PHYS112 : Labs

5 Ohm's Law

In addition to reading this assignment, you may need to refer to Appendix A on uncertainties and Appendix B on linear regressions.

Introduction

Ohm's Law

The current flowing through an Ohmic device is directly proportional to the potential difference

between its terminals,

$\begin{displaymath} V=iR \end{displaymath}$

(10)

The constant of proportionality

is called the resistance of the device. This relationship is known as Ohm's Law. You will investigate the dependence of current on potential difference in resistors and light bulbs. Your job will be to determine whether or not these devices are Ohmic and if so to determine their resistances.

**Figure 8:** A pair of resistors connected in series and in parallel.
$\includegraphics{Ohm-seriesparallel.eps}$

Equivalent Resistance

A pair of resistors can be connected either in series or in parallel as shown in Figure 8. Note that two wires emerge from each pair. The equivalent resistance $R_\mathrm{eq}$ of a pair of resistors is the resistance between these two wires. The equivalent resistance of two or more resistors connected in series is given by

$\begin{displaymath} R_{eq} = R_1 + R_2 + R_3 + \ldots \end{displaymath}$

(11)

and that of two or more resistors connected in parallel is given by

$\begin{displaymath} \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \end{displaymath}$

(12)

You will test Eqs. 11 and 12 empirically by measuring the current drawn by series and parallel pairs of resistors subject to a known potential difference.

Resistor Color Codes

Most resistors you will encounter are marked with a set of bands, according to a standard color code, which you can use to determine their resistances. There are ten colors corresponding to numerical digits 0-9 (See the table below.), and gold and silver bands indicating 5% and 10% accuracy in the coded resistance, respectively. Starting at the far end of the resistor from the gold/silver band, the first two bands are the first two digits in the resistance. The third band gives the power of ten by which you multiply the first two digits to obtain the resistance.

color	black	brown	red	orange	yellow	green	blue	violet	gray	white
digit	0	1	2	3	4	5	6	7	8	9
multiplier	1	10	100	1k	10k	100k	1M	10M	100M	1000M

$\displaystyle R = [\mathrm{band 1}][\mathrm{band 2}] \times 10^{[\mathrm{band 3}]}$	$\textstyle \pm$	$\displaystyle 5\% (\mathrm{gold})$	(13)
	$\textstyle \pm$	$\displaystyle 10\% (\mathrm{silver})$

For example, Blue Yellow Red Gold gives $R = 64 \times 10^{2} \Omega = 64 \times 100 \Omega = 6400 \Omega$ with a tolerance of 5%, or $\pm 320 \Omega$ .

Experiment

You have been supplied with three resistors, a light bulb, a power supply, several wires for connecting them, and a digital multimeter (DMM) for measuring potential differences (voltages), currents, and resistances.

Warnings

Before turning on the power supply or connecting it to a circuit, make sure that the switch for the voltage range is in the 0-8 V position, and that the Voltage Increase dial is turned all the way down (the counter-clockwise direction). Then slowly increase the voltage as needed. Do not put more than 3 V across a light bulb!
When using the DMM as an ammeter (i.e. to measure current) ...
1. make sure the red lead is plugged into the port on the DMM labeled for measuring current. (You want the one for smaller currents if there are two.)
2. always be certain to connect it in series with the device through which you are trying to measure the current. Never connect an ammeter in parallel with anything, or you will blow its fuse!
When using the DMM as a voltmeter or ohmmeter, make sure the red lead is plugged into the voltage port, and make sure to connect it in parallel with the device(s) of interest. (Connecting a voltmeter or Ohmmeter in series with something doesn't make any sense, but it also doesn't blow any fuses.)

Resistance

**Figure 9:** (a) A simple circuit consisting of a voltage source connected in series with a resistor. (b) The same circuit with an ammeter to measure the current through the resistor and a voltmeter to measure the voltage drop across its terminals.
$\includegraphics{Ohm-circuita.eps}$ $\includegraphics{Ohm-circuitb.eps}$

Set up the circuit shown in Figure 9a with one of your resistors.
Record the resistance and uncertainty indicated by the color code on the resistor.
Set the emf of the power supply to a value in the range $0 < \mathcal{E} < 3 \mathrm{V}$ .
Plug the red lead of the DMM into the voltage port, and connect the DMM in parallel with the resistor as shown in Figure 9b. Turn the knob on the DMM to a suitable voltage scale, and record the voltage across the resistor.
Plug the red lead of the DMM into the current port, and connect the DMM in series with the resistor as shown in Figure 9b. Remember never to connect an ammeter in parallel with anything! Turn the knob on the DMM to a suitable current scale, and record the current through the resistor.
Repeat steps 3-5 for four other source emfs in the range $0 < \mathcal{E} < 3 \mathrm{V}$ .
Repeat the whole process (steps 3-6) with the light bulb instead of a resistor. In choosing your source emfs be sure that all of your source emfs are below 3 V and that three of them are in the low-voltage range in which the filament does not glow.
Use the DMM to measure the resistances of the resistor and light bulb you used. Ask for help if you need it.

Equivalent Resistance

**Figure 10:** A voltage source connected in series with (a) two resistors connected in series and (b) two resistors connected in parallel.
$\includegraphics{Ohm-series.eps}$ $\includegraphics{Ohm-parallel.eps}$

Set up the circuit shown in Figure 10a.
Set the source emf to about 1 V.
Use the DMM to measure the voltages and across each of the resistors and the voltage across the pair.
Use the DMM to measure the current through the resistors. Remember never to connect an ammeter in parallel with anything!
Set up the circuit shown in Figure 10b.
Keep the source emf at about 1 V.
Use the DMM to measure the voltage across the resistors.
Use the DMM to measure the currents and flowing through each of the resistors and the current flowing into the pair. Remember never to connect an ammeter in parallel with anything!
Use the DMM to measure the resistances of the resistors.

Analysis

Resistance

Put your voltage and current data into a spreadsheet, and plot vs for the resistor and the light bulb.
Are your data compatible with a linear model? If so, use the LINEST function to find the slope of a linear fit to the data and its uncertainty (see Appendix B).

Equivalent Resistance

Use your voltage and current measurements ( and ) and Ohm's law (Eq. 10) to determine the equivalent resistances of the series and parallel pairs of resistors.
Use the resistances you measured with the DMM and Eqs. 11 and 12 to calculate the predicted equivalent resistances of your series and parallel pairs of resistors.
Is there an obvious relationship between your measured , , and in the series circuit? How about between , , and in the parallel circuit?

Before You Leave Lab

Discuss with your instructor preliminary responses to the individual assignment below.

Hand In ...

... a printout of your spreadsheet and responses to the following.

For the resistor you studied in detail, give the resistance indicated by the color codes, the resistance you measured with the DMM, and the resistance you extracted from your vs. graph with uncertainties. Are they consistent with each other within uncertainty?
Is the light bulb you studied an Ohmic device? Give reasoning based on your data.
Compare your measurement of the resistance of the light bulb with your graph of vs. . What might be going on here?
What relationships, if any, did you find between , , and in the series circuit and between , , and in the parallel circuit?
Do the equivalent resistances you determined from your voltage and current measurements agree with the predictions of Eqs. 11 and 12 within uncertainty? Include your results in your response.