( ) ( ) = q o. T 12 = τ ln 2. RC Circuits. 1 e t τ. q t

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1 Objectives: To explore the charging and discharging cycles of RC circuits with differing amounts of resistance and/or capacitance.. Reading: Resnick, Halliday & Walker, 8th Ed. Section Apparatus: Computer with Science Workshop 750 Interface w/ Voltage Output Voltage Sensor (CI-6503) RC Circuit Board (SE--9791) Extech basic multimeter Banana plug patch cords Background: When an uncharged capacitor is connected across a DC voltage source, it will begin to accumulate charge on its plates, but upon close inspection, you will notice that the rate at which it charges up decreases as time passes. Why is this, you might ask? If we stop to think about how charge accumulates on the capacitor plates we might realized that, at first, the capacitor is easy to charge because there is very little charge on the plates. But as charge accumulates on the plates, the voltage source must do more work to move additional charges onto the plates because the plates already have charge of the same sign on them. The electric repulsive force on a bit of charge delivered to a plate increases with time, and therefore so does the work required. As derived in class, we find that the capacitor charges exponentially, quickly at the beginning and more slowly as the capacitor becomes fully charged. Thus, the charge on the plates at any time is given by the expression: ( ) = q o 1 e t τ q t ( ) where qo is the maximum charge on the plates and τ is the capacitive time constant (τ=rc, where R is the resistance and C is the capacitance.) NOTE: Taking the extreme limits, notice that when t=0, q=0 which means there is not any charge on the plates initially. Also, notice that when t goes to infinity, q goes to qo which means it takes an infinite amount of time to completely charge the capacitor. The time it takes to charge the capacitor to half full is called the half-life and is related to the time constant in the following way: T 12 = τ ln 2 In this experiment the charge on the capacitor will be measured indirectly by measuring the voltage across the capacitor since these two values are proportional to each other: q=cv. 1

2 Procedure: PART I: Computer Setup 1) Connect the Science Workshop 750 interface to the computer, turn on the interface, and turn on the computer. 2) Open DataStudio and create a new experiment. 3) Click the SETUP button,, at the top and then pick a voltage sensor from the SENSORS categories and drag it to the Analog A icon on the picture of the interface. Double click on the sensor icon and set the sensor properties to low sensitivity and 500 Hz sampling rate. 4) Drag a graph icon from the DISPLAYS window onto the voltage in the DATA window. (You might also want to drag a digital display if you would like to get a readout of the output voltage.) PART II: Equipment Arrangement 1) Using the Board (PASCO SE-9791), build an initial RC circuit by connecting the red and black leads from the voltage sensor to the posts next to the capacitor (posts 5 & 6). Be sure to connect the black lead to the low potential side of the capacitor! You may wish to use alligator clips connected to the voltage sensor leads to allow for better connections at the posts. 2) Insert an AA battery into the slot on the RC circuit board and set the switch (S2), for the time being, to External. This will prevent the circuit from charging before we are ready to take data. 3) Set switch (S1) to Charge and briefly connect a jumper wire from coil A to coil C. This will initially allow any charge on the capacitor to quickly drain off through the (ideally) zero resistance jumper wire. 4) Now move your jumper cable to connect coils A and B, thereby completing the charging RC circuit. You are now ready to start taking data! 2

3 PART III: Data Recording: Charging RC Circuit: 1) Click start,, and then quickly flip switch S2 on the circuit board to battery. You should immediately notice the voltage vs. time plot begin to indicate an increase in the potential across the capacitor as shown in figure 3. 2) Collect data for at least a minute. By this time, the curve should be leveling off around the maximum voltage... the EMF of the battery. (Note: It might not be exactly 1.5 V as labelled on the battery. In your data analysis, be sure to use the correct value for Vmax, not the value written on the the battery s label!) 3) Save your data to a file called Charging RC Circuit Data (15k) or something similar. 4) Reset your RC circuit by discharging the capacitor as done earlier, and deleting the last run of data (under the Experiment menu). 5) Now insert one of your extra resistors (5.1kΩ, 10kΩ, 15kΩ, or 33kΩ) into the coils parallel to the existing 15kΩ resistor. By inserting an additional resistor in parallel to the existing resistor, how does the equivalent resistance of the circuit change? Does it increase or decrease? How, then, would you expect the time constant and the rate at which the capacitor is charged to change? How would you expect the curve to change? (These predictions should be recorded in your labbooks!) 6) Repeat your data collection with each of the 4 additional resistors added in parallel with the existing 15kΩ resistor. (You will only insert ONE additional resistor at a time!) 7) Save your data after each run using a name similar to the one in step 3, but with appropriate numbers indicating total resistance in the circuit. Discharging RC Circuit: 1) Now we need to start with a fully charged capacitor. Recall that the capacitor will charge immediately if there is no resistance in the circuit. You decide how to most effectively and quickly charge the capacitor. (HINT: You might want to flip the switch S1 to Charge to ensure that, once you do fully charge the capacitor, the charge stays there until you are ready to analyze the discharge.) Use the Extech multimeter to record what the maximum voltage across the capacitor actually is. 3

4 2) When you are ready to take data for the discharging circuit, click start,, and then quickly flip switch S1 to Discharge. You should immediately see the plot of voltage begin to drop quickly according to the expected exponential decay curve, starting form the maximum voltage determined in the previous step. Again, collect for about a minute and then stop the data collection. 3) Save your data to a file called Discharging RC Circuit Data (15k) or something similar. 4) Reset your RC circuit by fully charging the capacitor as done earlier, and deleting the last run of data (under the Experiment menu). 5) Now repeat for each of your extra resistors (5.1kΩ, 10kΩ, 15kΩ, or 33kΩ) inserted in parallel to the 15kΩ resistor, saving each data set with a unique and understandable file name. 6) Now you are ready to start analyzing your data. PART IV: Analyzing the Data 1) Use the handheld multimeter to measure the resistance of the resistor on the circuit board, and your additional resistors. Remember, just because some painted some orange and red lines on your resistors, doesn t necessarily means that those are the actual values of the resistors. You should always check for yourself. 2) Use the cursor and magnifier buttons at the top of the graph window to determine the time it takes for the voltage to rise to one-half it s maximum value. The Smart Tool is a very useful feature of the DataStudio software for making this type of measurement. Click on the Smart Tool icon and a cursor and cross hairs will appear on your plot. Place your mouse cursor over the crosshairs until you see a hand and crosshairs icon appear. You can not move the Smart Tool cursor to the point in your data where it just begins to rise. Now move your mouse cursor just slightly towards the upper right hand corner of the Smart Tool cross hairs. You will see the hand and crosshairs icon change to a hand and a Δ symbol. Now drag up and to the right while holding the mouse button down and you will see a box form with values listed next to the upper and right axes. If your are over a point in your data set, the values will be color coded to match your data, if not they will be black. Move the upper right hand corner to a point that is at half the maximum voltage and set the corner by releasing the mouse button. The color coded number across the top represents the time duration to rise from 0 V to half-maximum, or the half-life of the charging RC circuit! Record this half-life, t1/2, along with any uncertainty determined from the plot. 3) Use this half-life, and it s associated uncertainty, to determine the capacitive time constant of the circuit and it s associated uncertainty. 4

5 4) Repeat this procedure to determine the half-life s of all of your data plots (10 in all). It is probably best to record this data in your laboratory notebooks in some sort of organized table, indicating which data set it belongs to, the resistance of the circuit, the half-life, the time constant, and all associated errors. We have purposely not provided an example of such a table in this handout, as you need to learn how to design and create your own data tables, with the pertinent information included. 5) Plot your data of total resistance, R, vs. time constant, τ, using all 10 data sets. What kind of relationship do you expect between these (for example if you put R on the horizontal axis and τ on the vertical axis. 6) The capacitor in your circuit board is is VERY likely to NOT be 1000 µf as labeled! From this plot, determine the capacitance of the capacitor in the circuit. DISCUSSION: 1) Are your results expected? 2) Do the values you get for each of the slightly different cases listed above make sense? Explain. 3) If you wanted to charge your circuit very quickly and then discharge it slowly, what R s and C s could you use for each of these steps? Explain. 4) If, instead of extra resistors, we inserted extra capacitors in parallel with the existing capacitor using the coils at A and B, how would this affect our results? (The shape of the curves.) QUESTIONS: (Answer these questions in reference to your first circuit: one 15kΩ resistor and the capacitor. ) 1) The time to half-maximum voltage is how long it takes the capacitor to charge halfway. Based on your experimental results, how long does it take for the capacitor to charge to 75% of its maximum value? 2) After four half-lifes, to what percentage of the maximum charge is the capacitor charged? 3) What is the maximum charge for the capacitor in this experiment? 4) What are some factors that could account for the percent difference between the stated and experimental values? 5

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