Question

A one-meter rod of 2 cm diameter is drawn until its resistance is 100 times the initial resistance. Its length afterward is?

Answer 1

For its initial state, write how resistance depends on other characteristics:

$$R=\frac{\rho L}{A}=\frac{4\rho L}{\pi D^2}.$$

For the drawn state:

$$R_d=100R=\frac{4\rho L_d}{\pi d^2}.$$

Since we know the initial volume of the material, find the diameter of the thin wire:

$$V=L\frac{\pi D^2}{4}=L_d\frac{\pi d^2}{4},\\\space\\ \pi d^2=\frac{L\pi D^2}{L_d}.$$

Substitute this into the equation for R_d:

$$100R=\frac{4\rho L_d^2}{L\pi D^2},\\\space\\ 100·\frac{4\rho L}{\pi D^2}=\frac{4\rho L_d^2}{L\pi D^2},\\\space\\ 100·L=\frac{L_d^2}{L},\\\space\\ L_d=10L=10\text{ m}.$$