Physics 212 Lab 07

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Feb 20, 2024

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Phys 212 Lab: Understanding RC Circuits Name: Andrew Crilley Date: 10/26/2021 Name: Nathan Hon Lab Sect.: 17 Name: ____________________________ Lab Instructor: Monica Ramierez Goals:  To quantify the rate at which a capacitor charges and discharges when connected in series with a resistive element.  To explore how the typical charge/discharge timescale depends upon the resistance and capacitance.  To practice linearization of data as a data-analysis technique. Lab goals: Lab01 Make experimental measurements using standard equipment, including multi-meters and oscilloscopes. Lab02 Design and debug simple electrical circuits with specified parameters. Lab03 Record and process experimental data. Lab04 identify patterns in the data and devise an explanation for an observed pattern. Lab05 Present data using graphs and tables that are correctly labeled and correctly scaled. Lab06 use spreadsheets to create scatter plot to visualize data. Lab07 generate “best fit” slope and intercept to find the values of physical quantities. Equipment:  PASCO 550 Universal Interface  Voltage Probes  Phys212 LabKit Module  Power Supply Set to 3V  (7) Stackable banana-plug connecting wires  Software: Microsoft Excel, PASCO Capstone Reference: Here are the excel tutorials if you don’t’ remember the details on how to create a scatter plot, fitting the data or creating new data with a formula (you’ll need to do all 3).  scatter plot: https://support.microsoft.com/en-gb/office/present-your-data-in-a- scatter-chart-or-a-line-chart-4570a80f-599a-4d6b-a155-104a9018b86e  formulas: https://support.microsoft.com/en-us/office/overview-of-formulas-in-excel- ecfdc708-9162-49e8-b993-c311f47ca173  trendline: https://support.microsoft.com/en-us/office/add-a-trend-or-moving-average- line-to-a-chart-fa59f86c-5852-4b68-a6d4-901a745842ad

Introduction: 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 want to approach the problem quantitatively . When an initially uncharged capacitor of capacitance C is connected in series with a resistor of resistance R and a power supply or battery with EMF ε , the charge on the capacitor (and hence the voltage across the capacitor) builds up rapidly at first, and then more slowly, eventually reaching a final steady-state value after a long time. This charging process is described by the equations ) 1 ( ) ( / RC t e t V     and (Eq. 1) ) 1 ( ) ( / RC t e C t Q     . (Eq. 2) The quantity RC in these equations has dimensions of time and is called the time constant of the circuit. When an already-charged capacitor C with an initial charge Q = CV 0 is connected across a resistance R , it discharges. The charge on the capacitor and the voltage across the capacitor both drop rapidly at first, and then more slowly, eventually dropping entirely to zero. This discharging process is described by the equations RC t e V t V / 0 ) (   and (Eq. 3) RC t e Q t Q / 0 ) (   . (Eq. 4) Directions: Lab Activity 1: Charging and Discharging of RC Circuit  Instead of batteries, we will use 3V on the power supplies.  Connect the terminals of the 1-F capacitor to each other for a few seconds using some wire. (What are you doing to the capacitor by “short-circuiting” it?)  Set up the circuit shown in Figure 1 below. Use the round bulb as the resistor in the circuit. Recall that the 1-F capacitor has a polarity and needs to be hooked up properly to the Power supply at 3V. Do NOT close the circuit yet ( i.e. , leave the switch in the open position B . Note that the switch has two possible closed positions, labeled A and C . Also, make sure that the light bulbs are properly screwed into their sockets.

Figure 1 An RC charging/discharging circuit. Before you begin taking data, convince yourselves of what will happen when the switch is thrown to position C , and what will happen when the switch is thrown to position A . Also, please note that the 1-F capacitor has a definite polarity and must be connected the correct way. Please ask your teaching assistants if you are unsure about your wiring of the circuit.  Make sure that the voltage probes are properly connected to measure the voltage V C across the capacitor. (Does it matter which wire (red or black) is connected to which end of the C?)  Make sure that the PASCO interface is powered on (the power button should be glowing blue light)  Open PASCO (start -> All Programs -> PASCO scientific -> PASCO Capstone), and click on Hardware Setup in the upper left corner. You should see a picture of the PASCO interface box. Click on the part of the picture that corresponds to the channel A that you plugged your voltage probe into. A drop down menu should appear. Scroll down to “Voltage Sensor”, and click it. Now click the “x” on the box to the right of “Tools” in the upper left corner to close the Hardware Setup box.  Click “Table and graph” from the menu of QuickStart templates.  Table: (1) Click the “Select Measurement” button on the top of the first column of the table. Select “Time(s)”. (2) Click the “Select Measurement” button on the top of the second column of the table. Select “Voltage(V)”. PASCO is now configured to operate as a digital voltmeter.

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(3) Enter your table title in the upper left corner of the table.  Graph: (1) Click the “Select Measurement” button along x-axis of the graph. Select “Time(s)”. (2) Click the “Select Measurement” button along y-axis of the graph. Select “Voltage(V)”. (3) Enter your graph title in the lower left corner of the graph.  Firstly, click “Record”. And then flip the switch to “C”. o When the capacitor seems to have charged completely, flip and hold the switch to position A . This position of the switch is spring-loaded, and it will be necessary for you to continue to apply light pressure for the switch to remain closed in this position. o When the capacitor seems to have discharged completely, stop recording data.  You might wish to do this a few times in order to get a reasonable set of data. Note that you can delete any set of data in “Data summary” in the leftmost column. 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.)

Q2. Without doing any explicit fits to the data, use the graph displayed in the PASCO window to estimate the time constant of this circuit. Explain how you did this!  Next, repeat the experiment, but this time, measure the voltage V R across the bulb. When voltage id discharging and tao is equal to time, then Vc is equal to Vmax/e. Therefore, if we find Vc=Vmax/e on the graph and then find the corresponding time, that will be Tao. Q3. Insert the graph of measured voltage across the bulb V R vs. t here. Label the point at which you switched from charging the capacitor to discharging the capacitor. (Alternatively, you can sketch the graph qualitatively on the axes below, if you so wish.)

Q4. Explain qualitatively why the voltage across the bulb varies as observed. The resistor is at max voltage when the capacitor is at its lowest charge. As the capacitor charges itself, the resistor voltage decreases which is seen by the segment a of the graph. When the switch is flipped the current flows in the opposite direction and follows the same trend as segment a, until the capacitor runs out of charge. Lab Activity 2: Quantitative Analysis of an RC Circuit  Modify (re-wire) your circuit so that: (a) the 1-µF capacitor is used in place of the the 1-F capacitor, and (b) the 100-kΩ resistor (brown-black-yellow stripes) is used in place of the light bulb.  Repeat the first experiment that you carried out earlier: i.e., record the voltage V C across the capacitor as a function of time as it charges, but Stop taking data after the capacitor is completely charged. We now have data for the charging of this capacitor only.  Leave the switch in position C .

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 Export the measured data to an Excel spreadsheet for analysis as follows: o In the Table display, highlight the columns of data containing the time and voltage readings. o Copy this to the clipboard (Press Ctrl-C or select Copy from the Edit menu) o Open Microsoft Excel and start a new spreadsheet. o Paste (Ctrl-V) the copied data into the first two columns. o Check the units on the data columns so you can label your graphs properly  Now, go back to PASCO and record data again, rapidly flipping the switch to A and holding it in position until the capacitor has completely discharged.  Export this data to the Excel spreadsheet as well. Q5. Use Excel to make graphs of V C versus t during the charging and discharging of the capacitor. Plot this data with a scatter plot . Remember to label the axes on your graphs correctly with proper units. Include these graphs with your lab report. Q6. Without using your graph or data, calculate the theoretical value of the time constant  for this circuit. Then make an estimate of  from your plot and compare to two values.

Q7. Just because the data “looks” exponential doesn’t mean it really is, that is V C = V Co exp ( -t/  ). To verify that it is exponential and to calculate  , we need to find a way to plot the discharge data so that (1) the graph should be linear (not exponential) with time and (2) the slope of this line is 1/  . (a) How can you manipulate this expression to get a function that is linear with time? (Hint: what function “undoes” an exponential function?) (b) You may need to make new columns in your Excel file. (c) Be sure to only try to fit data recorded after the switch was thrown. (d) At some point V C becomes so small that the uncertainty in voltage measurements will be as large as or larger than Vc itself, so don’t included data once Vc has decay to essentially zero. Include this plot with your report. From the plot, determine the value of the time constant  . Explain below how you did this, including how you got a function that is linear with time. (Include a graph of your original data and your new graph.)

We linearized the graph by using the ln function to remove the e and then that resulted in the y- axis being ln(vc/max), the x-axis as time, and slope as -1/tau. We took the data from the pasco and used an excel function to create a graph as such. Q8. Similarly, plot the charging data in such a way that the graph would look linear if it were described by V C = V Co [ 1-exp ( -t/  )], where V Co is the peak capacitor voltage (not the initial capacitor voltage). That is, how can you manipulate this expression (Eq. 1) to get a function linear with time, as you did for the discharging data? Include this plot with your report. From the plot, determine the value of the time constant  . Explain below how you did this. (Be sure to only try to fit data recorded after the switch was thrown.)

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Q9. Compare the values of  from Q6 – Q8. Use a quantitative measure of their agreement (percent difference or standard error- which works better here?) There was a percent difference in the data of 367 Percent, this is because the initial estimate in question 6 was based off a one point, while in question 8, it is a more precise calculation as we linearized and used several points.