Understanding_RC_Circuits-FA22

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Phys 212 Lab: Understanding RC Circuits Name: ____________________________ Date: ______________ Name: ____________________________ Lab Sect.: __________ Name: ____________________________ Lab Instructor: ______________________ 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  Supercap Pack  (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  t rendline: 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 V ( t )= ε ( 1 − e − t / RC ) and (Eq. 1) Q ( t )= C ε ( 1 − e − t / RC ) . (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 V ( t )= V 0 e − t / RC and (Eq. 3) Q ( t )= Q 0 e − t / RC . (Eq. 4) Directions: Lab Activity 1: Charging and Discharging of RC Circuit  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 Batteries (check that the connector screws on the blue board are tight). 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. 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. QA. Before you begin taking data, predict (using sentences and a sketched graph of V vs t ) what you think will happen when the switch is flipped to position C. QB. Predict (using sentences and a sketched graph of V vs time) What will happen when the switch is flipped from position C to position A .  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.

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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. (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 for charging and discharging. Label the point at which you switched from charging the capacitor to discharging the capacitor. (Alternatively, you can insert a picture of the graph – make sure only to include relevant data range.) Explain what is happening using sentences. 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. Q2a Explain how you will do this. Q2b. What is time constant? Q2c. Next, repeat the experiment, but this time, measure the voltage V R across the bulb, estimate your time constant. Q3. Insert the graph of measured voltage across the bulb V R vs. t here (make sure only to include relevant data range) for charging and discharging. Label the point at which you switched from charging the capacitor to discharging the capacitor. (Alternatively, you can sketch the graph qualitatively.) Q4. Explain, with complete sentences, qualitatively why the voltage across the bulb varies as observed. 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 1-F capacitor, and (b) the 100-kΩ resistor (brown-black-yellow stripes- larger of 2 resistors) 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 .  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. You might need to increase the sampling rate in PASCO.  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 here: 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, Does the data really follow V C = V Co exp ( -t/  )? To verify that it is exponential and to calculate  , we need 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/  . Q7a.Explain below how can you manipulate this expression to get a function that is linear with time? (Hint: what function “undoes” an exponential function?) Q7b Write out your linearized version of the equation. Q7c Explain – what part of you data is relevant, and what parts will you not use in your plot?

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Q7d. Now, do the manipulation in Excel. You may need to make new columns in your Excel file. Include your new plot below. Q7e. From the plot, fit your data using a linear fit and determine the value of the time constant  . (Include a graph of your original data and your new graph.) 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). Q8a. Explain using words, how can you manipulate this expression in Q8 (Eq. 1) to get a function linear with time, as you did for the discharging data? Q8b. Explain what data you will include or remove and why. Q8c. Include this plot below. Q8d. 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.) Q9. Compare the values of  from Q6 ,Q7 and Q8 in a sentence below. Use a quantitative measure of their agreement (percent difference) in your discussion.