Question

A cylindrical wire has radius R and length L. If R and L both are doubled, the resistance and resistivity of the wire changes in what proportions?

Answer 1

Dear Student,
On changing the dimension of the wire resistivity does not change as we know that the resistivity depends upon the material of the wire from which it is made.
We know that the resistance of a wire = R
$$\begin{gathered} Resis\tan ce=R=\frac{\rho l}{A} \\ Here\rho =resistivity \\ l=lengthofthewire \\ A=areaofcrosssectionofthewire. \\ A=\pi r^2 \\ herer=radiusofthewire. \end{gathered}$$
Now since the volume of the wire remains the same when its length and radius is doubled
$$\begin{gathered} R^{\text{'}}=\frac{\rho l\text{'}}{A\text{'}}=\frac{\rho 2l}{\pi (2r{)}^2} \\ R\text{'}=\frac{2\rho l}{4\pi r^2}=\frac{R}{2} \end{gathered}$$
Thus the resistance will become half.
Regard.