Lect10 - Physics 212 Ph Lecture 10 Today's Concept:...

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Unformatted text preview: Physics 212 Ph Lecture 10 Today's Concept: Kirchhoff’s Rules Circuits with resistors & batteries Physics 212 Lecture 10, Slide 1 Music Music Who is the Artist? A) B) C) D) E) Norah Jones Diana Krall Jane Monheit Nina Simone Marcia Ball • A little gender equality • Anyone know the connection between Oscar Peterson and Diana Krall? Krall? • Both great Canadian jazz pianists – Peterson was one of Krall’s mentors Physics 212 Lecture 10 Your Your Comments: “I usually can't RESIST putting some lecture thoughts down, but usually can RESIST putting some lecture thoughts down but CURRENTLY CURRENTLY I need to DROP in my bed to GAIN some sleep. ;)” “Resistance Is Futile! All voltages will be assimilated. your individual Is Futile! All voltages will be assimilated your individual currents will be added to our own.” “I thought I had this, now you lost me. I don't understand understand how to figure out where the currents are going.” We’ll start simply, with the checkpoints, then to a calculation “h “how current flows and direction, how many fl di currents can there be” “The example with the wire connecting A and B between the resistors was confusing. I worked it out on paper and it seems like there should be a flow flow.” Thi This is a great example – we’ll do it at the end to make sure of the concepts “Saw an Explosions in the Sky concert after the exam last night. It was Physics 212 Lecture 10, Slide electrifying.” 3 Today’s Today’s Plan: • Summary of Kirchoff’s rules – these are the key concepts concepts • Example problem • Review Checkpoints Physics 212 Lecture 10, Slide 4 Last Time Resistors in series: Current through is same. Voltage drop across is IRi Reffective R1 R2 R3 ... Resistors in parallel: Voltage drop across is same. 1 Current through is V/Ri Reffective Solved Circuits R1 V 11 1 ... R1 R2 R3 R2 = R3 V I1234 R1234 R4 Physics 212 Lecture 10, Slide 5 5 New Circuit R1 R3 V1 V2 R2 How Can We Solve This One? R1 V1 V2 R3 R2 = V I1234 R12 THE ANSWER: Kirchhoff’s Rules Physics 212 Lecture 10, Slide 6 5 Kirchoff’s Voltage Rule V i 0 Kirchoff's Voltage Rule states that the sum of the voltage changes caused by any elements (like wires, batteries, and resistors) around a circuit must be zero. WHY? WHY? The potential difference between a point and itself is zero ! Physics 212 Lecture 10, Slide 7 Kirchoff’s Current Rule I in I out Kirchoff's Current Rule states that the sum of all currents currents entering any given point in a circuit must equal the sum of all currents leaving the same point. WHY? WHY? Electric Electric Charge is Conserved Physics 212 Lecture 10, Slide 8 Checkpoint Checkpoint 1 How many potentially different currents are there in the circuit shown? many potentially different currents are there in the circuit shown? A. 3 B. 4 C. 5 D. 6 E. 7 “There is only one node, splitting the initial current into two separate currents, meaning that there are 3 separate currents in the circuit.” A “The original+ 2 split+1 when they join back B together=4” C “Electric D potential decreases in every resistor.” Physics 212 Lecture 10, Slide 9 Checkpoint Checkpoint 1 How many potentially different currents are there in the circuit shown? many potentially different currents are there in the circuit shown? I1 I3 I2 I1 A. 3 B. 4 C. 5 I3 D. 6 E. 7 Look at the nodes! Top node: I1 flows in, I2 and I3 flow out A B Bottom C D node: I2 and I3 flow in, I1 flows out That That’s all of them! Physics 212 Lecture 10, Slide 10 Checkpoint Checkpoint 2 GAIN In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows. DROP If we are to write Kirchoff's voltage equation for this loop in the clockwise direction starting from point a, what is the correct order of voltage gains/drops that we will encounter for resistors R1, R2 and R3? A 60 drop, gain, gain A. drop, drop, drop B. gain, gain, gain C. B gain, drop, drop D. E. drop, drop, gain 50 C D 40 E 30 20 10 Physics 212 Lecture 10, Slide 11 0 Checkpoint Checkpoint 2 GAIN In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows. DROP If we are to write Kirchoff's voltage equation for this loop in the clockwise direction starting from point a, what is the correct order of voltage gains/drops that we will encounter for resistors R1, R2 and R3? A A. drop, drop, drop B. gain, gain, gain C. drop, gain, gain B gain, drop, drop D. E. drop, drop, gain C D E With the current Against the current VOLTAGE DROP VOLTAGE GAIN Physics 212 Lecture 10, Slide 12 1 2 1 Calculation Calculation 2V 1V I2 In this circuit, assume Vi and Ri are known. What is I2 ?? 1V • Conceptual Analysis: – Circuit behavior described by Kirchhoff’s Rules: • KVR: Vdrops = 0 • KCR: Iin = Iout •Strategic Analysis – – – Write down Loop Equations (KVR) Write down Node Equations (KCR) Solve Physics 212 Lecture 10, Slide 13 V1 R1 ++ - -+ + R3 - - V2 R2 I2 In this circuit, assume Vi and Ri are known. What is I2 ?? - V3 ++ I1 Calculation Calculation I3 - (1) Label and pick directions for each current (2) Label the + and – side of each element This is easy for batteries For resistors, the “upstream” side is + Now write down loop and node equations Physics 212 Lecture 10, Slide 14 V1 R1 ++ - -+ + R3 - - V2 R2 I2 - V3 ++ I1 Calculation Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? I3 - • How many equations do we need to write down in order to solve for I2? (A) 1 • Why?? – – (B) 2 (C) 3 (D) 4 (E) 5 We have 3 unknowns: I1, I2, and I3 We need 3 independent equations to solve for these unknowns (3) Choose loops and directions Physics 212 Lecture 10, Slide 15 V1 R1 ++ - -+ + R3 - - V2 R2 I2 - V3 ++ I1 Calculation Calculation I3 - In this circuit, assume Vi and Ri are known. What is I2 ?? • Which of the following equations is NOT correct? (A) (B) (C) (D) I 2 = I1 + I 3 - V1 + I1R1 - I3R3 + V3 = 0 - V3 + I3R3 + I2R2 + V2 = 0 - V2 – I2R2 + I1R1 + V1 = 0 • Why?? – – (4) Write down voltage drops (5) Write down node equation (D) is an attempt to write down KVR for the top loop Start at negative terminal of V2 and go clockwise • Vgain (-V2) then Vgain (-I2R2) then Vgain (-I1R1) then Vdrop (+V1) Physics 212 Lecture 10, Slide 16 R1 V1 R2 V2 R3 V3 I1 Calculation Calculation I2 In this circuit, assume Vi and Ri are known. I3 • We have the following 4 equations: 1. 2. 3. 4. I2 = I1 + I 3 - V1 + I1R1 - I3R3 + V3 = 0 - V3 + I3R3 + I2R2 + V2 = 0 - V2 – I2R2 - I1R1 + V1 = 0 • Why?? – – – What is I2 ?? We need 3 equations: Which 3 should we use? A) A) Any 3 will do B) 1, 2, and 4 C) 2, 3, and 4 We need 3 INDEPENDENT equations Equations 2, 3, and 4 are NOT INDEPENDENT • Eqn 2 + Eqn 3 = - Eqn 4 We must choose Equation 1 and any two of the remaining ( 2, 3, and 4) Physics 212 Lecture 10, Slide 17 R1 V1 R2 V2 R3 R 2R R V3 2V V V I1 I2 I3 I1 I2 I3 Calculation Calculation In this circuit, assume Vi and Ri are known. assume What is I2 ?? • We have 3 equations and 3 unknowns. I2 = I1 + I 3 V1 + I1R1 - I3R3 + V3 = 0 V2 – I2R2 - I1R1 + V1 = 0 (6) Solve the equations •The solution will get very messy! Simplify: assume V2 = V3 = V V1 = 2V R1 = R3 = R R2 = 2R Physics 212 Lecture 10, Slide 18 Calculation: Calculation: Simplify In this circuit, assume V and R are known. R 2R R 2V V V I1 I2 What is I2 ?? • We have 3 equations and 3 unknowns. I2 = I1 + I 3 -2V + I1R - I3R + V = 0 (outside) -V – I2(2R) - I1R + 2V= 0 (top) I3 current direction • With this simplification, you can verify: I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (-2/5) V/R Physics 212 Lecture 10, Slide 19 FollowFollow-Up 2V R V 2R a I1 • We know: I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (-2/5) V/R I2 b R V I3 • Suppose we short R3: (A) (B) (C) (D) Vab Vab Vab Vab remains the same changes sign increases goes to zero What happens to Vab (voltage across R2?) R Why? Redraw: Bottom Loop Equation: Vab + V – V = 0 Vab = 0 2R a 2V V I1 I2 b V c d I3 Physics 212 Lecture 10, Slide 20 A V B R R Is there a current flowing between A and B ? A) Yes B) No A & B have the same potential Current flows from battery and splits at A No current flows between A & B Some current flows down Some current flows right Physics 212 Lecture 10, Slide 21 Checkpoint 3a Consider the circuit shown below. Note that this question is not identical to the similar looking one I1 I2 you answered in the prelecture. Which of the following best describes the current flowing in the blue wire connecting points a and b? A. Positive current flows from a to b B. Positive current flows from b to a C. No current flows between a and b “current flowing through R is greater than current flowing through 2R, so current must flow from a to b” “potential at b is higher than that at a” “It has equal to the top and bottom of it” Physics 212 Lecture 10, Slide 22 Checkpoint 3a Consider the circuit shown below. Note that this question is not identical to the similar looking one I1 I2 you answered in the prelecture. I1 I I2 I4 I3 Which of the following best describes the current flowing in the blue wire connecting points a and b? Which of the following best describes theB.urrent flowing in the blue wire connecting points a and b? of the following best describes c Positive current flows from b to a A. Positive current flows from a to b the current flowing in the blue wire connecting points C. No current flows between a and b I1R – I2 (2R) = 0 I2 = ½ I1 I4R – I3 (2R) = 0 I4 = 2 I 3 a: I1 = I + I3 b: I + I2 = I4 I1 - I3 + ½ I1 = 2I3 I1 = 2I3 I = +I3 Physics 212 Lecture 10, Slide 23 Prelecture Prelecture What is the same? Checkpoint Current flowing in and out of the battery 2R 3 2R 3 What is different? Current flowing from a to b Physics 212 Lecture 10, Slide 24 I 2/ I 3 R 1/ I 3 a V 2/ I 3 2/ I 3 2 1/ I 3 2R b R 1/ I 3 V/2 2R 1/ I 3 1/ I 0 3 1/ I 3 2/ I 3 Physics 212 Lecture 10, Slide 25 Consider the circuit shown below. Checkpoint 3b In which case is the current flowing in the blue wire connecting points a and b the largest? which case is the current flowing in the blue wire connecting points the largest? A. Case A B. Case B C. They are both the same “since there is less resistance the voltage drop is less. so the current is greater” “The current will want to avoid the 4R more than it will want to avoid the 2R.” th 4R th it th 2R “Since the resistance on both top and bottom is the same, the current in a line 'ab'' (in both cases) is zero.” ab Physics Physics 212 Lecture 10, Slide 26 Consider the circuit shown below. Checkpoint 3b IA c IB c In which case is the current flowing in the blue wire connecting points a and b the largest? which case is the current flowing in the blue wire connecting points the largest? A. Case A B. Case B C. They are both the same Current will flow from left to right in both cases In In both cases, Vac = V/2 I2R = 2I4R IA = IR – I2R = IR – 2I4R IB = IR – I4R Physics 212 Lecture 10, Slide 27 Model Model for Real Battery: Internal Resistance + r V0 r R VL V0 R VL Usually can’t supply too much current to the th load without voltage “sagging” Physics 212 Lecture 10, Slide 28 Kirchhoff’s Laws (1) Label all currents Choose any direction (2) Label +/- for all elements Current goes + - (for resistors) Battery signs fixed! (3) Choose loops and directions Must start on wire, not element. (4) Write down voltage drops First sign you hit is sign to use. (5) Write down node equations R1 A + I1 - + V -1 B V2 + R2 I2 + + V3 I3 R3 R5 - + I4 R4 + +I 5 Iin = Iout (6) Solve set of equations Physics 212 Lecture 10, Slide 29 17 ...
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