Question

If a copper wire is stretched to make it \(0.1 \%\) longer, then what is the percentage change in its resistance? (A) \(0.2 \%\) (B) \(0.3 \%\) (C) \(0.4 \%\) (D) \(0.1 \%\)

Short answer

The percentage change in resistance when a copper wire is stretched to make it 0.1% longer is approximately \( \boxed{(A)} 0.2 \% \).

Step-by-step solution

Understand the resistivity formula

The resistivity formula is given by: \(R = \rho \frac{L}{A}\), where R is the resistance, \(\rho\) is the resistivity (constant for a given material), L is the length, and A is the cross-sectional area of the wire. Since we're dealing with percentage change, we can disregard the resistivity constant. So, the formula becomes: \(\frac{R1}{R2} = \frac{L1}{L2} \cdot \frac{A2}{A1}\).

Calculate the new length and area

Given that the length of the wire increases by 0.1%, we have: \(L2 = L1 + 0.001 L1 = 1.001 L1\). When the wire is stretched, its volume remains constant. So, we can write: \(A1 L1 = A2 L2\). Rearranging for A2, we get: \(A2 = \frac{A1 L1}{L2}\).

Substitute the new values into the formula

Now, plug the new values of L2 and A2 into the formula from step 1: \(\frac{R1}{R2} = \frac{L1}{1.001 L1} \cdot \frac{\frac{A1 L1}{1.001 L1}}{A1}\). Notice that both L1 and A1 cancel out: \(\frac{R1}{R2} = \frac{1}{1.001} \cdot \frac{1}{1.001}\).

Calculate the percentage change in resistance

The percentage change in resistance can be calculated as: \(\%\:change = \frac{R2 - R1}{R1} \cdot 100\%\). Substituting our values, we get: \(\%\:change = (\frac{1}{(1/1.001)^2} - 1) \cdot 100\%\). Calculate the result: \(\%\:change \approx 0.200 \% \). The percentage change in resistance is approximately 0.2%. Answer: \( \boxed{(A)} 0.2 \% \).