Question

A wire of length 2 m and resistance 8 \(\Omega\) is stretched to double its original length, keeping the volume constant. What is the new resistance of the wire?

• A. \( 16 \, \Omega \)
• B. \( 32 \, \Omega \)
• C. \( 8 \, \Omega \)
• D. \( 4 \, \Omega \)

Hint:

When a wire is stretched with constant volume, the resistance changes by the square of the factor by which the length changes. If the length is doubled, the resistance increases by \( 2^2 = 4 \) times.

Answer 1

The Correct Option is B

The resistance \( R \) of a wire is calculated using the formula: \[R = \rho \frac{l}{A}\] Where: - \( \rho \) represents the resistivity of the material. - \( l \) denotes the length of the wire. - \( A \) is the cross-sectional area of the wire. Initial conditions of the wire: - Length \( l = 2 \, \text{m} \) - Resistance \( R = 8 \, \Omega \) Upon stretching the wire to twice its original length, the new length \( l' \) becomes: \[l' = 2 \times 2 = 4 \, \text{m}\] The volume of the wire remains constant during this process. Therefore, the initial volume equals the final volume: \[V = A \cdot l = A' \cdot l'\] Substituting the known lengths: \[A \cdot 2 = A' \cdot 4\] This yields the new cross-sectional area: \[A' = \frac{A}{2}\] The new cross-sectional area is half the original area. The new resistance \( R' \) is determined by: \[R' = \rho \frac{l'}{A'}\] Substituting \( l' = 4 \, \text{m} \) and \( A' = \frac{A}{2} \): \[R' = \rho \frac{4}{\frac{A}{2}} = \rho \frac{4 \cdot 2}{A} = \rho \frac{8}{A}\] Given the original resistance \( R = \rho \frac{2}{A} = 8 \, \Omega \), we can express the new resistance in terms of the original resistance: \[R' = \rho \frac{8}{A} = 4 \cdot \rho \frac{2}{A} = 4 \cdot R\] Calculating the new resistance: \[R' = 4 \cdot 8 = 32 \, \Omega\] Alternatively, as resistance is directly proportional to \( \frac{l}{A} \), with \( l \) doubling and \( A \) halving, the resistance change is: \[R' = R \cdot \frac{l'}{l} \cdot \frac{A}{A'} = 8 \cdot \frac{4}{2} \cdot \frac{A}{\frac{A}{2}} = 8 \cdot 2 \cdot 2 = 32 \, \Omega\] The wire's new resistance is \( 32 \, \Omega \).