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Experiment 2
Ohm’s Law
2.1 Objectives
- Become familiar with the use of a digital voltmeter and a digital ammeter to measure DC voltage and current.
- Construct a circuit using resistors, wires and a breadboard from a circuit diagram.
- Construct series and parallel circuits.
- Test the validity of Ohm’s law.
- Reduce a complicated resistance circuit to a simple one-resistor equiv- alent circuit.
2.2 Introduction
In the US, most of us use electricity every day. That electricity is handled in circuits: a closed loop of conductors travelling from power plants to neighborhoods to households and back again. That closed loop, with all of its many parts, forms one huge electrical circuit. Today we’ll use the 3 essential parts of a circuit – power supply (or battery), wires, and resistors. We’ll learn how resistors affect the current of electrons that flows through them, and how connecting resistors in different ways changes their behavior.
- Ohm’s Law
2.3 Key Concepts
As always, you can find a summary on-line at HyperPhysics 1. Look for keywords: electricity and magnetism, ohm’s law, resistor, resistor combina- tions To play with constructing circuits and actually see how the electrons flow through a circuit, check out the online simulation “Circuit Construction Kit” 2 from the University of Colorado.
2.4 Theory
One of the fundamental laws describing how electrical circuits behave is Ohm’s law. According to Ohm’s law, there is a linear relationship between the voltage drop across a circuit element and the current flowing through it. Therefore the resistance R is viewed as a constant independent of the voltage and the current. In equation form, Ohm’s law is:
V = IR. (2.1)
Here, V is the voltage applied across the circuit in volts (V), I is the current flowing through the circuit in units of amperes (A), and R is the resistance of the circuit with units of ohms (Ω). Eq. 2.1 implies that, for a resistor with constant resistance, the current flowing through it is proportional to the voltage across it. If the voltage is held constant, then the current is inversely proportional to the resistance. If the voltage polarity is reversed (that is, if the applied voltage is negative instead of positive), the same current flows but in the opposite direction. If Ohm’s law is valid, it can be used to define resistance as:
R =
V
I
where R is a constant, independent of V and I. It is important to understand just what is meant by these quantities. The current (I) is a measure of how many electrons are flowing past a given point during a set amount of time. The current flows because of the
(^1) http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html (^2) http://phet.colorado.edu/en/simulation/circuit-construction-kit-dc
- Ohm’s Law
(a) Series (b) Parallel
Figure 2.1: Schematics of circuits illustrating resistors connected in series and in parallel.
Series Parallel V (^) S = V 1 + V 2 V (^) P = V 1 = V (^2) I (^) S = I 1 = I 2 I (^) P = I 1 + I (^2) R (^) S = R 1 + R (^2) R^1 P = (^) R^1 1 + (^) R^1 2 or R (^) P = (^) RR 1 1 +^ RR^2
Table 2.1: Equations for two resistors in series and parallel.
Using these relationships, a complex circuit can be redrawn as a circuit with a single resistor. You may wish to review the process of finding the equivalent resistance of circuits in your physics textbook.
2.5 In today’s lab
Today we’ll become accustomed to some standard electrical equipment: we’ll figure out how to use the circuit boards, resistors, and wires to create series and parallel circuits. Then we’ll learn how to measure the current through a wire, as well as the voltage between two points in a circuit. Finally, we’ll verify that the equations that are presented in the theory section are actually correct.
2.6. Equipment
2.6 Equipment
- DC Power Supply (Fig. 2.2)
- 2 digital multimeters (Fig. 2.3)
- breadboard (Fig. 2.6)
- several banana-to-banana wires
Safety Tips
- When plugging or unplugging wires, first turn off all electronics that are connected or will become connected to the circuit.
- If you are color blind or suspect that you are, you may find the color codes on the resistors difficult. Please consult your lab instructor for advice or help.
The DC Power Supply
A DC power supply is used to provide varying voltage to a circuit. The power supply used in this lab is shown in Fig. 2.2. The black and red connectors are the negative (−) and positive (+) output terminals, respec- tively. The voltage knob controls the power supply’s output voltage. The current knob sets a limiting current. Here, adjust the current control to its maximum setting (all the way clockwise) at all times. Note: Prior to making any change in the circuit, always turn the voltage knob to its minimum setting (all the way counterclockwise) and turn off the power supply! So the next time you turn on the power supply its output will be zero volts.
The Digital Multimeter
The digital multimeter is shown in Fig. 2.3. As its name suggests, a multi- meter has multiple functions. It can be used for several different purposes, two of which are a voltage measuring device (a voltmeter) and a current measuring device (an ammeter). We will use these functions in this exper- iment.
2.6. Equipment
(a) Voltmeter connected in parallel (b) Ammeter connected in series
Figure 2.4: Schematics of meters being connected in a circuit.
voltmeter) or current (as an ammeter), one cable is always connected to the COM plug. If the multimeter is used to measure current, the other lead is connected to either the 10A plug or the 400mA plug. A voltmeter must be connected in parallel (across) to the cir- cuit element of interest, as shown in Fig. 2.4(a). Since the voltmeter measures potential difference between two points, it is easy to connect. To measure the potential difference (voltage drop) across a resistor, use two cables to connect the plugs of the voltmeter to the circuit across the resis- tor (one cable before the resistor and a second cable after the resistor). A voltmeter typically has a very large internal resistance; therefore very little current will flow through it. Consequently, the current in the circuit will be approximately the same before and after the voltmeter is connected. To use the multimeter as an ammeter, the dial selector is set to one of the positions labeled “A”. Similar to the voltmeter settings there are AC and DC settings. Like the voltmeter, two cables must be connected to the ammeter. One of your cables MUST be connected to the plug labeled “COM”. The second cable can be connected to one of two possible plugs —- either the “10A” plug or the “400mA” plug. If you have a large amount of current (anything above 400 mA), you must connect the cable to the terminal marked “10A”. If you put it in the “400mA” terminal you could damage the multimeter. If you are unsure if you have too much current for the 400 mA plug, start with the 10A plug. If you do not get any reading at
- Ohm’s Law
Figure 2.5: Standard symbols
all (i.e. 0.00), you have a very small current and can then move the cable to the 400 mA plug. An ammeter must be connected in series with the circuit ele- ment of interest, as shown in Fig. 2.4(b). This means that unlike measur- ing voltage, if you want to measure current you must break the circuit and wire the ammeter in. All of the current must flow through the ammeter in order for it to be measured. If you use your finger to trace the path of a charge in Fig. 2.4(b) after it leaves the power supply, you will see that it must go through both the resistor and the ammeter. In contrast, tracing the path of a charge in Fig. 2.4(a) you will see that it has two “parallel” paths through which it can go (do not connect an ammeter in this man- ner). An ammeter typically has a very small internal resistance. Therefore, the current in the circuit is approximately the same before and after the ammeter is connected. Standard electronic symbols are shown in Fig. 2.5. The positive side of a battery or power supply is indicated with the longer vertical line.
The Breadboard
The breadboard is designed for quick construction of simple electronic cir- cuits and is shown in Figure 2.6. Electronic elements (e.g. resistors) are
- Ohm’s Law
Figure 2.7: These two circuits are equivalent — they have the same config- uration of elements and will act in exactly the same manner.
Suggestions for building circuits
The schematic representation of electronic circuits typically shows wires as straight lines and changes in the direction of the wires are indicated by abrupt bends in the wires. In practice, the flexible wires are not straight and as you might expect changes in direction are not abrupt 90 degree bends in the wires. Adding measuring devices (e.g. ammeters, voltmeters) to the circuit increases the circuit’s complexity. The following steps will guide you through the construction of a simple circuit that includes an ammeter and a voltmeter. To avoid confusion, all of the wires used in the following example have different colors. The figures in this guide show both the circuit represented schematically and how the circuit actually looks in practice.
- Start by building the circuit without any meters. Where two lines meet, you will need two wires. Although it may seem efficient to initially construct the circuit with the meters included, experience has shown that this method often leads to wiring errors. Figs. 2.8(a) (schematically) and 2.8(b) (in practice) show a simple circuit with a power supply and a single resistor. The green wire is connected to the positive terminal of the power supply and the white wire is connected to the negative terminal.
2.6. Equipment
(a) Schematic (b) In practice
Figure 2.8: Building a circuit that includes a power supply and resistor.
- Include an ammeter in the circuit to measure current. Attach a single wire to the “COM” input of your ammeter; in this example, this is the purple wire. Identify the element in your circuit through which the desired current is flowing (in this case the resistor). Unplug the wire (or wires) leading into one end of that element and plug all of them into either the 400mA or 10A input of your ammeter, depending on the size of the current you are measuring. In this example, there is only one wire leading to the resistor (the green wire) and we are using the 400mA setting of the ammeter. Plug the free end of the purple wire into the plug on the breadboard where you removed the circuit’s wire (or wires) — i.e. the place where the green wire was connected in Fig. 2.8(b). You have now forced all of the current carried by the wire (or wires) to go through the ammeter in addition to the circuit element of interest. The ammeter is now properly connected in series with the resistor. Figs. 2.9(a) (schematically) and 2.9(b) (in practice) show our simple circuit with a power supply, a single resistor, and an ammeter. Turn the dial to read mA or A. By default, it is set to read AC current. We have DC current, so press the yellow button to change the mode to DC. You’ll have to do this again if the multimeter turns off automatically. Note that the ammeter should
2.6. Equipment
(a) Schematic (b) In practice
Figure 2.10: Building a circuit that includes a power supply, voltmeter, resistor, and ammeter.
notation. Scientific notation uses a number between 0 and 9.9 multiplied by some power of ten. The resistor color code system uses a number between 01 and 99 multiplied by some power of ten. These color bands tell the value of the resistance. Starting from the end, the first band represents the first digit of the resistance value and the second band the second digit. The third band represents the power of ten multiplying the first two digits. The fourth band represents the tolerance. If the fourth band is absent, it means the tolerance is 20%. Table 2.2 is a color code chart, from which one can tell the resistance of a resistor.
Example
Suppose the color code on a resistor is yellow, violet, orange and gold like the resistor depicted in Fig. 2.11. What is its resistance and what is the uncertainty of this resistance? The value of the resistance can be found from the first three colors. From the table above, the first digit is 4 (corresponding to the yellow band), the second digit is a 7 (corresponding to the violet band) and the power of 10 multiplier is 3 (corresponding to the orange band).
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Table 2.2: Resistor color codes
Color 1st digit 2nd digit Power of 10 Tolerance black 0 0 0 - brown 1 1 1 - red 2 2 2 - orange 3 3 3 - yellow 4 4 4 - green 5 5 5 - blue 6 6 6 - violet 7 7 7 - gray 8 8 8 - white 9 9 9 - gold - - - 5% silver - - - 10% none - - - 20%
Figure 2.11: Example resistor.
- Ohm’s Law
rule is that the uncertainty is half of the value of the least significant digit. This value is 0.01 V so half of that is 0.005 V. Here’s why: The meter can only display two digits to the right of the decimal so it must round off additional digits. So if the true value of the voltage is between 7.445 and 7.454 V, the voltmeter rounds it to 7.45 V. Thus the average value and its uncertainty can be written as 7. 45 ± 0 .005 V. When you record this, be sure to write 7.45 V, not 7.450 V. Writing 7.450 V implies that the uncertainty is 0.0005 V. Note that in the previous example we assumed that the meter reading was steady. If instead, the meter reading is fluctuating, then the situation is different. For this case, you need to estimate the range over which the display is fluctuating and then estimate the average value. For example, if the display is fluctuating between 5.4 and 5.8 V, you would record your reading as 5. 6 ± 0 .2 V. The uncertainty due to the noisy reading is much larger than your ability to read the last digit on the display, so you record the larger error.
Combining uncertainties
Information on combining uncertainties is contained in Appendix A. As was done in Physics 251, KaleidaGraph can give you the uncertainty in the slope of a graph by choosing “Curve fit”, then “General fit” and finally “fit1”.
Note! You will be asked about the consistency of results, or to compare values. Whenever this is asked, it is meant to be a quantitative answer. See Appendix A for the instructions on determining consistency.
2.7. Procedure
2.7 Procedure
- The units of all quantities must be specified, i.e. Ω = Ohms, V = Volts and A = Amps.
- For unit abbreviations, the prefix “k” means “kilo” = 10^3 and “m” means “milli”= 10−^3
- Set the current control knob to its maximum setting at all times (full clockwise position).
Circuit with one resistor
- Construct the circuit shown in Fig. 2.12. Refer to the Sugges- tions section above. Choose a resistor that has a resistance of at least 1000 Ω, so that we can neglect the ∼ 6 Ω internal resistance of the ammeter. Choose a voltage setting on the power supply, and read off the voltage and current from the meters. Then use Ohm’s Law (Eq. 2.1) to experimentally 5 determine the resistance. Record your measurements in Data Table 1 in your Excel spreadsheet. Refer to Eq. A.2 (included at the end of this write-up) to calculate the uncer- tainty in your experimentally determined resistance. Compare your measured value (Ohm’s Law value) with the nominal value given by the color code (see Question 1). (^5) Here “experimentally” means that we are performing an experiment to measure the resistance, not that the method is “experimental” and thus not well-tested yet.
Figure 2.12: Schematic for Step 1
2.7. Procedure
Figure 2.14: Schematic for 2 resistors connected in parallel.
Two resistors connected in parallel
- Construct the circuit shown in Fig. 2.14. Use two different resistors having resistances of approximately 1 kΩ and 2 kΩ. Set the power supply voltage to the middle of its range and record your measured voltage and current in Data Table 4. For two resistors wired in parallel, the uncertainty of the equivalent resistance is given by Eq. 2.6.
|δR (^) P | = R (^2) P
δR (^1) R (^21)
δR (^2) R (^22)
- Make sure to answer questions 1-7.
