A Cu Wire is stretched by 0.1% of its original length. What is the percentage change in its resistance?
Answer should be clearly worked, with all steps.
Answer should be clearly worked, with all steps.
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Look up specific resistance. This is a physical constant for a material like copper. The resistance formula is according to that. Resistance increases with increasing length and increases with decreasing cross sectional area. We don't need the specific resistance, just understand the formula in this case. That is because we are only looking for the ratio (as a percentage), not the actual resistance.
R = rho(Length / area_cross_section)
Where R is resistance, length in meters, cross section in square meters, rho is the constant for specific resistance. In this case we have the change in length and change in cross section, while rho is constant..
Assume a wire is constant round cross section.
Stretch the length 0.1%. The cross section decreases by 0.1% so the volume is the same. (from Young's modulus etc)
E = (F*L1) / (A( L2-L1))
where:
E = Young's modulus, a physical constant for the material (not needed here).
F is the tensile force, A is the cross sectional area of the wire, L1 is the initial length and L2 is the stretched length after applying the tensile force F. This is just to show the cross section is inversely proportional to the stretch. No calculations needed, but you can rearrange the formula to show A in terms of stretch (L1-L2), where E, F and L1 are constant.
Diameter??
Cross section = pi(D^2)/4 or pi(r^2), but D = diameter not needed here.
Resistance increases 0.1 percent because of length increase, and increases 0.1% because of cross section decrease. Thus, the changes (as ratios) according to the specific resistance formula:
R = rho((length + 0.1%) / (area - 0.1%))
use ratio not percent
= (1 + 0.001) / (1 - 0.001)
= 1.001 / 0.999
= 1.002002...
as a percentage change = increase 0.2%.
R = rho(Length / area_cross_section)
Where R is resistance, length in meters, cross section in square meters, rho is the constant for specific resistance. In this case we have the change in length and change in cross section, while rho is constant..
Assume a wire is constant round cross section.
Stretch the length 0.1%. The cross section decreases by 0.1% so the volume is the same. (from Young's modulus etc)
E = (F*L1) / (A( L2-L1))
where:
E = Young's modulus, a physical constant for the material (not needed here).
F is the tensile force, A is the cross sectional area of the wire, L1 is the initial length and L2 is the stretched length after applying the tensile force F. This is just to show the cross section is inversely proportional to the stretch. No calculations needed, but you can rearrange the formula to show A in terms of stretch (L1-L2), where E, F and L1 are constant.
Diameter??
Cross section = pi(D^2)/4 or pi(r^2), but D = diameter not needed here.
Resistance increases 0.1 percent because of length increase, and increases 0.1% because of cross section decrease. Thus, the changes (as ratios) according to the specific resistance formula:
R = rho((length + 0.1%) / (area - 0.1%))
use ratio not percent
= (1 + 0.001) / (1 - 0.001)
= 1.001 / 0.999
= 1.002002...
as a percentage change = increase 0.2%.