Resistance Vs Temperature Experiment Lab Report

Name: Emily Gatlin Partne r: Whitney Heaston Date Perform ed: 11 t h February 2009 Date: 18 t h February 2009 T.A. John Caruth

Re s is tance Ve rs us Te mpe ra ture

OBJECTIVE The objective of the experiment uses the principle that temperature This demonstrates how to apply a conceptual theory to produce quantitative results that reflect the respective expected predictions particularly regarding the resistance of a material using resistivity. Lastly, the experiment uses a mastery of a temperature-sensor in order to measure the temperature to calculate its effect upon the resistance of a given material.

INTRODUCTION The Re s is t iv i ty of Con duc t in g M ate rial s The net flow of charge flowing through a given area A per unit time t or current is shown by the following equation:

I

where Q=charge that passes through the sp ecified area Q , t =time the charge takes to pass through t he area t

Charge carriers each possess a charge q and density n that move with an average drift velocity vd , which makes the current within a given area A to be:

Q  nqvd At

I 

Q  nqvd A t

Assume that the current density J is the amount of current I flowing per unit area A and J 

I then A

the current density is J  nqvd . Resistivity is the ratio of the electric field strength E expressed  

E . J

Within a current, electrons move and give up energy due to collisions and lose kinetic energy transferred to the material in the form of heat energy. The movement of electrons is completely random unless direction by the presence of an electric field. In an electric field, the electrons have a net movement in the opposite [1]

Resistance vs. Temperature

Emily A. Gatlin

direction to the electric field. Thus, the acceleration of the electrons in the presence of an electric field is

a

eE m

Consequently, the average drift-velocity is vd  a where   t or the average time between collisions. Thus, the formula arises from the subsequent substitutions:

m eE ne 2 vd    J  E  2 m m ne  The resistivity is directly proportional to the mass of the electron and inversely proportional to the number density of the conduction electrons, their respective charge squared and the average time between collisions. Moreover, the electric field strength E is the change in electric potential V per unit length l

V V V A E   l   l I l I A

and Ohm's law states R 

   R

V I

A l R  l A

The changes in temperature produce a proportional change in resistivity of the material   T . Thus, if the initial temperature is T0 the resistivity is 0 at the initial temperature. The change in resistivity from temperature creates a function where resistivity exists as a function of temperature (T )  0 1  (T T0 ) across a length of wire with a given area.

Area Length Thus, this enables the expression of resistance as a function with the dependent variable, temperature

R  0 1  (T T0 )

l l R  0 1  (T T0 ) A A [2]

Resistance vs. Temperature

Emily A. Gatlin

 R  R0 1  (T T0 ) where R0  resistance at T0

The Re s is t iv i ty of Se m i - Con duc t in g M ate r ial s In a non-conducting material, electrons remain confined to an energy state called the valance band and stay restricted from the conduction band that allows free movement of electrons. Raising the temperature of the material can allow electrons to move from the valance band to the conduction band. Thus, in semi-conducting materials, the band gap or the energy gap to move from the valance band to the conduction band is much smaller than for non-conductors. The number density of electrons in the conduction band is a function of absolute temperature in degrees Kelvin (K ) 3 E gap

E gap

 kT 2 ne  2 me  e 2kT 2  2  me  mass of the electron

 R  R0e 2 kT If R0 

Taking the natural logarithm of both sides yields E gap

Planck's constant   6.63 1034 J/s or 2

ln R  ln R0  ln e 2 kT E gap 1 ln R  ln R0  2k T

E gap  band gap energy k  Boltzmann's constant 1.3810-23 J/K  8.617 10-5 eV/K

Using an individual thermistor, the calculation of the temperature uses the Steinhart-Hart equation

For small changes, the power component allows the 3

use of constant N c so that N  2 kT m 2 then c e  2 2 

1  A  B ln R  C (ln R )3 T

E gap

nc  N c e

2 kT

T  temperature K , R  resistance of thermistor , A, B , C  curve fitting constants

E gap



ml N c e 2 A

m e 2kT N c e 2

PROCEDURE The apparatus in the experiment is a coil of copper wire, a thermistor, a temperature controlled water reservoir, the apparatus for the Wheatstone bridge measurements, digital volt-ohm meter, a digital thermometer and banana leads.

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Resistance vs. Temperature

Emily A. Gatlin

The Pasco temperature controlled water reservoir allows the samples to be in a uniform temperature bath. The Wheatstone bridge apparatus is set-up. After the

Rx

+

resistance at each

Rs

room

v

R1

point of water.

temperature beginning at

v

―

apparatus is set-up, the data for

temperature to the boiling

R2

The

heat switch on the heater for

the water is the control for the temperature manipulation of

the apparatus. Microsoft Excel® analyzes the

collected data. In the next section of the experiment, the resistance versus temperature for a 0.00

thermistor the following apparatus allows the measurement of the resistance versus temperature for the thermistor: the thermistor and digital multimeter replace the Wheatstone bridge and copper coil sample. Microsoft Excel calculates resistance of the thermistor in a separate spreadsheet.

D A T A/ A NA L YS IS Par t I. Resistance versus Temperature for Copper Wire T ( C) n1 Rs R O

23.30 31.10

6.61 6.70

1.00 1.00

1.949852507 2.030303030

35.10 40.20 45.40 50.80

6.67 6.74 6.77 6.81

1.00 1.00 1.00 1.00

2.003003003 2.067484663 2.095975232 2.134796238

55.10 60.50 65.50 70.20 75.70

7.13 6.86 6.99 6.90 7.98

1.00 1.00 1.00 1.00 1.00

2.484320557 2.184713376 2.322259136 2.225806452 3.950495050

80.80 84.10

6.80 6.90

1.00 1.00

2.125000000 2.225806452 [4]

Resistance vs. Temperature

Emily A. Gatlin

90.98 96.07

6.87 6.91

1.00 1.00

2.194888179 2.236245955

99.70

7.00

1.00

2.333333333

R0 = 1.94985251 = 0.00393000 A= 7.66E-03 B= 0.00

%

Calculated

Theoretical

R = AT + B 0.0061x + 1.9034 R0 1.909500000 A=R0a 6.10E-03 B= 1.9034 a= 0.003194554

4.500000000

Error =

18.71%

Resistance vs. Termperature

4.000000000 3.500000000 3.000000000 y = 0.0061x + 1.9034

2.500000000

Resistance vs. Termperature

2.000000000 1.500000000 1.000000000 0.500000000

0.000000000 0.00

20.00

40.00

60.00

80.00

100.00

120.00

The major sources of error during this section of the experiment probably originate from the human error of time differentiation as the temperatures in the water bath continually increased. In order to get the best data, we had to attempt to record the resistance at the exact instance of the corresponding temperature. Additionally, the standard resistances could contain error by presenting resistances that deviate from these values. Moreover, the calibration setup might be a little offset producing error. Additionally, the mechanical error present could also be a possible source of error in the experiment. Of course, the potential for human error always exists in addition to the aforementioned sources of error. [5]

Resistance vs. Temperature

Emily A. Gatlin

Par t II. Resistance Versus Temperature for a Thermistor O

T ( C) 20.10 25.10 30.10 35.10

R 30.60 21.60 18.30 15.30

T (OK) 293.10 298.10 303.10 308.10

1/ T (OK) 3.41180E-03 3.35458E-03 3.29924E-03 3.24570E-03

ln(R) 3.4210000 3.0726933 2.9069011 2.7278528

40.10 45.10 50.10 55.10

13.20 11.20 9.50 8.20

313.10 318.10 323.10 328.10

3.19387E-03 3.14367E-03 3.09502E-03 3.04785E-03

2.5802168 2.4159138 2.2512918 2.1041342

60.10 65.10 70.10 75.10 80.10

7.00 6.00 5.30 4.80 4.50

333.10 338.10 343.10 348.10 353.10

3.00210E-03 2.95770E-03 2.91460E-03 2.87274E-03 2.83206E-03

1.9459101 1.7917595 1.6677068 1.5686159 1.5040774

85.10 90.10 95.10 99.30

3.70 3.30 3.00 2.60

358.10 363.10 368.10 372.30

2.79252E-03 2.75406E-03 2.71665E-03 2.68601E-03

1.3083328 1.1939225 1.0986123 0.9555114

k=

8.62E-05

Theoretical egap=

0.622 0.594

Calculated m = Egap/2k =

3205.6

egap=

5.53E-01

% Error

11.15% 6.96%

As in the first part of the experiment, the same errors within the apparatus are possible along with the improper calibration of the apparatus initially along with the inability to collect the instantaneous data with the corresponding

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Resistance vs. Temperature

Emily A. Gatlin

temperature. Another possible source of data is the water bath still possessing excess heat from the previous section and producing skewed results. Overall, the error was significant, but the concepts still existed in the predicted relationships.

4.0000000

ln (R) Versus (1/T)

3.5000000 3.0000000

y = 3205.6x - 7.6468

2.5000000 2.0000000

ln (R) Versus (1/T)

Linear (ln (R) Versus (1/T))

1.5000000 1.0000000 0.5000000 0.0000000

0.000E+005.000E-04 1.000E-03 1.500E-03 2.000E-03 2.500E-03 3.000E-03 3.500E-03 4.000E-03

CONCLUSION The experiment provided a highly effective experiment to display the relationship between temperature and resistance. Although the data collected presented significant data, the overall pattern still existed as expected. The use of temperature as the independent variable allowed its manipulation to produce an effect on the resistance. In this experiment, the resistance graduation increases as the temperature increases. The comparison of the resistance of the thermistor showed the effect of temperature on a semiconductor. The resistance increases in a semiconductor with temperature, but at the direct correlation seen in the conductor. Clearly, both cases show that there is a relationship between the temperature and resistance of an object.

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