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Physics Laboratory Manual

n

L ABOR AT ORY

Loyd

28

Measurement of Electrical

Resistance and Ohm’s Law

OBJECTIVES

o Define the concept of electrical resistance using measurements of the voltage across and current

in a wire coil.

o Investigate the dependence of the resistance on the length, cross-sectional area, and resistivity of

the wire.

o Investigate the equivalent resistance of series and parallel resistors.

EQUIPMENT LIST

.

Resistance coils (standard set available from Sargent-Welch or Central Scientific consisting of 10 m and

20 m length of copper and German silver wire)

.

Direct current ammeter (0–2 A), direct current voltmeter (0–30 V, preferably digital readout)

.

Direct current power supply (0–20 V at 1 A)

THEORY

If a voltage V is applied across an element in an electrical circuit, the current I in the element is determined

by a quantity known as the resistance R. The relationship between these three quantities serves as a

definition of resistance.

COPYRIGHT ª 2008 Thomson Brooks/Cole

R¼

V

I

ðEq: 1Þ

The units of resistance are volt/ampere, which are given the name ohm. The symbol for ohm is O. Some

circuit elements obey a relationship known as Ohm’s Law. For these elements the quantity R is a constant

for different values of V. If a circuit element obeys Ohm’s Law, when the voltage V is varied the current I

will also vary, but the ratio V/I should remain constant. In this laboratory we will perform measurements

on five coils of wire to investigate if they obey Ohm’s Law. We also will determine the resistance of the coils.

The resistance of any object to electrical current is a function of the material from which it is constructed, the length, the cross-sectional area, and the temperature of the object. At constant temperature

the resistance R is given by

ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work

covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information

storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher.

279

280

Physics Laboratory Manual n Loyd

R1

R2

R3

Figure 28-1 Resistors in series.

R1

R2

R3

Figure 28-2 Resistors in parallel.

R¼r

L

A

ðEq: 2Þ

where R is the resistance (O), L is the length (m), A is the cross-sectional area (m2), and r is a constant

dependent upon the material called the resistivity (O$ m). Actually r is a function of temperature, and if

the temperature of the coils of wire rises as a result of the current in them, this may be a source of error.

Circuit elements in an electrical circuit can be connected in series or parallel. Three resistors (R1, R2,

and R3) are connected in series as shown in Figure 28-1. For resistors in series the current is the same for all

the resistors, but the voltage drop across each resistor is different. For resistors in series the equivalent

resistance Re of the three resistors is given by

Re ¼ R1 þ R2 þ R3

ðEq: 3Þ

The same three resistors are shown connected in parallel in Figure 28-2. For resistors in parallel

the current is different in each resistor, but the voltage across each resistor is the same. In this case the

equivalent resistance Re of the three resistors in terms of the individual resistors is given by

1

1

1

1

¼

þ

þ

Re R1 R2 R3

ðEq: 4Þ

One of the objectives of this laboratory will be to observe the behavior of resistors in series and parallel.

EXPERIMENTAL PROCEDURE

1. Connect the ammeter A, the voltmeter V, and the power supply PS to the first resistor as shown

in Figure 28-3. The basic circuit is the power supply in series with a resistor. To measure the current in

the resistor, the ammeter is placed in series. To measure the voltage across the resistor, the voltmeter

is placed in parallel.

2. Vary the current through resistor R1 in steps of 0.250 A up to 1.000 A. For each specified value of the

current, measure the voltage across the resistor and record the values in Data Table 1. The resistors

will heat up and may be damaged by allowing current in them for long periods of time. Measurements

should be made quickly at each value of the current. APPLY VOLTAGE ONLY WHEN DATA ARE

BEING TAKEN.

3. Repeat Step 2 for each of the five resistors. For each resistor the ammeter must be in series with that

resistor and the power supply, and the voltmeter must be in parallel with the resistor. Record all

values in Data Table 1.

4. Connect the first four resistors in series to measure the equivalent resistance of the combination. Use

two values of current, 0.500 A and 1.000 A, and measure the value of the voltage for each of these

values of current. Record the voltage in Data Table 2.

Laboratory 28 n Measurement of Electrical Resistance and Ohm’s Law

!

Ammeter

”

R1

”

!

281

!

R2

R3

R4

R5

”

Power Supply

Figure 28-3 Measurement of current and voltage for resistor R1.

!

R1

”

R2

!

Power

” Supply

!”

Ammeter

Figure 28-4 Resistors R1 and R2 in parallel.

5. Measure the voltage across the combination of R2, R3, and R4 in series for currents of 0.500 A and

1.000 A and record the values in Data Table 2.

6. Connect R1 and R2 in parallel as shown in Figure 28-4 and measure the voltage across the combination

for current values of 0.500 A and 1.000 A and record in Data Table 2.

7. Connect R1 and R3 in parallel as shown in Figure 28-5 and measure the voltage for current values of

0.500 A and 1.000 A and record in Data Table 2.

8. Connect R2 and R3 in parallel and perform the same measurements as described in Steps 6 and 7.

Record the results in Data Table 2.

COPYRIGHT ª 2008 Thomson Brooks/Cole

!

R1

R2

”

R3

!

!”

Ammeter

Figure 28-5 Resistors R1 and R3 in parallel.

Power

” Supply

282

Physics Laboratory Manual n Loyd

CALCULATIONS

1. The first four coils are made of copper with resistivity of r ¼ 1.72 & 10$ 8 O–m. The fifth coil is made of

an alloy called German silver with resistivity of r ¼ 28.0 & 10$ 8 O–m. The first, second, and fifth coils

are 10.0 m long, and the third and fourth coils are 20.0 m long. The diameters of the first, third, and

fifth coils are 0.0006439 m, and the diameters of the second and fourth coils are 0.0003211 m. Use these

values in Equation 2 to calculate the value of the resistance for each of the five coils and record the

results in Calculations Table 1 as the theoretical values for the resistance Rtheo.

2. If Equation 1 is solved for V, the result is V ¼ IR. There is a linear relationship between the voltage and

the current, and the slope of V versus I will be the resistance R. Perform a linear least squares fit to

the data in Data Table 1 with V as the vertical axis and I as the horizontal axis. Record in Calculations

Table 1 the slope of the fit for each resistor as the experimental value for the resistance Rexp. Also

record the value of the correlation coefficient r for each of the fits.

3. Calculate the percentage error in the values of Rexp compared to the values of Rtheo for the five resistors

and record the results in Calculations Table 1.

4. For the data of Data Table 2 calculate the values of the equivalent resistance for the various series and

parallel combinations listed in the table as the value of the measured voltage divided by the

appropriate current. Calculate and record the mean of the two trials as ðRe Þexp in Calculations Table 2.

5. Equations 3 and 4 give the theoretical expressions for equivalent resistance for series and parallel

combinations of resistance. Calculate a theoretical value for the equivalent resistance for each series

and parallel combination measured in Data Table 2. For the values of the individual resistances

R1, R2, and R3 in Equation 3 and 4, use the experimental values determined from the fit to the data on

the individual resistors. Record this theoretical value for the equivalent resistance in each case as

(Re)theo in Calculations Table 2.

6. Calculate the percentage difference between the values of ðRe Þexp and (Re)theo for each of the series and

parallel combinations measured and record the results in Calculations Table 2.

GRAPHS

1. Construct graphs of the data in Data Table 1 with V as the vertical axis and I as the horizontal axis.

Use only one piece of graph paper for all five resistors, making five small graphs on that one sheet.

Choose different scales for each graph if needed, but make the five graphs as large as possible while

still fitting on one page. Also show on each small graph the straight line for the linear least squares fit.

Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

L A B O R A T O R Y 2 8

Section . . . . . . . . . . . . . . . .

Date . . . . . . . . . . . . . . . .

Measurement of Electrical Resistance and Ohm’s Law

PRE-LABORATORY ASSIGNMENT

1. Three resistors R1, R2, and R3 are connected in series with R1 < R2 < R3. Choose all correct answers

below. The total resistance of the combination is (a) less than R1, (b) less than R3, (c) greater than R3,

(d) greater than 3 R3, (e) equal to R1 þ R2 þ R3.

2. Two resistors R1 and R2 are connected in parallel with R1 < R2. Choose all correct answers below. The
total resistance of the combination is (a) less than R1, (b) less than R2, (c) greater than R2, (d) greater
than 2 R2, (e) equal to (R1R2)/(R1 þ R2).
3. A wire of length L1 and diameter d1 has resistance R1. A second wire of the same material has length
L2 ¼ 2 L1 and diameter d2 ¼ 2 d1. The resistance of wire two is R2. Choose the correct value for R2.
(a) R2 ¼ R1, (b) R2 ¼ 2R1, (c) R2 ¼ ½R1, (d) R2 ¼ 4R1.
4. If a circuit element carries a current of 3.71 A, and the voltage drop across the element is 8.69 V, what is
the resistance of the circuit element? Show your work.
R ¼ ____________________ O
COPYRIGHT ª 2008 Thomson Brooks/Cole
5. A resistor is known to obey Ohm’s Law. When there is a current of 1.72 A in the resistor, it has
a voltage drop across its terminals of 7.35 V. If a voltage of 12.0 V is applied across the resistor, what
is the current in the resistor? Show your work.
I ¼ ____________________ A
283
284
Physics Laboratory Manual n Loyd
6. The resistivity of copper is 1.72 & 10$ 8 O–m. A copper wire is 15.0 m long, and the wire diameter is
0.0500 cm. What is the resistance of the wire? Show your work.
R ¼ ____________________ O
7. A wire of cross-sectional area 5.00 & 10$ 6 m2 has a resistance of 1.75 O. What is the resistance of a wire
of the same material and length as the first wire, but with a cross-sectional area of 8.75 & 10$ 6 m2?
Show your work.
R ¼ ____________________ O
8. Three resistors of resistance 20.0 O, 30.0 O, and 40.0 O are connected in series. What is their equivalent
resistance? Show your work.
R ¼ ____________________ O
9. Three resistors of resistance 15.0 O, 25.0 O, and 35.0 O are connected in parallel. What is their
equivalent resistance? Show your work.
R ¼ ____________________ O
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
L A B O R A T O R Y 2 8
Measurement of Electrical Resistance and Ohm’s Law
LABORATORY REPORT
Data Table 1
I (A)
VR1 (V)
VR2 (V)
VR3 (V)
VR4 (V)
VR5 (V)
R1
R2
R3
R4
R5
0.250
0.500
0.750
1.000
Calculations Table 1
Rtheo
Rexp
r
COPYRIGHT ª 2008 Thomson Brooks/Cole
% Error Rexp
285
286
Physics Laboratory Manual n Loyd
Data Table 2
Combination
I (A)
V (V)
I (A)
R1 R2 R3 R4 Series
0.500
1.000
R2 R3 R4 Series
0.500
1.000
R1 R2 Parallel
0.500
1.000
R1 R3 Parallel
0.500
1.000
R2 R3 Parallel
0.500
1.000
V (V)
Calculations Table 2
Combination
(Re)exp1 (O)
R1 R2 R3 R4 Series
R2 R3 R4 Series
R1 R2 Parallel
R1 R3 Parallel
R2 R3 Parallel
SAMPLE CALCULATIONS
1. Rtheo ¼ rL/A ¼
2. (Re)exp ¼ V/I ¼
3. % Error ¼
4. (Series) (Re)theo ¼ R1 þ R2 þ R3 þ R4 ¼
5. (Parallel) (Re)theo ¼ (1)/(1/R1 þ 1/R2) ¼
6. % Difference ¼
(Re)exp2 (O)
ðRe Þexp (O)
(Re)theo (O)
% Diff
Laboratory 28 n Measurement of Electrical Resistance and Ohm’s Law
287
QUESTIONS
1. Do the individual resistors you have measured obey Ohm’s Law? In answering this question, consider
the least squares fits and the graphs you have made for each resistor. Remember that linear behavior
of V versus I is the proof of ohmic behavior.
2. Evaluate the agreement between the theoretical values for the individual resistances and the experimental values.
3. Does your agreement between the experimental and theoretical values of the series combinations of
resistors support Equation 3 as the model for series combination of resistors? The agreement is not
expected to be perfect, but you are to determine if the agreement is reasonable within the expected
experimental uncertainty.
COPYRIGHT ª 2008 Thomson Brooks/Cole
4. Evaluate the agreement between the experimental and theoretical values of the parallel combinations
of resistors. Do the results support Equation 4 as the model for the parallel combination of resistors
within the expected experimental uncertainty?
5. The first and second coils have the same length, and the third and fourth coils have the same length.
They differ only in the cross-sectional area. According to theory, what should be the ratio of the
resistance of the second coil to the first and the fourth coil to the third? Calculate these ratios for your
experimental results and compare the agreement with the expected ratio.
288
Physics Laboratory Manual n Loyd
6. The first and third coils have the same cross-sectional area, and the second and fourth coils have the
same cross-sectional area. They differ only in length. According to theory, what should be the ratio of
the resistance of the third coil to the first and the fourth coil to the second? Calculate these ratios for
your experimental results and compare the agreement with the expected ratio.
...
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