6. RC Circuits* a (b) Which way do the electrons move in the upper wire? c R 6-1. * William A Schwalm RC Circuits

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1 6. RC Circuits* Objective: You will become familiar with the fact that the current flowing to or from a capacitor plate is the same as charge flow onto or off from the plate. You will see that in simple RC circuits the voltages and currents vary exponentially in time. As an application your team is asked to solve the practical problem of measuring the resistance of a multi-meter. The learning objectives are the following: 1. To be able to explain how charge on either plate of a capacitor relates to the current flowing to or from the plate. 2. To understand why the internal resistance of a meter must be taken into account when measuring voltage across a circuit element with large resistance. 3. To recognize the time scale for changes in voltage and current in a circuit comprising a resistor and capacitor; to see that it is controlled by the product RC, which is a time constant. 4. To apply this understanding to a practical problems in electrical measurement. (These concepts are described in your textbook.) Reading assignment: Review capacitors, circuits and RC circuit. Read the following sections. (Section numbers may be slightly different depending on the edition of your textbook: Check the section titles.) Knight, Jones and Field : 21.7 Capacitance and Capacitors, 23.7 RC circuits. Serway and Vuille (212): 16.6 Capacitance, 16.7 Parallel Plate Capacitor, 18.5 RC Circuits Serway and Jewett (252): 26.1 Definition of capacitance, 28.4 RC Circuits Pre-lab exercises: Bring the answers to the following questions with you to lab. The instructor will check them when you enter. You will turn them in finally with your report. 1. Before S is closed, the capacitor C is uncharged. When it is closed, a charge begins to build up on the upper plate. (a) What is the sign of the charge building up on the upper plate? b B S C a (b) Which way do the electrons move in the upper wire? c R d (c) Does a charge accumulate on the lower plate as well? If so what sign does it have? (d) As the charge builds up on the lower plate, which way do the electrons move in the lower wire? * William A Schwalm

2 (e) In both the top and the bottom wires the current flow is in the direction opposite to that of the electron motion. Why is that? (f) To charge the capacitor, current flows out of the positive terminal of the battery (top). But an equal current must flow into the negative terminal. Since no charge can flow across empty space between the two capacitor plates (disks in the figure) how does charge get from the positive battery terminal all the way around to the negative terminal? Explain. 2. Consider again the same circuit as in problem 1. (a) After the switch S has been closed, and the capacitor is partly charged (not fully charged) how is the voltage difference V a V d across the capacitor related to the charge Q on the upper plate of the capacitor? Write an equation. (b) At the same instant as in part a), how is the voltage difference V d V c across the resistor related to the current I flowing through the resistor? Give an equation. (c) How is the voltage difference V b V c related to the battery voltage V o? (d) Use your responses to the parts above to come up with an equation involving Q and I and the battery voltage V o. (e) Looking at the equation you have written, explain why it is that, as the charge Q builds up on the capacitor plate, the rate of charging decreases. (What variable represents the rate of charging?) 6-2

3 (f) If you were to wait a very long time after closing S, what would be the limiting values of the charge Q on the upper capacitor plate, the voltage V d at the other plate, and the current I? 3. (Part c is for Phys 212 and 252) Suppose a multi-meter is used to measure voltage differences in the circuit below, represented as a schematic diagram. Suppose the battery is 10 Volts and each resistor is 47 mega Ohms. (a) Without applying the meter, what should be the voltage difference between points b and c? (b) If you want to measure the voltage difference from point b to point a (a positive value) where would you connect the V wire and where would you connect the COM wire? Indicate these connections with lines. a R 1 (+) b V OFF V c R 2 A A V COM A 10A (c) Now suppose you actually make this measurement and find V V 4.89 Volts. a b What is the resistance of the volt meter? (Actually, the effective volt-meter resistance will depend on which scale you have it set to, so this would be for a particular meter setting.) 6-3

4 Scenario: The development team at Ryan Measurement Technologies is assigned to write instructions for using standard resistors and a multi-meter to measure capacitance. As usual they have passed this responsibility to your group. The idea is simple, but to prepare for the job, you will need to explore some basic concepts. So what you need to produce for the company is a short paragraph and the necessary data that would tell other employees how to use an off-theshelf multi meter with a set of standard resistors and a stop watch to measure capacitance. In-class questions for group discussion: Answer these on the white board when you are prompted to do so by your instructor. Later the responses of your group will be submitted as part of your report. 1. What is the definition of capacitance? Don t look it up, try to come up with it. Discuss it in your group and try to figure it out. Give a formula and explain what each part means. Also, what would be the difficulty in trying to measure capacitance by just measuring each quantity in its definition? 2. At your work station you will be provided with a battery, 2 light bulbs (which are sort of resistors) 2 switches and a big capacitor. Set up the following circuit and use it to charge and then discharge the capacitor through the light bulbs. Develop an explanation of what you see. The teaching assistant will call on one of you to explain. (+) 3. Exponential decay: (Parts b and c for Phys 212 and 252) Several important processes in nature are exponential, meaning that when you graph them you see the graph of an exponential function. One such process is radioactive decay. A certain isotope has a half life of 100 days. That means if you start with nuclei of the isotope at t = 0, then 100 days later you have (about) left, and so on. Starting with nuclei, make a graph (with actual numbers, not just a cartoon) on your white board showing number verses time. Let the time axis on the graph extend out to two years. Include axis scales and indicate the points where you actually calculated values using tiny circles. (a) Copy your sketch here neatly using a ruler. 6-4

5 (b) The time constant for any exponential decay is defined using the exponential function. So if y decays exponentially in time then t / y yo e where e is the base of natural logarithms, e = or e is about 3. (What is y o?) This defines the time constant : When t increases by one time constant, y decreases to about a third of what it was. Use this idea to estimate the time constant for radioactive decay of the isotope from the graph in part a). Explain. Outline your reasoning here. (c) Recall that the log of a product is the sum of the logs, and that the log of x p is just p times the log of x. The natural logarithm ln N of a number is defined in such a way that the natural log of e is just 1, so t / t t ln yoe ln yo ln e ln yo So if you were to graph natural log of y versus time t, you would get a straight line. What would the slope and intercept be? Therefore, if you were to plot the natural log of the number of nuclei surviving at time t versus t, how could you determine the time constant? Discuss this and report your answers on the white board. Write it here too. (d) Recall that electric potential, or voltage, is potential energy per charge. That means, for example, that the unit of potential, i.e. the Volt, is really a Joule/Coulomb. Use dimensions (i.e. units) to do the following. The fundamental dimensions are length (L), mass (M), time (T) and charge (Q). Thus for example the dimensions of potential would be Energy/Q = M L 2 T -2 Q -1. First, from Ohm s law, find the dimensions of resistance R. Then from any equation you know, find the units of capacitance C. Finally, show that RC has units of time. RC is the time constant in an RC circuit. Thus RC. 6-5

6 Problem 1 You need to find a way to use a stop watch and a standard capacitor to measure the resistance of a multi-meter used as a volt meter. You may consider the following circuit, where the voltage source is not really a battery but the DC power supply that you connect to through the wall terminals. Measurement plan: Work out a measurement plan, using the capacitor and stopwatch, for measuring the resistance of the meter. Tell how to connect the meter, what to do with the switch, what data to collect, what the group members will do and so on. Then tell how the data will be used, graphed, analyzed etc. in order to get the required value. V Record your plan here: Implementation: Carry out your plan. Collect and record the data here. Also, make notes you will need later to figure out what the data mean. Analysis: Show your analysis here and calculate the best value you can for the meter resistance. Be aware that your employer expects your team to work out the best solution you can, not the easiest or quickest. 6-6

7 Product: Write up a short description of how to measure the resistance of a voltmeter using a standard capacitance and a stop watch for use by other employees of Ryan Measurement Technologies. This should be one or two short paragraphs of careful professional writing. Be sure to include a critical discussion of anything unexpected that might limit the usefulness of this method of measuring capacitance. Discuss possible sources of error, as usual. Problem 2 Now you are given some unknown capacitance values to measure. You are to measure them using a stop watch and the same multi-meter. Measurement plan: Discuss the measurement plan with your group members. Then record your measurement plan here. Give all necessary details. (A drawing might be nice.) Implementation: Do it and record the necessary data here. 6-7

8 Analysis: Work out the values of the unknown capacitors here. Show the steps of your calculations. Explain your work succinctly where appropriate. Summary: After discussing with your team members, comment on which parts of the exercises relate to which of the learning objectives and how they relate. 6-8