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Problem Set 2 Solutions p. 1 of 6 Fall 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2009 Problem Set 2 Solutions Problem 1: RC Circuit In the circuit shown, the switch S has been closed for a long time. At time t=0 the switch is opened. It remains open for “a long time” T, at which point it is closed again. Write an equation for (a) the charge stored on the capacitor and (b) the current through the switch as a function of time. (a) The capacitor begins uncharged. When the switch is opened at t=0 we have an RC circuit with R = 150 k Ω and C = 10.0 μ F, so τ = RC = 1.50 s. The final voltage (after an infinite time) on the capacitor will be the battery voltage (10.0V) so we can write the equation for voltage on the capacitor during charging as: ( ) ( ) [ ] 1.50 s 11 0 . 0 V 1 f o r tt CF VV e e tT τ −− =− = < During discharge the capacitor starts at its value at t = T (which we can get with the equation above) and then drives through the 100 k Ω resistor and the switch. The time constant is thus now only 1.00 s. So the voltage goes like: () ( ) ( ) [] 1.00 s 1.50 s 0 10.0 V 1 for T C VV e e e == Of course, we were asked for charge, not voltage, for which we use QC V = . (b) When the switch is open (between t = 0 and T) there is no current through it. When it is closed, however, current flows both from the battery AND from the capacitor, both in the same direction (from top to bottom). So they add. The battery just drives a current by ohm’s law through the 50.0 k Ω resistor. The capacitor current we can get from the above voltage and the 100 k Ω resistor. So add them and we have: ( ) 1.00 s 1.50 s 10.0 V 10.0 V 1f o r 100 k 50 k T I ee t T + ΩΩ Note that we could replace V/k Ω with mA, but there is no particular need to do so. Exam Problem Problem 2: RL & RC Circuits In experiments #2 & 3 you looked at series RL & RC circuits, saw how to measure the time constants in these circuits and how, from plots of current and voltage, to determine the inductance or capacitance and the resistance in the circuit. In this problem we repeat
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Problem Set 2 Solutions p. 2 of 6 Fall 2009 part of that experiment on a different circuit board (values are different than what were used in the lab). You set up a series RL or RC and drive it with a power supply, which feeds in a square wave potential (on then off). Using this setup, you record the voltage and current from the power supply: 0 1 02 03 04 05 06 07 08 0 0 1 2 3 4 5 0 10 20 Current (A) Time (ms) Voltage (V) From the information above determine the time constant of the circuit and the resistance and capacitance or inductance (tell us which!) that makes up the circuit. Note that although your numbers will be approximate (you have to read them off of the plot) I always choose very nice numbers for this type of problem. Since current is initially zero and then flows, this must be an RL circuit.
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This note was uploaded on 05/06/2010 for the course 8 8.02 taught by Professor Hudson during the Fall '07 term at MIT.

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