Answer to Question #287789 in Electrical Engineering for Joe

Question

A one meter of annealed copper 2.5 cm in diameter is drawn until its resistance is 100 times the initial resistance, its diameter afterward is?

Answer 1

The volume of this specimen is constant, the volume is the cross-sectional area A times length L:

$$V=A_1L_1=A_2L_2,\\\space\\ \dfrac{\pi d_1^2}{4}L_1=\dfrac{\pi d_2^2}{4}L_2→d_1^2L_1=d_2^2L_2,\\\space\\ [L_2]=L_1\dfrac{d_1^2}{d_2^2}.$$

The resistance:

$$R_2=100R_1,\\\space\\ R=\dfrac {\rho L}A:\\\space\\ 100\dfrac {\rho L_1}{A_1}=\dfrac {\rho L_2}{A_2}→100\dfrac { L_1}{A_1}=\dfrac {L_2}{A_2}→100\dfrac {L_1}{d_1^2}=\dfrac {L_2}{d^2_2},\\\space\\ d_2=\dfrac{d_1}{10}\sqrt{\dfrac {[L_2]}{L_1}},\\\space\\ d_2=\dfrac{d_1}{10}\sqrt{\dfrac {\bigg[L_1\dfrac{d_1^2}{d_2^2}\bigg]}{L_1}}=\dfrac{d_1^2}{10d_2},\\\space\\ d_2=\dfrac {d_1}{\sqrt{10}}=0.79\text{ cm}.$$

The new diameter is 7.9 mm.