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Ohm's Law and Electrical Circuits

Introduction

In this experiment, you will measure the current-voltage characteristics of a resistor and check to see if the resistor satisfies Ohm's law. In the process you will learn how to use the multimeter to measure voltage, current, and resistance. You will then test some of the laws of circuit theory. When a potential difference, V, is applied across a conductor, an electrical current, I, will flow from the high potential end to the low potential end. In general the current will increase with the applied voltage (potential difference). A plot of the current as a function of the voltage is called the current-voltage (I-V) characteristic. If the I-V characteristic is a straight line, as in Fig. 1, then we say that the piece of conductor satisfies Ohm's law: V = IR, where R is a constant defined to be the resistance and has units of volts/ampere, or Ω (ohm).
Figure 1

Figure 1: I-V curve for an ohmic material

In an electrical circuit, the wires that are used to connect the circuit elements do have resistance. However, the resistances of the wires are usually negligible compared with the resistances of the circuit elements. There are specific elements called resistors that control the distribution of currents in the circuit by introducing known resistances into the circuit. The currents and voltages at different parts of the circuit can be calculated by using circuit theory that will be discussed later. There are many kinds of resistors but the most common ones are the carbon composite resistors shown below. These resistors are small brown cylinders with colored bands. The color bands follow a color code giving the resistance to within a specified manufacturing tolerance.
Figure 2

Figure 2

In this lab you will be studying only simple DC circuits consisting of a power source and one or more resistors connected with wires whose resistances are negligible compared to those of the resistors. The basic theory for analyzing the circuit is summarized by two laws known as Kirchhoff's Rules: Two types of resistor connections are usually found in a circuit, the series and the parallel connection shown in Fig. 3.
Figure 3

Figure 3

Using Kirchhoff's Rules it can be shown that the three resistors in series are equivalent to a single resistor with equivalent resistance, R, given by:
( 1 )
R = R1 + R2 + R3  (resistors in series)
 
Likewise, the three resistors connected in parallel are equivalent to a single resistor with equivalent resistance, R, given by:
( 2 )
1
R 
 = 
1
R1
 +
1
R2
 +
1
R3
   (resistors in parallel)
 

Apparatus

The apparatus for this experiment consists of a regulated power supply and two multi-meters. These pieces of equipment are described below.

Regulated Power Supply

Figure 4

Figure 4

The regulated power supply and its symbol in a circuit are shown above. This power supply converts the output from a regular 110 V, 60 Hz AC outlet into a constant DC power source with variable voltage from 0 to 20 V. It produces a maximum current of 0.5 A. Turning the control knob on the device can vary the output voltage. It is good practice to always start from the zero voltage and gradually increase it to the desired value. The output is obtained through the red and black jacks. By convention, the red jack is the positive terminal and the black jack is the negative terminal.

Measuring Currents, Voltages, and Resistances

When the multi-meter is set to measure current it serves as an ammeter, when it is set to measure voltages, it serves as a voltmeter, and when it is set to measure resistances, it serves as an ohmmeter. The symbols for the ammeter, voltmeter and the ohmmeter are given below.
Figure 5

Figure 5

To measure the current flowing through an object such as a resistor, the ammeter is connected in series with the object as shown in Fig. 6a. Ammeters have very low resistance so that when they are placed in a circuit, they do not significantly affect the total circuit resistance and hence the current to be measured. To measure the voltage across an object such as a resistor, the voltmeter is connected in parallel with the object as shown in Fig. 6b. Voltmeters have very large resistance so that only a small portion of the circuit's current will be diverted through the voltmeter. To measure the resistance of an object such as a resistor, the ohmmeter is connected to the object as shown in Fig. 6c. If the resistor is connected to a circuit, then one end of the resistor must be disconnected from the circuit while making this measurement. The battery in the multi-meter supplies the current necessary for measuring the resistance so that no external power supply is needed.
Figure 6

Figure 6

Making Simultaneous Current and Voltage Measurements

Figure 7

Figure 7

There are two ways of making simultaneous measurements of A and V as shown in Fig. 7a and Fig. 7b. In Fig. 7a, the ammeter measures the current in the resistor R, but the voltmeter does not measure the voltage across the resistor, VR. Instead it measures the voltage across the resistor plus the voltage on the ammeter, VA. Since VR + VA = IR + IRA, where RA is the resistance of the ammeter, the voltmeter measurement will be approximately equal to VR if R is much larger than the resistance of the ammeter. Ammeters typically have resistances of 0.001 Ω or less. Using method (a) to measure the voltage on a resistor with a small resistance, say 0.1 Ω, would produce an error in the voltage of IRA / IR = 0.001/0.1 or a 1% error. On the other hand, for a large resistance, say R = 1000 Ω, the error reduces to
IRA / IR = 0.001/1000 or 0.0001%.
The method illustrated in Fig. 7a should therefore be used to measure large resistances.
In Fig. 7b, the voltmeter measures the voltage across the resistor R, but the ammeter does not measure the current through the resistor, I. Instead it measures the current through the resistor plus the current through the voltmeter, IV. The sum of these currents is given by:
( 3 )
I + IV =
VR
R
 +
VR
RV
 
where RV is the resistance of the voltmeter. Therefore, the ammeter measurement will be approximately equal to I if R is much smaller than RV. Voltmeters typically have resistances of 100,000 Ω or more. Using method (b) to measure the current on a resistor with large resistance, say 1000 Ω, will produce an error in the measured current of IV / I = R/RV = 1,000/100,000 or a 1% error. For a small resistance, say R = 0.1 Ω, the error reduces to
R/RV = 0.1/100,000 or 0.0001%.
The method illustrated in Fig. 7b should be therefore be used to measure small resistances.

Procedure

Resistance Measurements

Current-Voltage Characteristics of a Resistor

This part of the experiment requires you to simultaneously measure the current and voltage on a resistor. The resistors used in this experiment have resistances of about 1000 Ω. Therefore, the method shown in Fig. 7a should be used to simultaneously measure I and V.

Kirchhoff's Rules

In this experiment you will verify Kirchhoff's rules on a simple circuit shown below.
Figure 8

Figure 8

When you are finished with the experiment, please clean up your work area and return all the wires and clips to their storage bins. Be sure that you and your instructor initial your data sheet(s), and that you hand in a copy of your data before you leave the lab.

Data Analysis

Resistance Measurements

For this part we shall denote the calculated equivalent resistance by RT, and the measured equivalent resistance by R.

Current Voltage Characteristics of a Resistor and a Light Bulb

Kirchhoff's Loop and Junction Rules

Discussion

Summarize the results for the section on Resistance Measurements. Which of the connection, series or parallel, gave the least total resistance? Why? Does your measured value of the total resistance of the series connection and the parallel connection agree with the calculated equivalent resistance? Describe the current-voltage characteristics of the resistor studied in the section on Current-Voltage Characteristics of a Resistor. Is the current zero when the voltage is zero? If not, explain the discrepancy. Did the result agree with Ohm's law? What is the value of the resistance obtained from the least square fit and how does it compare with the value measured with the ohmmeter? Compare the V-I plots for the resistor and light bulb. From the "shape" of V-I plot for the light bulb, what can you conclude about the resistance of the light bulb? In the section on Kirchhoff's Rules, did your current measurements satisfy the junction rule? Did your voltage measurement satisfy the loop rule? Do your comparison quantitatively, taking uncertainties in the measurements into account. In addition to the uncertainty of the measuring instrument, does the connection of the ammeter or voltmeter to the circuit cause additional uncertainty? If so, are these uncertainties significant?