Cross sectional area - Sample Essay
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 i?? 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 = ? i?? 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 if it adheres to the theoretical equation linking resistance and length of a wire. Its unit is therefore (? i?? m)/m2 = ? m-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. The other variables will be kept (or assumed to be) constant. Materials Used – Metal wire of length greater than a meter. – Direct power supply – Voltmeter (i?? 0. 02 volts)
– Ammeter (i?? 0. 02 Amperes) – Ruler (i?? 0. 001 meters) – Scotch tape – 4 alligator clips – 5 connecting wires Method 1. Measure a wire length of approximately 1. 2 meters, not starting from its beginning. 2. Attach the segment of the wire measured to a firm table using scotch tape. 3. Use 3 connecting wires and 2 alligator clips to connect a power supply and an ammeter in series with each other, connecting all that across the metal wire. 4. Using 2 more connection wires and alligator clips, connect a voltmeter across the wire. 5. You now have completed a circuit that should look like the one below: 6. 6.
Using the power supply, induce current flow through the circuit. 7. Make sure the potential difference across the metal is held constant at 0. 30 volts using the power supply settings. Verify voltage constancy with the voltmeter. 8. Record the current reading shown on the ammeter, together with the length of metal wire forming part of the circuit. 9. Repeat stages 9 – 11 ten times, increasing the distance between the two pairs of alligator clips by 10 cm each time. Doing this increases the metal wire length forming part of the circuit. 10. The resistance of the metal wire is obtained by dividing the voltage used in every measurement (0.
30 v) by the current measured in every measurement. You should get different resistance values for different lengths of wire used. This calculation assumes the wire to be ohmic. Results The method above was conducted. A table comparing the length of the metal wire, its induced current, and its consequent resistance is shown below. Note that the potential difference across the metal wire in all the measurements is 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, if assuming 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 ? 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 i?? 0. 001) meters [m] Resistance [? ] 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: 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. The ones above and below it (Y2 and Y3 respectively) are regression lines that pass through the error bars to give the greatest and smallest possible gradients for the data (i. e. greatest and smallest value of constant k). Note that the three lines of regression do not intersect in their middle as is common. That is because the individual resistance error bars plotted increase with each data point. They don’t stay constant throughout.
The regression line passing through the plotted points has equation y = 0. 6358x + 0. 0293. The coefficient of correlation of the regression line is approximately 0. 999, indicating that the regression line equation is an extremely reliable indicator of the relationship between the resistance and length of the metal wire, based on the experimental results. This relationship seems to be proportional in its nature. However, it is not directly proportional. The regression line does not pass through the origin. It is for this reason that the other possible lines of linear regression come in handy.
Through their examination, alternative y- intercepts could be searched for in hope of substantiating the direct proportionality relationship sought after. The equation dominating Y2 is y = 0. 7x + 0. 02. Concurrently, Y3 is dominated by equation y = 0. 5222x + 0. 0578. Both equations represent linear regression lines that do not pass through the origin, but above it. They intersect the Y – axis at positive values. No matter how one examines this data, the metal wire seems to contain electric resistance when it has no length, which is a debatable result. The existence of a systematic error in the experiment is a more sensible conclusion.
Conclusion The experiment yielded results that point at a proportional relationship between the length of the metal wire, and its electric resistance. As hypothesized, increasing wire length means increasing its electrical resistance. The linear line of regression passing through the plotted data points substantiated this. The two extreme linear lines of regression fitted to the error bars also reinforced the linearity. In all cases, a linear, or proportional, relationship appears to exist between the variables examined. The constant of linear proportionality, k, fluctuates from 0. 52 ? m-1 to 0. 7 ? m-1, depending on the error bars used.