A wire of length L𝐿 and cross-sectional area A𝐴 has resistance R𝑅.Part AWhat will be the resistance Rstretched𝑅stretched of the wire if it is stretched to twice its original length? Assume that the density and resistivity of the material do not change when the wire is stretched.Express your answer in terms of the wire's original resistance R𝑅 .
Question
A wire of length L𝐿 and cross-sectional area A𝐴 has resistance R𝑅.Part AWhat will be the resistance Rstretched𝑅stretched of the wire if it is stretched to twice its original length? Assume that the density and resistivity of the material do not change when the wire is stretched.Express your answer in terms of the wire's original resistance R𝑅 .
Solution
The resistance (R) of a wire is given by the formula:
R = ρL/A
where: ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
When the wire is stretched to twice its original length, the volume of the wire remains constant. Therefore, the cross-sectional area (A_stretched) of the stretched wire will be half of the original cross-sectional area (A). This is because the volume (V) of the wire, which is given by the product of the cross-sectional area and the length (V = AL), remains constant.
So, the resistance (R_stretched) of the stretched wire will be:
R_stretched = ρ(2L)/(A/2) = 4ρL/A = 4R
Therefore, the resistance of the wire when it is stretched to twice its original length is four times its original resistance.
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