Physics Lab 202P-8 Understanding RC Circuits NAME: LAB PARTNERS: LAB SECTION: LAB INSTRUCTOR: DATE: EMAIL ADDRESS: Penn State University Created by nitin samarth Physics Lab 202P-8 Page 1 of 16
Physics Lab 202P-8 Software List Science Workshop Microsoft Excel Equipment List (all items marked with * are in the student kit, others are supplied at the time of the lab) Science Workshop Interface + voltage probes One 1 µf capacitor. One 1 F "super-capacitor" A two position switch. A 100 kω resistor *Two 1.5 V batteries + a battery holder *Flashlight bulb (use one of the "long" bulbs in your kit) + bulb holder *Hookup wires with alligator clips Penn State University Created by nitin samarth Physics Lab 202P-8 Page 2 of 16
Prelab checkbox: Satisfactory Unsatisfactory Physics Pre-lab 202P-8 Understanding RC Circuits Name: Section: Date: (Read this & answer the questions before coming to lab) Summary of relevant concepts: When an initially uncharged capacitor of capacitance C is connected in series with a resistance R and a battery of emf E, the charge on the capacitor and the voltage across the capacitor gradually change from 0 to a final steady state value. This charging process is described by the following equations: V (t) = E (1 - e -t/rc ) Q (t) = CE(1 - e -t/rc ) The quantity "RC" has dimensions of TIME & is called the "time constant" of the circuit. When a capacitor C with an initial charge Q 0 = CV 0 is connected across a resistance R, it discharges, and the charge on the capacitor and the voltage across the capacitor gradually become 0. This discharging process is described by the following equations: V(t) = V 0 e -t/rc Q(t) = CV 0 e -t/rc Penn State University Created by nitin samarth Physics Lab 202P-8 Page 3 of 16
Pre-lab Questions: Consider the RC circuit shown below. The capacitor initially has no charge on it; hence the initial voltage across it is also 0. The battery is ideal and has an emf E. The switch is closed at time t = 0. Answer the questions that follow. C E - + R Q1. Without resorting to the charging equations for an RC circuit, how would you convince a friend that: (a) immediately after the switch is closed, the voltage across the capacitor is still is 0? (Start with Q= CV: take a time derivative. If V were to suddenly jump from 0 to some finite value, what would the current in the circuit equal?) This is a crucial property to remember: you CANNOT CHANGE the VOLTAGE on a capacitor SUDDENLY! (b) A long time after the switch has been closed, the voltage across the capacitor is E? Penn State University Created by nitin samarth Physics Lab 202P-8 Page 4 of 16
Q2. Again, without resorting to explicit charging equations, how would you explain to someone what the value of the CURRENT in the circuit is: (a) immediately after the switch is closed? (b) a very long time after the switch has been closed? Penn State University Created by nitin samarth Physics Lab 202P-8 Page 5 of 16
Q3. Now, look at the charging equation: V (t) = E (1 - e -t/rc ) Suppose you carried out an experiment in which you measure V as a function of time t. How would you plot your data so that: (a) You could easily verify that the data obeyed the above equation? (b) You could readily extract the time constant τ = RC from your data? (Note that the easiest type of data to fit is a STRAIGHT LINE! So, the idea is to transform the equation above into a form that allows you to see whether the data follow a straight line.) Penn State University Created by nitin samarth Physics Lab 202P-8 Page 6 of 16
Q4. Using the charging equation, figure out the value of the voltage across the capacitor at the following times: (a) t = 0 (b) t = RC (c) t = 3RC (d) t = 10 RC Penn State University Created by nitin samarth Physics Lab 202P-8 Page 7 of 16
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Lab Activity: RC Circuits. Activity 1: Learning to make measurements with the Science Workshop Interface In an earlier lab, we examined how capacitors are used to store energy. However, our understanding of the charging and discharging process was only treated at a qualitative level. In this lab, we approach the problem quantitatively. For the experiments, we need to use a software program called "Science Workshop" located in the "Physics" group on the START menu. This program allows us to measure and store the time variation of the voltage across any two points in a circuit by sending the signal through an interface. The measured data can then be displayed in a table and/or as a graph. Our aim is to acquire voltage measurements in an RC circuit as a function of time. After exporting the data into an Excel spreadsheet, we will carry out a quantitative analysis of the data. Data acquisition and analysis using such computer-interfaced instruments is an integral aspect of all modern science and technology laboratories. Before we proceed to the actual experiments, take some time to familiarize yourself with Science Workshop by following the procedure below: Start Science Workshop: You should see the active window shown below. Drag this symbol to ANALOG CHANNEL A. Click & drag the "analog plug" icon shown above to analog channel A Click & drag the "Graph" icon shown above to analog channel A Click & drag the "Table" icon shown above to analog channel A When you do this, you should see the screen shown at the top of the following page. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 9 of 16
Now, you are ready to use the program. Double click anywhere on the x-axis of the graph. When you do this, you will get a screen that allows you to set the parameters for the x-axis. You can leave the settings unchanged for now: but remember that you will need to change them later in this lab. Double click anywhere on the y-axis of the graph. When you do this, you will get a screen that allows you to set the parameters for the y-axis. Since the maximum voltage of your two batteries in series is around 3 V, set the maximum value of the voltage for 4 V. The default rate at which the program measures data is 10 Hz (i.e. 10 readings per second), which is more than sufficient for your first experiment. This is the default rate and can be changed by clicking the "sampling options" icon. Try this. Note that you will have to change this rate later in the lab. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 10 of 16
Now, we are ready to make a trial set of measurements! Set up the circuit shown below, making sure that you connect the batteries to each other with the correct polarities (i.e. "positive" end of one battery connected to the "negative" terminal of the second battery)! Note that the voltage probes from the Science Workshop interface box should already be connected to Analog Channel A. Interface box bulb I 1 F A B Connect the terminals of the capacitor to each other for a few seconds using some wire. Hook up the circuit shown above but do NOT complete the circuit yet -- i.e. leave the switch OPEN. Note that the switch has two possible closed positions, labeled A and B in the diagram below. Make sure that the voltage probes are properly connected to measure the voltage V C across the capacitor. Now, click on the "REC" icon on your program screen and immediately complete the circuit by closing the switch in position A. You will see your data getting recorded in the table and it will also be displayed on the graph. After the capacitor seems to have charged completely, flip the switch to position B. Stop recording data by clicking the "STOP" icon when the capacitor has discharged completely. Note now that you have (a) a graph that displays V C vs. t and (b) a table that shows the actual voltage readings at different times. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 11 of 16
Q1. Sketch (qualitatively) below the time variation of the measured voltage across the capacitor. Label the point at which you switched from charging the capacitor to discharging the capacitor. (Alternatively, you can print the graph and attach it to your lab report, if you so wish.) V C (V) TIME (s) Q2. Without doing any explicit fits to the data, use the graph displayed in the Science Workshop window to estimate the time constant of this circuit. Explain how you did this! Penn State University Created by nitin samarth Physics Lab 202P-8 Page 12 of 16
Next, repeat the experiment but this time measure the voltage V R across the BULB. Q3. Sketch (qualitatively) below the time variation of the measured voltage V R across the bulb. Label the point at which you switched from charging the capacitor to discharging the capacitor. (Alternatively, you can print the graph and attach it to your lab report, if you so wish.) V R (V) TIME (s) Q4. Explain why the voltage across the bulb varies as observed. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 13 of 16
Activity 2: Quantitative Analysis of an RC Circuit In the circuit that you set up earlier, replace: (a) the 1 F capacitor by the 1 µf capacitor; (b) the bulb by a 100 KΩ resistor. Change the x-axis parameters in the Science Workshop graph to read "milliseconds" instead of "seconds" Change the "sampling options" data rate from 10 Hz to 100 Hz. Then, repeat the first experiment that you carried out earlier: i.e. record the voltage V C across the capacitor, but STOP taking data after the capacitor is completely charged. We know have data for the CHARGING of the capacitor only. Leave the switch in position A. Export the measured data to an Excel spreadsheet for analysis as follows: Highlight the column of data containing voltage readings Copy this to the clipboard (ctrl-c or from the edit menu) Open Microsoft Excel and start a new spreadsheet Paste (ctrl-v) the copied data into a column -- preferably the second column in your empty spreadsheet. Now, go back to Science Workshop and record data again, this time flipping the switch from A to B (i.e. you are now DISCHARGING the capacitor). Export this data also to an Excel spreadsheet. Note that there is no immediate way in which to export the "time" data. So, we need to manually make another column for time in seconds -- preferably the first column in each of your spreadsheets. The easiest way to do this is to use the "ROW( )" function in Excel. If you insert this function into any cell with the argument empty (i.e. leave the parentheses empty), it simply returns the value of the row. So, use the following menu sequence: insert function ROW( ) into the first cell in column 1. To copy this function into the whole column, just copy the first cell where you placed the function; then, highlight all the desired cells in a column and hit "ctrl-v" or "paste." Go ahead & do this: you should now have your first column filled with integers that indicate the number of the particular row. Next, recall that the data was taken at a rate of 100 Hz (i.e. every 0.01 second). Note that we need to decide where we want to define t = 0 (this is when you closed the switch and started to charge the capacitor). Do this by simply looking at the numbers in your data. Then, replace the "ROW()" function by something like [ROW() - N]*0.01, where N is the number of the row that contains the t = 0 measurement. Copy and paste this function into all cells in the first column. We now have two columns of data each for charging the capacitor and discharging the capacitor: column 1 contains time in seconds, starting with t = 0 and column 2 contains the voltage measurements. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 14 of 16
Q5. Use Microsoft Excel to make plots of V C vs t during the charging and discharging of the capacitor. Remember to label the axes on your graphs correctly with proper units, etc. Include these plots with your lab report. Q6. From the graphs that you just made, estimate the time constant of your circuit, explaining how you do this. Q7. Next, use the results of the last pre-lab question to plot the charging data in such a way that tests whether it obeys the capacitor charging equation. Include this plot with your report. From the plot, determine the value of the time constant. Explain below how you did this. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 15 of 16
Q8. Finally, plot the discharging data in such a way that tests whether it obeys the capacitor discharging equation. Include this plot with your report. From the plot, determine the value of the time constant. Explain below how you did this. Penn State University Created by nitin samarth Physics Lab 202P-8 Page 16 of 16