between the terminals equivalent to the potential drop over the resistor R 2

Between the terminals equivalent to the potential

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— between the terminals (equivalent to the potential drop over the resistor R 2 ). Let’s calculate these quantities. Shorting the terminals out is equivalent to putting a 0 Ohm resistor in parallel with R 2 . The current that flows is what you would get if R 2 were not even present: I SC = V b R 1 . On the other hand, if we leave the terminals open, the current that flows through the circuit is I = V b / ( R 1 + R 2 ). The open circuit voltage is thus given by V OC = V R 2 = IR 2 = V b R 2 R 1 + R 2 . The Th´ evenin equivalent resistance is defined as the ratio of these quantities: R T V OC I SC . For this particular circuit, it takes the value R T = V b R 2 R 1 + R 2 × R 1 V b = R 1 R 2 R 1 + R 2 . The solution for the charge on the capacitor is thus finally given by Q ( t ) = CV b R 2 R 1 + R 2 [1 - exp( - t ( R 1 + R 2 ) /CR 1 R 2 )] . 86
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9.5 Th´ evenin in general Th´ evenin’s theorem is much more general than this simple example. In general, it means that you can be given any grungy mess of resistors and batteries and reduce it to a single battery with EMF V OC and a single resistor R T : anything like this Huge mess of resistors and batteries can be reduced to this V OC R T If you know what the big mess inside the “black box” is made out of, then you can figure out V OC and R T as we did for the example above. In some situations, you don’t know. For example, you might have a piece of lab equipment that you need to use for some experiment, and you need to know how to treat it in a circuit. In that case, you measure V OC ; you short out the terminals and measure I SC ; and then you know R T = V OC /I SC . Th´ evenin’s theorem works only when all the elements inside the box obey Ohm’s law. This means that the relationship between current and voltage for every element is linear . This in turn means that the relationship between current and voltage for any combination of circuit elements in the box must be linear. (Why? Add a bunch of lines together: you get a line!) The procedure for determining the Th´ evenin equivalence of a circuit simply assumes that you have some linear relationship between the voltage and current at the terminals. V OC tells us where this relationship crosses the voltage axis; I SC tells us where it crosses the current axis. R T is the slope. (Negative slope, really, since the line rises in the “wrong” direction; same difference.) 87
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V I OC V I SC Slope = R T 88
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  • Spring '08
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