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### Lecture_18_Part1

Course: PHYS 272, Winter 2007
School: Purdue
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Word Count: 1049

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Analysis Macroscopic 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. Resistance Many elements in a circuit act as resistors: prevent current from rising above a certain...

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Analysis Macroscopic 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. 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 V and I in such elements. Need to combine the properties of material and geometry. Conductivity Combining the properties of a material Geometry Conventional current: I = q nAv = q nAuE Different properties of the material Group the material properties together: I = ( q nu ) AE I J = = ( q nu ) E = E (A/m 2 ) Current density: A Conductivity = q nu A V m Exercise The conductivity of tungsten at RT is =1.8. 107 (A/m2)/(V/m) and it decreases 18 times at a temperature of a glowing filament (3000 K). The tungsten filament has a radius of 0.015 mm. What is E required to dive 0.3A through it? J= I = E A E= I A 0.3 A E= = 424 V/m 7 2 -10 2 1.8 10 (A/m )/(V/m) / 18 7.1 10 m ( )( ) Conductivity with two Kinds of Charge Carriers I Cl = q1 n1 Av1 = q1 n1 Au1E I Na = q2 n2 Av2 = q2 n2 Au2 E I = I Na + I Cl = q2 n2 Au2 E + q1 n1 Au1E J= I = ( q2 n2u2 + q1 n1u1 ) E = E A = q2 n2u2 + q1 n1u1 Resistance V = - E dl f i V = EL I J = = E A V E= L I = AE V A 1 I= V = V = L R R R Conventional current: I = V Widely known as Ohm's law L Resistance of a long wire: R = A George Ohm (1789-1854) Units: Ohm, Resistance combines conductivity and geometry! Microscopic and Macroscopic View Microscopic v = uE Macroscopic J = E i = nAv = nAuE V I = q nAv = R Can we write V=IR ? A = 0.002 mm2 Exercise: Carbon Resistor Conductivity of Carbon: = 3.104 (A/m2)/(V/m) L=5 mm L R= A What is its resistance R? R= ( 3 10 ( 0.005 m ) 4 2 (A/m )/(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? V I= R 1.5V I= 0.018 A = 18 mA 83 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. Ohmic Resistors Ohmic resistor: resistor made of ohmic material Ohmic materials: materials in which conductivity is independent of the amount of current flowing through V I= R R= L A 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) V I= R V R= I V-A dependence: 3V 100 mA 1.5 V 80 mA 0.05 V 6 mA R 30 19 8 I Clearly not ohmic! V Semiconductors Metals, mobile electrons: slightest V produces current. If electrons were bound need to apply some field to free some of them and start flow current Semiconductors: n depends exponentially on E = q nu Conductivity depends exponentially on E Conductivity rises (resistance drops) with rising temperature Nonohmic Circuit Elements Semiconductors Capacitors V I= R |V|=Q/C, function of time Batteries: double current, but |V|emf, hardly changes V 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 from the definition of resistance: L R= A Exercise: Voltage Divider Know R , find V1,2 R1 emf R2 V2 V1 Solution: V I= R emf Requivalent V = IR emf = R1 + R2 1) Find current: I = 2) Find voltage: R1 V1 = IR1 = emf R1 + R2 R2 V2 = IR2 = emf R1 + R2 3) Check: V1 + V2 = emf R1 R2 emf + = emf R1 + R2 R1 + R2 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 from the definition resistance: L R= A 1 A = R L Aequivalent = A1 + A2 + A3 Two Light Bulbs in Parallel R1 = 30 What is the equivalent resistance? 1 Requivalent 1 1 = + R1 R2 R1R2 R1 + R2 R2 = 10 What is the total current? V 3V = = 0.4 A I= 7.5 R Requivalent = Requivalent 300 2 = = 7.5 40 V V 3V 3V + = + = 0.4 A Alternative way: I = I1 + I 2 = R1 R2 30 10 Work and Power in a Circuit Current: charges are moving work is done Work = change in electric potential energy of charges U e = q V Power = work per unit time: P= U e q V q = = V t t t I Power for any kind of circuit component: P = I V Units: AV = CJ J = =W sC s Power Dissipated by a Resistor emf R Know V, find P P = IV V I= R ( V )2 P= R Know I, find P P = IV V = IR P = I 2R In practice: need to know P to select right size resistor capable of dissipating thermal energy created by current. Energy Stored in a Capacitor Q = C V V = Q dQ C Q Q C dE = dQV = Q Q Q 1 E = dE = dQ = QdQ C C0 0 0 Q E= 2C 2 Q Alternative approach: 0E 2 Energy density: 2 E = V / s 0 Q2 = 2C 2 E= C ( V ) 2 V A V Energy: 0 ( 2 ) As = 0 ( ) 2s 2s 2 2 A C= 0 s = C ( V ) 2 2 Real Batteries: Internal Resistance Drift speed of ions in chemical battery: v ~ (FNC - eEC ) In usual circuit elements: J = E In a battery: force I FNC J = = - EC ~ unit charge A e rint - internal resistance emf EC = FNC I F s s - , assuming uniform field: V = EC s = NC - I e A e A Vbattery = emf - rint I Real Batteries: Internal Resistance Vbattery = emf - rint I ideal battery Model of a real battery Round trip (energy conservation): emf - rint I - RI = 0 I= emf R + rint R rint0.25 1.5 V 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
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