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What is the relationship between the length of an electrically conductive metal wire, and its electric resistance?

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Introduction

Research Question

What is the relationship between the length of an electrically conductive metal wire, and its electric resistance?

Hypothesis

It is predicted that as the length of the wire increases, so will its electric resistance. The length of the wire used and its resistance will be directly proportional. Therefore, one should expect a relationship of the form

R = k · L

Between the two variables, where R is wire resistance, L is wire length, and k is a constant.

The hypothesis springs from the assumption that the wire will follow the theoretical relationship between resistance and length of a wire:

   R    =     ρ · L

A

Dictated by many scientific experiments. This equation holds that the resistance of a metal wire is proportional to its length and resistivity (p), and inversely proportional to its cross sectional area (A). Note that resistivity is a constant dependent on the material the wire is made of. Although not specifically tested in this experiment, the constant predicted in the first equation should in fact represent                                

image00.png

if it adheres to the theoretical equation linking resistance and length of a wire. Its unit is therefore (Ω·m)/m2  = Ωm-1

Variables

  • Length of the wire.
  • Wire temperature
  • Wire cross sectional area
  • Electric current flowing through wire
  • Potential difference across the wire
  • Resistance of the wire

The length of the wire will be changed by fixed values and is hence an independent variable. The resistance of the wire will be affected by these changes, making wire resistance a dependent variable that shall be examined. Resistance in itself will vary the current flowing through the wire. Current is thus also a dependent variable.

...read more.

Middle

0.22

0.400

1.07

0.28

0.500

0.88

0.34

0.600

0.74

0.41

0.700

0.63

0.48

0.800

0.55

0.55

0.900

0.49

0.61

1.000

0.46

0.65

*Values for resistance are given to two significant figures - the same as the significant figures for voltage values. Uncertainty in resistance will be calculated later. The resistance was calculated as specified in step 13 in the method.

For further investigation, it is necessary to calculate the uncertainty range for the resistance values. As previously mentioned, the resistance of the wire is attainable through the equation R = V/I, ifassuming the wire to have ohmic behavior. R is wire resistance, V is potential difference across wire, and I current through wire. The relative uncertainty for resistance is therefore

ΔR/R = ΔV/V  +  ΔI/I

Hence, the absolute resistance uncertainty is

ΔR =R · (ΔV/V  +  ΔI/I ).

Since the potential difference, its uncertainty, and the uncertainty in current are constants throughout, then:

ΔR =R · (0.02/0.3  +  0.02/I )

This equation is applicable to the resistance values obtained in the preceding table. Below is a table showing the same resistance values as before, with individual uncertainty for each value. Uncertainties are expressed in two significant figures, just like the actual resistance values. Each value is compared with the respective wire length used:

(Length ± 0.001) meters [m]

Resistance [Ω]

0.100

0.10 ± 0.01

0.200

0.15 ± 0.01

0.300

0.22 ± 0.02

0.400

0.28 ± 0.02

0.500

0.34 ± 0.03

0.600

0.41 ± 0.04

0.700

0.48 ± 0.05

0.800

0.55 ± 0.06

0.900

0.61 ± 0.07

1.000

0.65 ± 0.07

From the data collected, a graph of wire resistance vs. wire length was constructed. It includes error bars for both length and resistance, though the former are too small to be seen. The graph is shown on the next page:

image02.png

Although one set of data points was plotted, three regression lines were drawn. The one in the middle (Y1) is the regression line for the points plotted.

...read more.

Conclusion

Another improvement could deal with minimizing the existence of additional resistance in the circuit besides that of the metal wire examined. This could be done in several ways. The connecting wires used could be shorter, meaning their resistance would be smaller. A power supply and ammeter with smaller internal resistances could also be used. In either of the cases, additional resistance in the circuit would decrease, and the systematic error in resistance minimized.

Finally, though not too necessary, it is possible to improve the experiment by minimizing resistance distortions due to temperature increases. A simple way would be to wait at least 3 minutes between each current flow induced through the wire. This way, the whole circuit together with the wire itself would have time to cool down. Wire temperature would hence not be a factor distorting the resistance values being measured, as it would be more or less the same in all measurements. Nevertheless, it may be that the distortions to resistance values caused by temperature increase are so small that this improvement on the whole is futile.

Another minor improvement to the method would be the use of a switch. This way, current flow could be initiated and stopped on demand (i.e. immediately). It would be of course necessary to use a switch that offers little resistance. Otherwise, the current construction of the circuit is preferable.

...read more.

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