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17.5 series and parallel resistors ;Kirchhoffs rules Statement of Kirchhoffs Rules Junction Rule ( I = 0) – The sum of the currents entering any point must equal the sum of the currents leaving that junction A statement of Conservation of Charge I 1 = I 2 + I 3

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– You must go around the loop in one direction – The sum of the measured will equal zero Loop Rule ( V = 0) The sum of the potential changes around any closed circuit loop must be zero (a)The voltage across a battery is taken to be positive (a voltage rise) if traversed from – to + and negative if traversed in the opposite direction. (b)The voltage across a resistor is taken to be negative (a drop) if the loop is traversed in in the direction of the assigned current and positive if traversed in the opposite direction V ba = - IR V ba = IR

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Example Calculate the current I flowing into the node (3+ I ) A = 2 A I = 2 -3 = -1 A The current flowing into the node is – 1 A which is the same as +1 A flowing out of the node Example Calculate the current I defined in the diagram I +2 A = - 4 A I = (- 4 – 2 ) A = - 6 A I is in the opposite direction I + I = 6 A I = ( 6 – 6 ) A = A

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Kirchhoffs rules : The sum of the potential changes around any closed circuit must be zero There are two ways to connect circuit elements. 1)Series combination: R3R3 R2R2 R1R1 +- ε I III V1V1 V3V3 V2V2 ( a ) RsRs +- V I ε ( b ) Apply the Loop Rule Figure (a) three resistors in series ( b) the equivalent resistance R s leads to the same current I, The current is the same in resistors because any charge that flows through one resistor flows through the other but the potential differences across them are not the same

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2) Parallel combination (a ) ( b ) I Figure ( a ) three resistors in parallel. ( b ) the equivalent single resistance R p produces the same current I V RpRp +- I I AB I ε R3R3 R2R2 R1R1 + - I I3I3 I1I1 I2I2 AB I ε

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Example (a ) find the equivalent resistance of the resistors in figure a ( b ) the current I in each resistor (a ) (b ) ( c ) Solved in the text book

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T.Norah Ali Almoneef8 V I 30 From the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it? The resistor with the largest resistance (30 ) Which resistor has the greatest current flow through it? Same for all because series circuit If we re-ordered the resistors, what if any of this would change? Nothing would change Conceptoal question

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A) find the current in the circuit shown in the figure. B ) find the potential difference across each circuit element In the figure, we had a 3kΩ, 10 kΩ, and 5 kΩ resistor in series, Example

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From the figure find ( a ) I ( total current ), R p ( total resistance ) ( b ) I 1, I 2, I 3

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Four resistors are connected as shown in figure. Find the equivalent resistance between points a and c. A.4 R. B.3 R. C.2.5 R. D.0.4 R. E.Cannot determine from information given. Example

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V I 30 From the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it? The resistor with the largest resistance (30 ) Which resistor has the greatest current flow through it? Same for all because series circuit If we re-ordered the resistors, what if any of this would change? Nothing would change Conceptual questions

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I Total resistance would increase Total current would decrease Voltage across each resistor would decrease (All voltage drops must still sum to total in series circuit; Kirchhoffs law of voltages) Current through each resistor would be lower (total current decreased, but same through each one ) If we added a resistor in series with these, what would happen to the total resistance, total current, voltage across each resistor, and current through each resistor?

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from the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it? All the same in parallel branches Which resistor has the greatest current flow through it? The path of least resistance (10 ) What else can you say about the current through each branch? They will sum to the total I (currents sum in parallel circuits; Kirchhoffs law of current) Conceptual questions

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If we added a resistor in parallel with these, what would happen to the total resistance, total current, voltage across each resistor, and current through each resistor? Total resistance would decrease Total current would increase Voltage across each resistor would still be V Current through each resistor would be higher and would sum to new total

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17.12 Kirchhoffs rules in complex circuits Kirchhoffs rules permit us to analyze any dc circuit.including circuits too complex Using the two rules (1) the sum of all the potential drops around any closed path in a circuit is equal to zero. (2) The current entering any point = The current leaving. Example Find the current in the circuit shown in the figure Solved in the text book

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A) 2 A B) 3 A C) 5 A D) 6 A E) 10 A 5 A 8 A 2 A P What is the current in branch P? The current entering the junction in red is 8 A, so the current leaving must also be 8 A. One exiting branch has 2 A, so the other branch (at P) must have 6 A. 5 A 8 A 2 A P Junctio n 6 A S Conceptual questions

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2 V 2 6 V 4 V I1I1 I3I3 I2I2 Which of the equations is valid for the circuit below? A) 2 – I 1 – 2I 2 = 0 B) 2 – 2I 1 – 2I 2 – 4I 3 = 0 C) 2 – I 1 – 4 – 2I 2 = 0 D) I 3 – 4 – 2I = 0 E) 2 – I 1 – 3I 3 – 6 = 0 Conceptual questions

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ΔV ab if one battery is reversed? a b ΔV ab = 27V Δv ab = 9v quiz Calculate ΔV ab

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quiz Calculate the current in the circuit.

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Find the current I, r and ε. I = 3 A r =2 =-5 V quiz

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Calculate the currents I 1, I 2, and I 3 in the three branches of the circuit in the figure. Quiz I 1 = A. I 2 = 2.6 A. I 3 = 1.7 A.

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summary Kirchhoffs Rules Loop Rule Series combination: Parallel combination

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Home work 45,46,71

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