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CU-Boulder PHYS 1120 - Circuits

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Crkt-1 More Circuits In an electrical circuit, circuit elements such as resistors and batteries can be connected together in series or in parallel. Resistors in series are connected like links in a chain; resistors in parallel are side-by-side, like so: Rtot R1 R2 R3 =in series: R1 Series: , Rtot > R1, R2, R3 tot 1 2 3RRR=+ +R Parallel: tot12 31R , Rtot < R1, R2, R3 11 1RR R=++Resistors in series act like a single large resistor. Resistors in parallel act like a single small resistor. Proof: Resistors in Series: Itot = I1 = I2 , ∆Vtot = ∆V1+∆V2 ⇒ tot12 1 2tot 12tot tot 1 2VVV V VRRII II∆∆ + ∆∆∆== =+=+R Resistors in Parallel: Itot = I1 + I2 , ∆Vtot = ∆V1 = ∆V2 ⇒ tot12 1 2tot tot 1 2 tot 1 2III I I 1 1 1VVVVRR+==+⇒ =+∆∆∆∆ R R2 R3 Rtot =in parallel: Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-2 Examples of parallel resistors: 1) Two 100 Ω resistors in parallel: tot1 1 100R51122100 100100===Ω⎛⎞⎟⎜+⎟⎜⎟⎜ΩΩ⎝⎠0= 2) 10 Ω in parallel with 0 Ω wire: tot11R011010===∞+ 100 Ω 50 Ω = 100 Ω 0 Ω !! 0 Ω=R2 Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-3 Key points: • The current is the same for resistors in series. Current is not "used up". Think of the water pipe analogy: two gravel plugs in series, same flow (same gal/min) through both plugs (assuming no leaks or bubble in the pipe) • Adding another resistor in series always increases the total resistance. • The voltage difference across each resistor is the same for resistors in parallel. Both resistors in parallel have the same big small small bigVVIR IR∆ == =. • Adding another R in parallel always decreases the total resistance. Like adding another pipe along side the original pipe ⇒ allows more flow ⇒ smaller total resistance gravel plug gravel plug R1 I I R2 Rbig bigger plug, less flow Rsmall Ismall Ibig ∆V same across both R's smaller plug, more flow lo P hi P ∆pressure same across both plugs Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-4 Kirchhoff's two rules for analyzing circuits (Kirchhoff is really spelled that way: 2 h's, 2 f 's ) Kirchhoff's Current Rule (also called the Junction Rule) The total current into any junction = total current out (junction = place where 3 or more wires meet) 3I This is also called Conservation of Current. In steady-state, the charge is not building up rcuit irchhoff's Voltage Rule (also called the Loop Rule) in a circuit = sum of the voltage drops I2 12II=+ anywhere, it is just flowing along at a steady rate. So the current into any portion of the cimust equal the current coming out of that portion, otherwise charge would be building up in thatpart of the circuit. KThe sum of the voltage rises around any complete loop around the same loop. Voltage rises and drops must sum to zero, since we must return to the same voltage after one complete loop. NNN12 12rise12fallfallV VIRIR I(RR) IRR=+=+⇒ =+Remember: voltage is a kind of "electrical pressure" or "electrical height" . If you go around a complete circuit and return to the same place, you are back at the same pressure (or height). So rises must equal drops. R2 VI med V lo V R 1+ – hi V Pump lo pressurehi P med P I1 I1 I3 junction junction Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-5 Ammeters and Voltmeters An ammeter measures thI Ie current through itself ifference between its terminals. t place the ammeter iA voltmeter measures the voltage d To measure the current through a resistor R, mus n series with R. To measure the voltage across R, must place voltmeter in parallel with R. has z internal resistance rinternal = 0, so current I is not affected. An ideal voltmeter has rinternal = ∞, so no current flows through ⇒ currents and voltages in rest of circuit are not affected. R I A V An ideal ammeter ero A V A Ii = 0 Vrint = ideal ammeter: rint = 0 ideal voltmeter: rint = ∞ Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-6 Circuits with multiple loops and batteries Have a circuit with known V's and known R's. Seek the I's. Procedure: I. Guess direction of I through each R. Draw I arrows, label each (I1 thru R1, etc). (Directions of currents not always obvious, so just guess. If you guess wrong, value of current I will come out with a negative value.) II. K's Current Law gives 1 or more equations: (eq'n 1) [3 unknowns ( I1 , I2 , I3 ) ⇒ will need 3 eq'ns to solve] 12II+=3I III. K's Voltage Law gives an equation for each complete loop in the circuit. 3 loops in this circuit. Only need 2 more equations, so only 2 of the 3 loop equations are needed. Loop 1: (eq'n 2) 1113VIRIR=+330Loop 2: (eq'n 3) 2223VIRIR=+Loop 3: (Moving CW around loop 3) NN111222rise drop rise dropVIRIRV+ − + − = Don't need the equation from loop 3, because already have 3 equations. Remember! In a resistor, if we move in the direction of the current, V drops, ∆V is negative ; if we move in the direction opposite current, V rises, ∆V is positive. In a battery, if move from (–) to (+) terminal, V rises ; if move from (+) to (–) terminal, V drops. 1 2 3 hi V lo V Ι hi V lo V I3 R1 = 10 Ω I1 R2 = 10 Ω I2 V2 = 10 V R3 = 30 Ω V1 = 10 V Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-7 We now have 3 equations in 3 unknowns ( I1 , I2 , I3 ) : (1) 12II+=3I333 33 3)(2) 1113VIRIR=+(3) 2223VIRIR=+The physics part of this problem is over; now we have a messy algebra problem. How do we solve? Eqn (1) says we can substitute ( I1 + I2 ) for I3 . ⇒ Eliminate I3 in equations (2), (3): 111 12VIR (II)R=++ 222 12VIR (II)R=++Rearrange: A. B. 1113 2VI(RR)IR=++21322VIRI(RR=+ +Now have 2 equations (A, B) in 2 unknowns (I1, I2). Now combine these to eliminate either I1 or I2. For instance, can solve eqn A for I1 : 123113VIRIRR−=+ ( eqn C ) Then plug this into eqn B: 12323213VIRVRI(RRR⎛⎞−⎟⎜⎟=+⎜⎟⎜⎟⎜+⎝⎠23R)+ Can solve this for I2 (messy!). Then plug into (C) to get I1. Then plug back into (1) to get I3. Last update: 9/28/2009 Dubson Phys1120 Notes, ©University of ColoradoCrkt-8 Household Wiring cold hot (120VAC) Wall socket = 3-prong plug (1 -5 V) The short slot is the dangerous


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CU-Boulder PHYS 1120 - Circuits

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