to drive 0.3A through it? I J = =! E A I E =! A E = 0.3 A ( 1.8! 10 7 (A/m 2 )/(V/m) / 18) 7.1! 10 "10 m 2 ( ) = 424 V/m
27 Conductivity with two Kinds of Charge Carriers I Cl = q 1 n 1 Av 1 = q 1 n 1 Au 1 E I Na = q 2 n 2 Av 2 = q 2 n 2 Au 2 E I = I + I = q2 n2au2e + q n Au Na Cl E J = I A = ( q 2 n 2 u 2 + q 1 n 1 u 1 )E =! E! = q + 2 n2u2 q1 n1u 1
28 Resistance # V f!! = "! E dl i!v = EL E =! V L I J = =! E I =! AE A I = " A L! V = 1 R! V =! V R Conventional current: I =! V R Widely known as Ohm s law Resistance of a long wire: L R =! A George Ohm ( ) Units: Ohm, Ω Resistance combines conductivity and geometry!
29 Microscopic and Macroscopic View Microscopic Macroscopic v = ue i = nav = naue J =!E I = q nav =! V R Can we write V=IR? Current flows in response to a ΔV
30 Exercise: Carbon Resistor A = mm 2 Conductivity of Carbon: σ = (A/m 2 )/(V/m) L=5 mm What is its resistance R? R = L R =! A ( m) 3! 10 4 (A/m 2 )/(V/m) ( )( 2! 10 "9 m 2 ) = 83 # (V/A) What would be the current through this resistor if connected to a 1.5 V battery? I! V V =! A = 18 ma R = 1.5 I 0. 83" 018
31 Constant and Varying Conductivity Mobility of electrons: depends on temperature ( = q nu & $ % A V ' m #! " Conductivity and resistance depend on temperature. Conductivity may also depend on the magnitude of current.
32 Ohmic Resistors Ohmic resistor: resistor made of ohmic material Ohmic materials: materials in which conductivity σ is independent of the amount of current flowing through I =! V R L R = not a function of current! A Examples of ohmic materials: metal, carbon (at constant T!)
33 Is a Light Bulb an Ohmic Resistor? Tungsten: mobility at room temperature is larger than at glowing temperature (~3000 K) I =! V R R =! V I V-A dependence: 3 V 100 ma 1.5 V 80 ma 0.05 V 6 ma R 30 Ω 19 Ω 8 Ω I ΔV
34 Semiconductors Metals, mobile electrons: slightest ΔV produces current. If electrons were bound we would need to apply some field to free some of them in order for current to flow. Metals do not behave like this! Semiconductors: n depends exponentially on E! = q nu Conductivity depends exponentially on E Conductivity rises (resistance drops) with rising temperature
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