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Chapter 20 A Microscopic View of Electric Circuits Announcements: March 31th last day for EXAM 1 free response re-take. Second midterm April 3rd, Thursday, 8-9:30 pm, room 203 No lecture on April 9th Capacitance of parallel plate capacitor: A=area, s-separation between plates Select a correct formula: 1. C = A*0/s 2. C = A/0s 3. C = 0/(A*s) 4. C = s/0A Ohmic Resistors Ohmic resistor: resistor made of ohmic material Ohmic materials: materials in which conductivity is independent of the amount of current flowing through DV I= R R= L sA not a function of current Examples of ohmic materials: metal, carbon (at constant T!) Is a Light Bulb an Ohmic Resistor? Tungsten: mobility at room temperature is larger than at ‘glowing’ temperature (~3000 K) DV I= R V-A dependence: 3V 100 mA 1.5 V 80 mA 0.05 V 6 mA R 30 19 8 DV R= I I V Semiconductors Metals, mobile electrons: slightest V 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 s = q nu Conductivity depends exponentially on E Conductivity of semiconductor rises (resistance drops) with rising temperature Nonohmic Circuit Elements Semiconductors Capacitors DV I= R |V|=Q/C, function of time Batteries: double current, but |V|emf, hardly changes DV I= R has limited validity! Ohmic when R is indeppendent of I! Conventional symbols: Series Resistance Vbatt + V1 + V2 + V3 = 0 emf - R1I - R2I - R3I = 0 emf = R1I + R2I + R3I emf = (R1 + R2 + R3) I emf = Requivalent I , where Requivalent = R1 + R2 + R3 For resistors made of the same material and with the same A it follows straight from the definition of resistance: L R A Parallel Resistance I = I 1 + I2 + I3 emf emf emf I R1 R2 R3 1 1 1 emf I emf Requivalent R1 R2 R3 1 Requivalent 1 1 1 R1 R2 R3 For resistors made of the same material and with the same A it follows straight from the definition of resistance: L 1 A Aequivalent A1 A2 A3 R A R L Question I1 I2 Identical bulbs are connected to identical set of batteries. They produce light. Compare I1 and I2 : A. I1 = I2 B. I1 slightly less than I2 C. I1 = 2*I2 D. I1 slightly less than 2*I2 Real Batteries: Internal Resistance Drift speed of ions in chemical battery: v ~ FNC eEC In usual circuit elements: J E rint - internal resistance In a battery: force I FNC J EC A e unit charge EC emf FNC I FNC s s V E s I , assuming uniform field: C e A e A Vbattery emf rint I Real Batteries: Internal Resistance ideal battery Vbattery emf rint I Model of a real battery Round trip (energy conservation): emf rint I RI 0 R emf I R rint Situation 1: disconnect R (R=) I=0 V = emf emf Vbattery emf rint I R rint Situation 2: R is in place I Situation 3: short battery (R=0) I = emf /rint V = 0 Real Batteries: Internal Resistance ideal battery Vbattery emf rint I emf rint I RI 0 emf I R rint rint0.25 1.5 V R R 100 10 1 0 Ideal 0.015 A 0.15 A 1.5 A infinite Real 0.01496 A 0.146 A 1.2 A 6A VR=RI 1.496 V 1.46 V 1.2 V 0V Ammeters, Voltmeters and Ohmmeters Ammeter: measures current I Voltmeter: measures voltage difference V Ohmmeter: measures resistance R Using an Ammeter Connecting ammeter: An ammeter must be inserted into the circuit in series with the circuit element whose current you want to measure. An ammeter must have a very small resistance, so as not to alter significantly the circuit in which it has been inserted. 0.150 Conventional current must flow into the ‘+’ terminal and emerge from the ‘-’ terminal to result in positive reading. Exercise: Connecting Ammeter Is it correct connection? A) Yes B) No Voltmeter Voltmeters measure potential difference VAB – add a series resistor to ammeter V I R Measure I and convert to VAB=IR Connecting Voltmeter: Higher potential must be connected to the ‘+’ socket and lower one to the ‘-’ socket to result in positive reading. Voltmeter: Internal Resistance VAB in absence of a voltmeter R1 emf VAB B rint R2 A A R2 emf R1 R2 VAB in presence of a voltmeter VAB R2||int R2||int R1 R2||int emf R2 rint R2 rint Internal resistance of a voltmeter must be very large Quantitative Analysis of an RC Circuit Vround _ trip emf RI VC 0 Q 0 C dQ emf Q / C I dt R emf I Initial situation: Q=0 0 R emf RI Q and I are changing in time dI d emf d Q dt dt R dt RC Q VC C d dt dI 1 dQ dt RC dt dI 1 I dt RC RC Circuit: Current dI 1 I dt RC 1 1 dI dt I RC I t 1 1 dI dt I I RC 0 0 t ln I ln I 0 RC I t ln I0 RC I e I0 t RC Current in an RC circuit I I 0e t / RC What is I0 ? Current in an RC circuit I emf t / RC e R RC Circuit: Charge and Voltage What about charge Q on capacitor? I dQ dt dQ Idt Current in an RC circuit I I 0e Current in an RC circuit emf t / RC I e R Check: t=0, Q=0, t emf Q Idt R 0 t / RC t t / RC e dt 0 Q C emf 1 et / RC Q V C t--> inf, Q=C*emf RC Circuit: Summary Current in an RC circuit I emf t / RC e R Charge in an RC circuit Q C emf 1 et / RC Voltage in an RC circuit V emf 1 et / RC The RC Time Constant Current in an RC circuit emf t / RC I e R When time t = RC, the current I drops by a factor of e. RC is the ‘time constant’ of an RC circuit. e t / RC e 1 1 0.37 2.718 A rough measurement of how long it takes to reach final equilibrium What is the value of RC? About 9 seconds Exercise: A Complicated Resistive Circuit Find currents through resistors I2 loop 1: emf r1I1 R1I1 R4 I 4 R7 I1 0 Loop 2 loop 2: I1 R2 I 2 r2 I 2 emf R6 I 2 R3 I 3 0 I3 Loop 1 Loop 3 I4 Loop 4 loop 3: I5 R4 I 4 R3 I 3 R5 I 5 0 nodes: I1 I 2 I 3 I 4 0 I3 I 2 I5 0 I 4 I 5 I1 0 Five independent equations and five unknowns