Resistance Measurements
OBJECTIVE
•
•
To understand the relationship between the current, voltage, and resistance in a
circuit.
To study series and parallel resistance circuits.
INTRODUCTION
Resistance of an electrical element is defined by R=V/I or R=ρL/A, where R is the
resistance of the circuit element, V is the potential drop across the element, I is the
current passing through the element, ρ is the resistivity of the elements material
makeup, L is the length of the element, and A is the cross-sectional area of the element.
There are several ways to measure the value of the resistance; the most direct of which
is to measure the potential drop and the current of the element with the aid of a
voltmeter and ammeter. Additionally, more than one resistor can be part of a circuit.
The resistance may be in a series or a parallel combination. Both of these methods will
be studied and applied to the determination of several unknown resistors, series
combinations, and parallel combinations.
APPARATUS
o computer
o Vernier computer interface
o Logger Pro
o one Vernier Differential Voltage Probe
o wires
o three known resistors & one unknown
resistor
o adjustable 5 volt DC power supply
o one Vernier Current Probe
THEORY
As previously stated, resistance R [Ω] of any circuit element is given by:
R=
V
I
Equation 1
Where, V [V] is the electric potential between two points across the element and I [A] is
the quantity of charge moving through the element in a unit time. A resistor generally
obeys this relationship over a wide range. However, it can vary by approximately ±10%
from the known value of the resistor; somewhat due to temperature variations of the
resistor itself. We call this the precision of the resistor.
Resistance Measurements - Page 1
As the name suggests, a resistance device is one which "resists" the flow of charges in
an electrical circuit, thereby reducing the current flow in the aforementioned circuit.
In general, the resistance of any material is given by:
R=
ρ L
A
Equation 2
Where, ρ [Ωm] is the resistivity of the material {varies based on what the material is
made of}, L [m] is the length of the material, and A [m2] is the cross-sectional area of the
material, through which the current flows.
Resistors may be connected together in a circuit to either increase their resistance
(series combination) or to decrease their resistance (parallel combination).
A series connection of resistors R A , R B , & R C is illustrated in Figure 1 below.
I
RA
VA
I
RB
VB
I
RC
I
Figure 1
VC
As indicated, the current flows through the resistors, thus, each resistor has the same
value of the current. However, as the potential difference is measured across each
resistor, each resistor will have a different potential drop (voltage).
The total voltage is, thus:
V TOTAL = V A + V B + V C
Equation 3
Using this relationship, Equation 1, and the statement that the current through each
resistor is the same yields:
V TOTAL = R A I + R B I + RC I
= I ( R A + R B + RC )
V TOTAL
= R A + R B + RC
I
R EQ SERIES = R A + R B + RC
Equation 4
This defines the relation that the equivalent resistance of a series combination is the
sum of the given resistance elements.
Resistance Measurements - Page 2
A parallel combination of resistors R A , R B , & R C is illustrated in Figure 2 below.
I
IA
RA
IB
RB
IC
RC
I
Figure 2
V
In the parallel combination, the resistors are all connected to the same potential
difference and, thus, each has the same voltage across them. However, as the current
approaches the junction of the three resistors, it must split, and a fraction of the total
current passes through each of the three resistors.
Thus, the total current in the circuit is given by:
I TOTAL = I A + I B + I C
Equation 5
Using this relationship, Equation 1 and the statement that the voltage across each
resistor is the same yields:
I TOTAL =
V
+
V
+
V
R A R B RC
1
1
1
=V (
+
+
)
R A R B RC
I TOTAL = 1 + 1 + 1
V
R A R B RC
1
1
1
1
=
+
+
R EQ PARALLEL R A R B RC
Equation 6
This gives the relation that the equivalent resistance of a parallel combination is the
inverse of the sum of the inverses of each of the given resistance elements.
Resistance Measurements - Page 3
Voltmeter-Ammeter:
The most direct way to determine an unknown resistance, or to verify the series/parallel
relationships, is to measure the actual current passing through the system and voltage
across the resistor (Ohm's Law). As indicated, a voltmeter (potential difference
measuring device) must be connected across the resistor (in parallel) in order that is
measures a difference from one side to another (again, the potential difference) and an
ammeter (current measuring device) must be connected along the same part of the
circuit that passes through the resistor (in series) in order to measure the current
actually passing through the resistor; the ammeter does not keep any of the current nor
does the resistor so it does not matter if the ammeter is placed before or after the
resistor…only that it be in-line with it. Figure 3 below is an illustration of this type of
measuring arrangement.
Figure 3
Resistance Measurements - Page 4
INITIAL SETUP
•
Using the equipment we have available, a current probe will serve as the
ammeter in the circuit and a differential voltage probe will serve as the voltmeter
in your circuit.
A. Connect the Current Probe to Channel 1 and the Differential Voltage Probe to
Channel 2 of the computer interface.
B. Open the file “22 Ohms Law” in the Physics with Vernier folder. A graph of potential
vs. current will be displayed. The meter displays potential and current readings.
C. With the power supply turned off, connect the power supply, one of the known
resistors, wires, and clips as shown in Figure 3. Take care that the positive lead from
the power supply and the red terminal from the Current & Voltage Probe are
connected as shown in Figure 3.
•
Note: Attach the red connector of the voltage probe closer to the positive side of
the power supply.
D. Click
to zero both sensors. This sets
. A dialog box will appear. Click
the zero for both probes with no current flowing and with no voltage applied.
•
This can have consequences in making the y-intercept ≠ 0; or V = I * R + ?.
E. Have Dr. Arts check the arrangement of the wires before proceeding.
EXPERIMENTAL PROCEDURE
a) With one of the known resistors in place, adjust the power supply to approximately
0.5 VDC.
b) Monitor the voltage and current. Click
to allow the V & I probes to talk to
the computer. When the readings are relatively stable click
.
c) Increase the voltage on the power supply to approximately 1.0 VDC. When the
readings are stable click
.
d) Increase the voltage by approximately another 0.5 V. When the readings are
stable click
. Repeat this process until you reach a voltage of approximately 5.0
V.
e) Click
and set the power supply back to 0 V.
• Indicate a best-fit-straight-line on the graph and compute the slope and yintercept.
• At this point you will want to print ONLY your graph (NO data tables).
o Be sure your graph has a clear title indicative of the resistor(s) connected .
o Auto scale the graph before printing!
Resistance Measurements - Page 5
f) Connect the second of the known resistors and repeat a-e.
g) Connect the third of the known resistors and repeat a-e.
h) Connect the unknown resistor and repeat a-e.
i) Using you knowledge of resistor combinations, predict how series resistors would
affect current flow.
o What would you expect the effective resistance of two equal resistors in series to
be, compared to the resistance of a single resistor?
j) Connect the three known resistors in series and repeat a-e. As shown in the
illustration, be sure to connect the voltmeter across the combination, not across the
individual resistors.
Resistance Measurements - Page 6
k) Using you knowledge of resistor combinations, predict how parallel resistors would
affect current flow.
o What would you expect the effective resistance of two equal resistors in parallel
to be, compared to the resistance of one alone?
l) Connect the three known resistors in parallel and repeat a-e. As shown in the
illustration, be sure to connect the voltmeter across the combination, not across the
individual resistors.
Resistance Measurements - Page 7
REPORT ITEMS
(To be submitted and stapled in the order indicated below)
(-5 points if this is not done properly)
COVER PAGE
•
•
•
•
Lab Title
Each lab group member's first and last name printed clearly
Group Color
Date
DATA (0 points)
• All data collected was in the form of graphs
DATA ANALYSIS (0 points)
• All data analysis will be graphical analysis
GRAPHS (worth up to 10 points)
•
•
•
•
Three known resistor graphs
One unknown resistor graph
One series combination graph
One parallel combination graph
GRAPH ANALYSIS (worth up to 40 points)
•
The answers to the questions below should be completed on the back of each
respective printout
For EACH of the known resistances:
• Are the results that you found in each of these trials consistent with what you
expected to find?
• Is there a specific relationship between the variables (V, I, & R) or only a general
one?
• Based on your graph of V vs. I for each resistor:
o Explain the significance of the slope. What should it be?
o How would you expect the slopes of the graphs of each of the resistors
individually to look in comparison to each other?
• Based on your data and graphs, do your resistors follow Ohm’s law? Explain.
Resistance Measurements - Page 8
For the unknown resistor:
• What does your data indicate as to the value of your unknown resistor?
• Within the errors you found for your other resistors and combinations (± the
actual percentage values), what do you feel is the range in which your resistor
would most likely fall?
• Based on your graph of V vs. I for this resistor:
o Explain the significance of the slope.
• Based on your data and graph, does your unknown resistor appear to follow
Ohm’s law? Explain.
For the series combination:
• Do the values (V vs. I) of the series combination, in theory, correspond to your
expectations; i.e. in comparison to any of the individual resistors, what
expectation about the collected data did you have? Explain any variations.
• Based on your graph of V vs. I for this resistor:
o Explain the significance of the slope. What should it be?
 Show this calculation on the back of the graph.
• Based on your data and graph, does your series combination appear to follow
Ohm’s law? Explain.
For the parallel combination:
• Do the values (V vs. I) of the parallel combination, in theory, correspond to your
expectations; i.e. in comparison to any of the individual resistors, what
expectation about the collected data did you have? Explain any variations.
• Based on your graph of V vs. I for this resistor:
o Explain the significance of the slope. What should it be?
 Show this calculation on the back of the graph.
• Based on your data and graph, does your parallel combination appear to follow
Ohm’s law? Explain.
CONCLUSION (worth up to 20 points)
•
See the Physics Laboratory Report Expectations document for detailed
information related to each of the four questions indicated below.
1. What was the lab designed to show?
2. What were your results?
3. How do the results support (or not support) what the lab was supposed to
show?
4. What are some reasons that the results were not perfect?
Resistance Measurements - Page 9
QUESTIONS (worth up to 20 points)
• DO NOT forget to include the answers to questions that were asked within the
experimental procedure
1) It was stated that the voltmeter (differential voltage probe) must be placed in parallel
with the resistor in order to measure the potential difference across the resistor.
However, the voltmeter, by the very nature that it is an electrical device, also has
resistance.
• What then, qualitatively speaking (really big, really small, doesn't matter,
etc.), must the resistance of the voltage probe be so as not to adversely affect
the operation of the circuit?
• Consider the parallel resistance equation as you attempt to answer the question!!
2) It was stated that the ammeter (current probe) must be placed in series with the
resistor in order to measure the current passing through the resistor. However, the
ammeter, by the very nature that it is an electrical device, also has resistance.
• What then, qualitatively speaking (really big, really small, doesn't matter,
etc.), must the resistance of the current probe be so as not to adversely affect
the operation of the circuit?
• Consider the series resistance equation as you attempt to answer the question!!
3) How do you suspect that the lights, TV, appliances, etc. of your home are connected
together; in series or in parallel? Explain your reasoning.
Resistance Measurements - Page 10