# Kirchhoff s Rules and RC Circuits

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1 PHYSICS II LAB 5 SP212 Kirchhoff s Rules and RC Circuits Pages are Appendixes added for extra information. Pages 1 9 only are the Lab instructions. I. Introduction A. Today s Lab will investigate Kirchhoff s junction law and Kirchhoff s voltage law, as well as experiment with parallel and series circuits. The final part of the lab is to study a capacitor and resistor together in a circuit (RC circuits). II. Needed Equipment A. Protek DMM, 3 resistors (each with different resistance value), multiple banana leads, two D cell batteries with holders, two light bulbs, Tektronix power supply (PS), one cylindrical capacitor, Labpro with 3 ammeters and a difference voltage meter, and a pencil/pen. III. Turn in your Pre lab/homework problem if assigned. IV. Procedure A. Preliminary Measurements 1. Setup (if directed by your instructor) To test our three ammeters and our voltage probe to ensure that they work properly before we start the experiment. Your instructor will show you how to do this using the Equipment Test Procedure.pdf file. If something isn t working correctly, tell your instructor and get a new sensor. SP212/SP Manual.php Page 1 of 12

2 2. Measure Resistance Use your Protek DMM to measure and record the resistance of each of the resistors on your lab bench. (Remember you can look at the resistor color code to determine the expected value of the resistance printed on each resistor, but there is a tolerance, this is why we measure its resistance when possible). 3. Your instructor will remind you how to deal with uncertainty from the DMM and Logger Pro. Link to the color code and DMM technical manuals; SP212/SP Manual.php B. Kirchhoff s Junction Rule, aka Kirchhoff s Current Rule: 1. Build the circuit shown below for this part. 2. Use the power supply for the emf, and set its output voltage to be about 3V. Zero your voltage and current probes, take some data, and then test to see if the current into the junction is equal to the current out of the junction. Also test to see if the potential differences across the resistors are the same, and if they are the same as the potential difference across the emf. Attach your graph to the spreadsheet. 3. Discussion: In a short paragraph consisting of complete sentences, discuss whether or not your measurements support or refute the formula for adding resistances in parallel. Also discuss whether or not they support the Junction Rule. Page 2 of 12

3 C. A Two Loop Circuit with an EMF in each loop: 1. Build the circuit shown below for this part. 2. Note that it is the circuit from the pre lab problem, with slightly different values for the circuit elements. We want to predict and measure the currents in this circuit, and compare the results. 3. After you build the circuit, take data, and then gently disconnect the D cell and power supplies. The idea is to avoid running the D cell down, but to leave the circuit almost intact, so you can take another data set if you need to. Enter the measured values of the emfs in the appropriate cells in your spreadsheet. 4. Using the same equations in your TI N spire CX calculator as you used to solve the homework problem, modify the values of the emfs and resistances as necessary and predict the currents in the three resistors. Enter these predictions in the appropriate cells in your spreadsheet, along with the measured resistances (from preliminary data) and measured currents. 5. Discussion: Do your measurements support or refute Kirchhoff's Rules for analyzing circuits? Please write a comprehensive and comprehensible paragraph Page 3 of 12

4 D. An RC Circuit (Skip to E if you completed this section in Lab 4.) 1. Measure the resistance of your 39 Ohm resistor with your Protek DMM. Look up the uncertainty code; it may be different from that for the Amprobe DMM used in the supplied example data. 2. The capacitance of the large blue capacitor is given in the spreadsheet template, along with its uncertainty. 3. Measure the time constant of this circuit. i. To make the measurement, follow these instructions: ii. Build the circuit shown below. For the resistor in the circuit, R, we will use a 39 resistor. PS is Tektronix power supply (5V (far left hand side)). Finally C is the 25,000 F (0.025 F) capacitor similar to the one shown on above. iii. Program Logger Pro by setting up one voltage and one current sensor, setting the duration of the experiment to 20 seconds, and data rate to 20 samples/second. iv. Start with the switch in the discharge position. Make sure the capacitor is completely discharged by leaving the switch in the discharge position for a long time. Zero all sensors. v. Then begin collecting data, and then throw the switch to the charging position. vi. After the capacitor is completely charged, flip the switch to the discharge position, and complete the data run. Page 4 of 12

5 vii. Fit exponentials to the charging and discharging curves, and find the time constant for each curve from the fits. Average the results for your best estimate of the measured value of the time constant, and use the standard deviation as your estimate of the experimental uncertainty. viii. After attaching your graph to the spreadsheet, annotate it: 1. Give it a title, including the number and name of the exercise, and number and name of the experiment and part. 2. Write the final result of the measurement, with uncertainty on your graph. 3. Label, as I have done in the sample data, when charging and discharging are taking place. 4. Point out anything else that is of interest or importance to your reader. 4. Calculate the theoretical time constant value for this circuit and calculate the uncertainty in your predicted value. τ = RC 5. Discussion: Compare your measured value for the time constant with the theoretically predicted value, that is, explain whether or not the two values agree. Write in complete sentences. In addition to comparing predicted and measured values for the time constant, explain the signs and shapes of each of the four curves in your graph. Use complete, grammatically correct sentences structured into a coherent paragraph. E. Resistors in Series (If time permits) 1. Setup the 22 Ohm and 39 Ohm resistor in series and use the Protek DMM to measure the resistance of the series combination. 2. Calculate the theoretical value of the series combination using your measured values for the individual resistances of the 22 Ohm and 39 Ohm resistors to calculate the theoretical resistance you should get when you connect them in series. Don t forget the uncertainty! 3. Compare your results to the theory; does it look like the overall resistance in series is the sum of the individual resistances? In other words, does your experiment match theory? Page 5 of 12

6 F. Kirchhoff s Loop Rule, aka Kirchhoff s Voltage Rule(If time permits): 1. Build the circuit shown below. 2. Use the power supply for the emf, and set its output voltage to be about 3V. The exact value doesn t matter; you will measure it when you take the data. When using the LabPro to measure voltages and currents, the uncertainty should be calculated as 2% of the mean value, plus the standard deviation 3. Zero the probes in the usual way, and collect data. 4. Use the measured current, along with the measured resistances, to calculate the potential differences across each of the resistors. Then, test to see whether the sum of the potential differences equals zero when you walk around this circuit. 5. Discussion: In a short paragraph consisting of a few complete sentences, discuss whether or not your measurements support or refute the formula for adding resistances in series. Also discuss whether they support or refute Kirchhoff s loop rule. G. Resistors in Parallel(If time permits): 1. Similarly to Part B1, predict and measure the resistance resulting from the parallel combination of your 22 Ohm and 39 Ohm resistors 2. Compare your results to the theory; does it look like the overall 1 resistance in parallel is? In other words, does your experiment match theory? R R 1 2 Page 6 of 12

7 H. Bulbs in Series and Batteries in Series(If time permits): 1. Quickly setup the same setup you saw last week to help refresh your memory about how bright the bulb was. 2. Then connect two bulbs in series with two batteries in series as shown in the next diagram. 3. Discussion: Remove one banana lead while answering these questions to avoid draining the battery needlessly. In a short paragraph consisting of a few complete sentences, compare the brightness of these bulbs with the brightness of a single bulb in the same circuit (as observed in step 1). 4. Discussion: What happens if you remove one bulb? Why? I. Bulbs in Parallel and Batteries in Series(If time permits):: 1. Connect two bulbs in parallel. Connect the parallel combination of bulbs to two batteries in series as shown in the next diagram. 2. Discussion: Remove one banana lead while answering these questions to avoid draining the battery needlessly. In a short paragraph consisting of a few complete sentences, discuss the brightness of these bulbs with the brightness of the bulbs in Part H2. 3. Discussion: What happens if you remove one bulb? Why? Page 7 of 12

8 4. Discussion: What would happen if you connected your two bulbs in parallel to two batteries in parallel as shown in the next diagram? Perhaps if there is time remaining, your instructor will let you come back and connect this circuit to see if your prediction was accurate. However, for now, after answering the above discussion questions move on to the next section. V. Lab Report to hand in: 1. Annotated Graph and discussion from Part B. 2. Spreadsheet, Graph, and discussion from Part C. 3. Spreadsheet, Graph, and discussion from Part D (if completed during this lab instead of lab 4.) 4. Spreadsheet and discussion from Parts E and Part G (if directed by your instructor). 5. Spreadsheet, Annotated Graph and discussion from Part F (if directed by your instructor). 6. Discussions from Part H and I (if directed by your instructor). Don t forget uncertainties, units, annotations and discussions. Page 8 of 12

9 VI. Clean Up (ensure all equipment is off and all wires are disconnected!) A. Golden Rule: Do unto others as you desire them to do unto you. This applies as much here in the lab as it does in the Fleet. As future Naval Officers, how can you expect your enlisted sailors to maintain a clean work area if your stateroom, work areas, mess area, etc is a pig sty? So as officers it is imperative that we clean up after ourselves not only to follow the Golden Rule, but also to lead by example for the enlisted personnel under our charge. 1. End of Lab Checkout: Before leaving the laboratory, please tidy up the equipment at the workstation and quit all running software. 2. The lab station should be in better condition than when you arrived and more importantly, should be of an appearance that you would be PROUD to show to your legal guardians during a Parents Weekend. 3. Have your instructor inspect your lab station and receive their permission to leave the Lab Room. 4. You SHALL follow this procedure during every lab for SP212! Many thanks to Dr. Huddle, Dr. Katz, Dr. Mungan, and Dr. Fontanella for their assistance in producing this Laboratory procedure; specific references can be supplied on request. LCDR Timothy Shivok Page 9 of 12

10 Appendix A At the end of this activity, you should: List of Objectives for this lab. 1. Be able to determine the equivalent resistance of Resistors in series. Then observe that the experimental results match theory. 2. Be able to determine the equivalent resistance of Resistors in parallel. Then observe that the experimental results match theory. 3. Develop an ease with Kirchhoff s first rule, aka Kirchhoff s Current Law (KCL) or junction rule, including being able to solve for various currents. 4. Observe that the algebraic sum of all the currents entering a node must be equal to the algebraic sum of all the currents exiting a node. 5. Develop an ease with Kirchhoff s second rule, aka Kirchhoff s Voltage Law (KVL) or loop rule, including being able to solve for various voltages and/or currents. 6. Observe again, that the algebraic sum of all the voltage drops and voltage sources in a closed loop must add up to zero. 7. Develop an ease with simultaneous equations (in your calculator or preferably by hand) and solve a complex circuit involving two emf sources and parallel resistors. Then observe that the experimental results match theory. 8. Observe the effects batteries and light bulbs in Series or Parallel arrangements. 9. Observe the voltage across the capacitor and current through the resistor as functions of time in a simple RC series circuit when the capacitor is charging. 10. Observe the voltage across the capacitor and current through the resistor as functions of time in a simple RC series circuit when the capacitor is discharging. Page 10 of 12

11 Appendix B Additional introductory material A. In the laboratory four, during the Simple Circuits portion, Kirchhoff's second rule, the loop rule, was introduced. In today's laboratory, we will review the second rule and study Kirchhoff's first rule, the junction rule. These rules represent fundamental principles that apply to any circuit. They are fundamental because the first rule is an alternative statement of conservation of charge and the second rule is based on conservation of energy. B. In the first part of this lab, we will study resistors connected in series and in the second part we will study resistors connected in parallel. Combinations of resistors are sometimes known as resistor networks. One of the important characteristics of a resistor network is the equivalent resistance, R eq. R eq can be thought of as the resistance of a single resistor that can replace the network. In each case, in order to study Kirchhoff's rules, we will add a source of emf to the resistor network and measure the voltages and currents associated with the resultant circuit. C. In another part of this lab, we will study a circuit containing more than one emf source and multiple resistors. Discussion A. In class, we learned to calculate the equivalent resistance of series resistors. R eq = R 1 + R 2 + R B. We also learned to calculate the equivalent resistance of parallel resistors. C. Additionally, we learned both KCL and KVL: a. That the algebraic sum of all the currents entering a node must be equal to the algebraic sum of all the currents exiting a node. b. That the algebraic sum of all the voltage drops and voltage sources in a closed loop must add up to zero. Page 11 of 12

12 D. RC Circuits The final part of the lab is to study a capacitor and resistor together in a circuit. Theory When a resistor, capacitor and emf, ε, are connected in series (the switch in position 1, to charge), the capacitor charges according to The quantity RC capacitor is is known as the time constant. Consequently, the voltage across the Finally, the associated charging current is If the battery is then removed from the circuit (by moving the switch to position 2, to discharge) and the capacitor remains connected across the resistor, the capacitor discharges according to where the voltage across the capacitor is and the current during discharge is We will measure the voltage across the capacitor and the current in the resistor for both the charging capacitor and the discharging capacitor. Page 12 of 12

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