The resistivity of a wire is \( \rho \), its volume is \( 3\,\mathrm{m^3} \) and its resistance is \( 3\,\Omega \). Its length is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 3/\sqrt{\rho} \)
The resistance of a wire is \( R=\rho l/A \). For a wire of length \( l \) and cross-section \( A \), the volume is \( V=Al \). So \( A=V/l \), and the resistance can be written in terms of the length and the volume.
\( \begin{aligned} R &= \frac{\rho\,l}{A} \\ &= \frac{\rho\,l^{2}}{V} \end{aligned} \)
With \( R=3\,\Omega \) and \( V=3\,\mathrm{m^{3}} \) (SI units), the square of the length is:
\( \begin{aligned} l^{2} &= \frac{RV}{\rho} \\ &= \frac{3\times3}{\rho} \\ &= \frac{9}{\rho} \end{aligned} \)
\( \Rightarrow\; l=\displaystyle\frac{3}{\sqrt{\rho}} \)