lec17_021607_-_Currents_and_Resistors_II

lec17_021607_-_Currents_and_Resistors_II - Currents and...

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Unformatted text preview: Currents and Resistors II 18th century BCE oracle bones establishing the solar year @ 365 days and the lunar month @ 29 days Find I and V through each resistor -1 Sample Circuit Parallel: R236 = 1 + 1 + 1 2 3 6 Constant V 1 = 1 Series: Constant I R15 = 1 + 5 = 6 -1 6 Parallel: R36 = + 3 6 Constant V 1 1 = 2 Friday, 16 February 2007 1 Friday, 16 February 2007 2 Find I and V through each resistor 2 Sample Circuit Series: Constant I Sample Circuit Find I and V through each resistor 6 R27 = 2 + 7 = 9 Parallel: V2 = V3 = V6 = 4V I6 = V6 4V 2 = = A R6 6 3 2 A 3 9 I total = Req = 18V = 2A 9 I3 = V3 4V 4 = = A R3 3 3 1 Now work backwards ... 2 Series: I total = I 2 = I 7 = 2A Series: I 6 = I1 = I 5 = V5 = I 5 R5 = V2 = I 2 R2 = (2A )(2 ) = 4V Friday, 16 February 2007 V7 = I 7 R7 = (2A )(7 ) = 14V 3 V1 = I1 R1 = 2 2 A (1 ) = V 3 3 2 10 A (5 ) = V 3 3 4 Friday, 16 February 2007 1 Find I and V through each resistor Sample Circuit Find I and V through each resistor Sample Circuit Parallel: V1 = V2 = V3 = V6 = 2 V 3 2 2 A, V 9 3 4 A, 4V 3 1 2 A, V 3 3 2 V V2 3 1 I2 = = = A R2 2 3 2 V V3 3 2 I3 = = = A R3 3 9 2 V V6 3 1 I6 = = = A R6 6 9 Friday, 16 February 2007 5 2 10 A, V 3 3 2A, 14V Friday, 16 February 2007 1 2 A, V 9 3 6 Kirchhoff's Kirchhoff' Rules Junction Rule: Derives from Conservation of Charge At any junction point in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction! Kirchhoff's Rules Kirchhoff' Sign Conventions: Cross a battery from the negative terminal to the positive terminal: Cross a resistor in the direction of positive current flow: Cross from the negative plate of the capacitor to the positive plate of a capacitor: And vice versa for all of the above! 7 Friday, 16 February 2007 8 Loop Rule: Derives from Conservation of Energy When any closed-circuit loop is traversed, closedthe sum of the changes in potential must equal zero! Friday, 16 February 2007 2 Battery Charging I1 Using Kirchhoff's Rules Kirchhoff' I1 R1 I3 R3 A I2 I3 A I5 B R5 I4 R4 I2 R2 1. Guess direction of Currents 1 R3 R2 2 B 3 2. Junction Rule A: I1 + I2 + I3 = 0 3. Loop Rule 1 +1 2 + I2R2 = 0 2 3 C D B: 0 = I1 + I2 + I3 1. Guess direction of Currents 2. Junction Rule A: I1 = I3 + I5 B: I5 + I2 = I4 9 Friday, 16 February 2007 I2R2 + 2 + I3R3 = 0 +1 + I3R3 = 0 = 2 + 1 C: I3 + I5 = I1 D: I4 = I2 + I5 10 Friday, 16 February 2007 Using Kirchhoff's Rules Kirchhoff' I1 A I5 B R5 I2 Big Picture 1. Electrostatics 2. Circuits 4 R1 I3 R3 I4 R4 R2 1 C 3 D 2 Capacitors and Dielectrics: Storing Charge Current and Resistance Direct Current Circuits (Steady State and RC) 3. Loop Rule 1 +1 I1R1 I3R3 = 0 2 3 4 +2 I2R2 I4R4 = 0 +4R4 + I5R5 I3R3 = 0 +2 2R2 + I5R5 + I1R1 1 = 0 = 3 + 2 1 11 3. Magnetics 4. Optics Friday, 16 February 2007 12 Friday, 16 February 2007 3 RC Circuit - Discharge t = 0: the time after the switch has been 0: thrown, after the E-field has been established Ein the circuit, when dQ/dt < > 0, but before any 0, charge has physically moved. At t = 0: 0: Q = Qo I = Io Vo , capacitor = Qo C I =- Qf RC Circuit - Discharge Q = Qo e I =- tf - ( RC ) tf = Qo e t - tf dQ Q = dt (RC ) t dQ dt =- (RC ) Q 1 - dQ Q -t = -Qo - e = o e dt RC Vo , resistor = I o R Vo , capacitor = Vo , resistor f dQ dt = - Q (RC ) Qo o ln Qf Qo - =- (RC ) - tf Qo = Io R C Q Q Io = R o = C (RC ) Friday, 16 February 2007 Qf Qo =e - ( RC ) tf Q = Qo e ( RC ) tf I = I oe - t = Qo e Where = RC is the time constant for the RC circuit 13 Current can be derived from charge, so only the charge expressions are on the equation sheet! Friday, 16 February 2007 14 RC Circuit - Charge Kirchhoff: At t = 0: 0: Vo , capacitor = 0 Q=0 I = Io Vo , resistor = I o R - IR - - = Io R At t = : I =0 Io = Q dQ = R C dt 1 (C - Q )dt = RdQ C t Q =0 C C Q dQ - = R C C dt RC Circuit - Charge (C - Q ) e - e t t = t - C e (C - Q ) C - Q (C - Q ) = C e Current: I = - t Q = C 1 - e - t = Qf 1- e - t R V f , resistor = 0 Qf = V f , capacitor = C Qf = C dQ I =+ dt Friday, 16 February 2007 dt dQ = RC C - Q 0 dt = RC Qf dQ C - Q 0 t C = ln RC C - Q t C e = C - Q dQ dt t - 1 RC (- C )e =- RC = I oe 15 Friday, 16 February 2007 - t RC 16 4 ...
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