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Topic 5.2 Extended A – Resistances in series and parallel The easiest way to picture a resistor is as a road construction zone in a highway: Here is the highway, supporting the maximum traffic flow. Here is the construction zone, constricting the flow of traffic: Of course, what will really happen is that the blocked lane of traffic will merge with the moving one. Cutting the speed by about a factor of two. NOTE: If instead of a two lane highway you had a three lane highway, a roadblock in one lane would slow the traffic by about a third. A roadblock in two lanes would slow the traffic by about two thirds. FYI: The roadblocks are one after the other, AKA INLINE. We say that they are in series.

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Now imagine two parallel two-lane highways: Each highway begins by supporting the original flow of traffic from the previous slide. Topic 5.2 Extended A – Resistances in series and parallel Each highway has its traffic flow cut in half. But there are TWO highways. FYI: The roadblocks are in parallel. Thus the traffic flow is equivalent to a single unblocked highway.

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Topic 5.2 Extended A – Resistances in series and parallel We can think of resistances in series in another way: Placing resistors in series increases the EFFECTIVE LENGTH of the resistor. L L L 3L SERIES R L Placing resistors in parallel increases the EFFECTIVE AREA of the resistor. A A A PARALLEL R 1 / A 3A FYI: Combining two or more resistors is series produces a resistance that is larger than any single resistor. FYI: Combining two or more resistors is parallel produces a resistance that is smaller than any single resistor.

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Now for some formulas... Consider three resistors in SERIES, as shown: Topic 5.2 Extended A – Resistances in series and parallel + - SERIES R1R1 R2R2 R3R3 V1V1 V2V2 V3V3 Note that the current I is the same everywhere. Recall that the sum of the voltages in a loop equals the terminal voltage of the battery: Thus V V = V 1 + V 2 + V 3 = IR 1 + IR 2 + IR 3 = I(R 1 + R 2 + R 3 ) = IR s where R s is the equivalent series resistance. R s = R 1 + R 2 + R 3 Equivalent Series Resistance R s

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Consider three resistors in PARALLEL, as shown: Topic 5.2 Extended A – Resistances in series and parallel + - PARALLEL R1R1 R2R2 R3R3 A1A1 Note that the voltage V is the same everywhere. Because of the conservation of charge, the sum of the currents going through the resistors must equal the total current: V V = V 1 = V 2 = V 3 VRpVRp Equivalent Parallel Resistance R p A2A2 A3A3 ApAp I1I1 I2I2 I3I3 I I = I 1 + I 2 + I 3 From Ohm's law, = + + V1R1V1R1 V2R2V2R2 V3R3V3R3 VRpVRp VR1VR1 VR2VR2 VR3VR3 1Rp1Rp 1R11R1 1R21R2 1R31R3

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Find the equivalent resistance (from A to B) of the following resistor setup: Topic 5.2 Extended A – Resistances in series and parallel The resistors are in SERIES so that 25 50 100 R s = R 1 + R 2 + R 3 R s = 25 + 50 + 100 R s = 175 A B

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Find the equivalent resistance of the following resistor setup: Topic 5.2 Extended A – Resistances in series and parallel The resistors are in PARALLEL so that 25 50 100 A B 1Rp1Rp = + + 1R11R1 1R21R2 1R31R3 1Rp1Rp 1 50 1 25 1 100 1Rp1Rp = + + 2 100 4 100 1 100 1Rp1Rp = 7 100 R p = 14.28

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Find the equivalent resis- tance of the following resistor setup: Topic 5.2 Extended A – Resistances in series and parallel Two resistors are in PARALLEL so that 25 50 100 A B 1Rp1Rp = + 1R11R1 1R21R2 1Rp1Rp = 5 100 R p = 20 1Rp1Rp = + 1 25 1 100 4444 20 50 A B REDUCED FORM The resistors are in SERIES: R s = R 1 + R 2 R s = 50 + 20 R s = 70

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The current through the circuit with three resistors in SERIES... Topic 5.2 Extended A – Resistances in series and parallel + -...will stop if any one resistor is removed. + - A1A1 A2A2 A3A3 ApAp The current through the circuit with three resistors in PARALLEL... Will NOT stop if any one resistor is removed.

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