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Name: Lab Partners:
Date:
Pre-Lab Preparation:
Fundamentals of Circuits IV:
Kirchhoff’s Circuit Rules
(due at the beginning of lab)
Directions: Read over the lab and then answer the following questions about the procedures.
Question 1 What can you measure with a multimeter?
Question 2 How should a multimeter be connected to measure the potential difference (voltage) across a resistor - in series, or in parallel with the resistor?
Question 3 A resistor has four colored stripes in the following order: orange, white, red and gold. What is the resistance, and what is the tolerance of the resistor?
Question 4 How will you measure the resistances to construct a rule for adding resistors in parallel?
Question 5 What is Kirchhoff ’s loop rule?
Question 6 What is Kirchhoff ’s junction rule?
PHYS-204: Physics II Laboratory i
Question 7 Which of the two Kirchhoff ’s rules is based on conservation of charge?
PHYS-204: Physics II Laboratory ii
(a)
(b) A
V
Figure 1: (a) Multimeter with voltage, current and resistance modes, and (b) symbols used to indicate a multimeter used as an ammeter, a voltmeter, and ohmmeter respectively.
will also illustrate the important fact that when the meter is properly connected, it will not alter the voltages or currents being measured. In order to do this you will need the following items:
- Digital multimeter
- Two 1.5 V D cell batteries with holders
- Connecting wires
- Two #14 bulbs in sockets
Activity 1.1: Measuring Current with a Multimeter
Fig. 2 shows a circuit (a), and two possible ways (b) and (c), that you might connect a multi- meter to measure the current through a bulb.
Prediction 1.1 Which of the diagrams in Fig. 2, (b) or (c), show the correct way to connect the multimeter to measure current through a bulb? Explain why you feel it should be connected this way. (The rule is: Any current through the multimeter must be the same as the current through the bulb in order to make the measurement.)
Step 1: Connect the circuit in Fig. 2 (a). Use two batteries in series to make a 3 volt battery. Note the brightness of the bulbs.
Step 2: Set the multimeter to read current, and connect it as shown in Fig. 2(b). Is the brightness of the bulbs the same as it was before connecting the meter?
Step 3: Now connect the meter as in Fig. 2(c). Is the brightness of the bulbs the same as when there was no meter in the circuit?
Question 1.1 If the multimeter is connected correctly to measure current it should not cause any significant change in the current through the bulb. Which circuit in Fig. 2 shows the correct way to connect a multimeter? Explain why, based on your observations.
(a)
−
(b)
−
A
(c)
−
A
Figure 2: A circuit with two light bulbs and a battery (a), and two possible but not necessarily equivalent ways to connect a multimeter to measure current - (b) in series with a bulb, and (c) in parallel with a bulb.
Question 1.2 Does the multimeter set to measure current act as a large resistance, or a small resistance? Explain, based on your observations. Why does it need to be designed this way?
Activity 1.2: Measuring Voltage with a Multimeter
Fig. 3 shows two possible ways that you might connect a multimeter to measure the voltage across a bulb.
(a)
−
V
(b)
−
V
Figure 3: Two possible but not necessarily equivalent ways to connect a multimeter to measure voltage - (a) in series with a bulb, and (b) in parallel with a bulb.
A carbon resistor is usually marked with colored bands. These bands tell you the value of the resistance and the tolerance (the accuracy of the given value of the resistor). The value of the resistance is given to two significant Fig. accuracy. The first two bands indicate the two significant digits of the value of the resistance, with different colors indicating numbers 0 through 9 according to the following table.
A table representing the resistor code Black 0 Blue 6 Brown 1 Violet 7 Red 2 Gray 8 Orange 3 White 9 Yellow 4 Green 5
Table 1: Resistor Codes
The third band indicates the number of zeros that come after the significant digits. As an example, the resistor shown in Fig. 5 has orange as its first band and white as its second band, giving 3 and 9 as the first two digits of the resistance. The third band is red, so the 3 and 9 are followed by two zeroes, giving a resistance of 3900 Ohms.
Figure 5: An example of a color-coded carbon resistor. The resistance of this resistor is 3900 Ω.
The fourth band indicates the precision of the resistor: silver indicates a tolerance of 10% gold indicates a tolerance of 5%, and no band or a red band indicates 20%). The precision of the resistor in Fig. 5 is 10%, meaning the resistance is somewhere between 3900 − 390 = 3510 Ohms, and 3900 + 390 = 4290 Ohms. In the next activity you will use the multimeter to measure the resistance of several carbon resistors and then compare your measured values with the those given by the colored bands. Note: Resistance must be isolated from the other electrical components before its resistance is measured. This also prevents damage to the multimeter that may occur if a voltage is applied across the meter’s terminals while in the resistance mode. For this activity you’ll need:
- several color-coded carbon resistors
- digital multimeter
Activity 1.3: Reading Resistor Codes and Measuring Resistance
Step 1: Choose three resistors of different values, and read their color codes. Record the resistances and tolerances in the first two columns of Table 2.
Step 2: After setting the multimeter to measure ”resistance”, measure the resistance of each resistor, and enter the values in the table in the column labeled ”Measured R.”
R from code (Ω) Tolerance from code Measured R (Ω) Percent discrepancy
Table 2: Resistance, Tolerance & Accuracy
Step 3: Determine the percentage difference between the coded value of each resistor and the value you measured. In general, percentage difference is given by the expression
Percent difference =
Accepted value - Experimental value Accepted value
∣ ×^100
Question 1.5 How do the values of the resistors measured with the multimeter compare to the values indicated by the color code? Assuming that your measured values are correct, are the values indicated by the color code within the stated tolerance?
Investigation 2:
Series and Parallel Combinations of Resistors
Several resistors can be wired in series to increase the effective length and in parallel to increase the effective cross sectional area of the resistive graphite, as shown in Fig. 6.
Resistors in Series
R 1 R 2
Resistors in Parallel
R 1
R 2
Figure 6: Carbon resistors wired in series and in parallel.
You will next explore the equivalent resistances of various combinations of resistors con- nected in series or in parallel. You will need the following:
Question 2.3 How do the measured values compare with your predictions for two and three resistors in series? If the measured values do not agree with your predicted ones, devise a new rule that describes the equivalent resistance when n resistors are wired in series. Use the notation Req to represent the equivalent resistance and R 1 , R 2 , R 3 ,... Rn to represent the values of the individual resistors.
Question 2.4 Does your rule agree with your observations in this activity, and in pre- vious labs, that the current through two identical resistors connected in series is half the current through a single resistor connected to the same battery? Explain why this might be so.
Activity 2.2: Equivalent resistance of resistors
connected in parallel
Prediction 2.2 If you have the values for three different carbon resistors, what would be the total resistance to the flow of electrical current if the resistors are wired in parallel? Explain, based on your previous observations with batteries and bulbs.
Question 2.5 Using the method you gave as your prediction 2.2, calculate the equivalent resistance if R 1 and R 2 are connected in parallel:
Predicted resistance of R 1 and R 2 in parallel = Ω.
Step 5: Connect R 1 and R 2 in parallel. Measure the resistance of the combination using the multimeter
Measured resistance of R 1 and R 2 in parallel = Ω.
Question 2.6 Using the method you gave as your prediction 2.2, calculate the equivalent resistance if all three resistors are connected in parallel.
Predicted resistance of R 1 , R 2 , and R 3 in parallel = Ω.
Step 6: Draw a diagram for the resistance network for the three different resistors wired in parallel. Above each resistor symbol write the measured value of the resistance.
Step 7: Connect all three resistors in parallel. Measure the resistance of the combination using the multimeter.
Measured resistance of R 1 , R 2 , and R 3 in parallel = Ω.
Question 2.7 How do the measured values compare with your predictions for two and three resistors in parallel? If the measured values do not agree with your predicted ones, devise a new rule that describes the equivalent resistance when n resistors are wired in par- allel. Use the notation Req to represent the equivalent resistance and R 1 , R 2 , R 3 ,... ...Rn to represent the values of the individual resistors.
Question 2.8 Does your rule agree with your observations in this activity, and in pre- vious labs, that the total current through two identical resistors connected in parallel is twice the current through a single resistor connected to the same battery? Explain why this might be so.
Activity 2.3: Extension: Other Combinations of Resistors in Series
and in Parallel
Step 1: Measure the resistance of the 22 Ω = R 4 and 68 Ω = R 5 resistors. Write down the measured values of each.
R 4 = Ω R 5 = Ω
Step 2: Connect R 1 , R 4 and R 5 in series. Measure the resistance of the combination using the multimeter.
Measured resistance of R 1 , R 4 and R 5 in series = Ω
Question 2.9 Using the method you gave as your prediction 2.1, calculate the equivalent resistance if R 1 , R 4 and R 5 are connected in series. Show your calculation. Does this value agree with your measured value, within a few per cent?
Predicted resistance of R 1 , R 4 and R 5 in series = Ω
(a) V 1 V 2
−
−
R 3
R 1
R 2
junction 1
junction 2
(b) V 1 V 2
−
−
R 3
R 1
R 2
branch 1
branch 3
branch 2
Figure 7: Junctions and branches of a circuit, as referred to in Kirchhoff’s Rules
from the − to the + terminal of a battery. (If you are going through a battery in the opposite direction, assign a negative potential difference to the trip across the battery terminals.)
- Find each of the junctions and apply the junction rule to it. You can place currents leaving the junction on one side of the equation and currents coming into the junction on the other side of the equation.
In order to illustrate the application of the rules, let’s consider the circuit in Fig. 8.
Question 3.1 Are the resistors R 1 and R 2 in series? in parallel? Why or why not?
In Fig. 8, the directions for the currents through the branches and for I 2 are assigned arbitrarily. If the internal resistances of the batteries are negligible, then by applying the loop and junction rules we find that:
Loop 1: V 1 − I 2 R 2 − I 1 R 1 = 0 (1)
Loop 2: −V 2 + I 2 R 2 + I 3 R 3 = 0 (2)
And applying the junction rule gives us
Junction 1: I 1 + I 3 = I 2 (3)
V 1 V 2
−
−
I 1
I 3
R 3
I 3
R 1
I 1
R 2
I 2
Figure 8: A complex circuit in which loops 1 and 2 share the resistor R 2. The currents I 1 and I 3 flow through R 2 in opposite directions and the net current through R 2 is denoted by I 1.
It is not obvious that the loops and their directions can be chosen arbitrarily. Let’s ex- plore this assertion theoretically for a simple situation and then more concretely with some specific calculations. In order to do the following activity you’ll need a couple of resistors and a multimeter as follows:
- 1 carbon resistor, 39 Ω
- 1 carbon resistor, 68 Ω
- Circuit board with variable resistor
- 1 multimeter
- 1 battery, 6V
- connecting wires
Activity 3.1: Applying the Loop and Junction Rules again.
Fig. 9 shows the same circuit as Fig. 8, but with different choices of directions for the loops and current through R 2.
Step 1: Use the loop and junction rules along with the new direction chosen for I 2 to rewrite the three equations relating values of battery voltages, resistance, and current in the circuit shown in Fig. 9.
Step 2: Show that if the I 2 of Fig. 9 = −I 2 of Fig. 8, then the three equations you just constructed can be rearranged algebraically so they are exactly the same as equations (1), (2), and (3).
Measured Measured Measured Theoretical Percent discrepancy R (Ω) ∆V (volts) I = ∆V /R (amps) I (amps)
Table 3:
Activity 3.2: Testing Kirchhoff’s Rules with a real circuit
Step 1: Use the resistance mode of the digital multimeter to measure the resistance between the center connection to the variable resistor and one of the other connections. What happens to the resistance as you turn the knob on the variable resistance clockwise? Counterclockwise?
Step 2: Set the variable resistor so that there is 25 Ω between the center connection and one of the other connections.
Step 3: Construct the circuit pictured in Fig. 8, using the variable resistor set at 25 Ω as R 2.
Step 4: Use the multimeter to measure the current in each branch of the circuit and enter your data in table 2. Compare the measured and calculated values of the current by computing the percent difference in each case. Note: The most accurate and quickest way to measure current with a digital multimeter is to measure the potential difference across each of the resistors and use Ohm’s law to calculate I from ∆V and R.
Question 3.3 How well do your measured currents agree with the theoretical values that calculated using Kirchhoff ’s rules? The multimeter is accurate to about 3%. Do your calculated and measured values agree to this accuracy?
Prediction 3.1 What do you predict will happen to each of the currents as the resistance of the variable resistor is decreased? That is, will the currents I 1 , I 2 , and I 3 increase or decrease? Explain your predictions.
Step 5: Decrease R 2 by turning the knob on the variable resistor. How good were your pre- dictions?
This laboratory exercise has been adapted from the references below.
References
[1] David R. Sokoloff, Priscilla W. Laws, Ronald K. Thornton, and et.al. Real Time Physics, Active Learning Laboratories, Module 3: Electric Circuits. John Wiley & Sons, Inc., New York, NY, 1st edition, 2004.
[2] Priscilla W. Laws. Workshop Physics Activity Guide, Module 4: Electricity and Magnetism. John Wiley & Sons, Inc., New York, NY, 1st edition, 2004.
[3] Lilian C. McDermott, et.al. Physics by Inquiry, Volumes I & II. John Wiley & Sons, Inc., New York, NY, 1st edition, 1996.
