Question

Find the ratio of the diameter of iron to copper wire, if they have the same resistance per unit length (as they might in household wiring).

Answer 1

Answer:

The ratio of the diameter of iron to Cu is;

$\frac{d{Fe} }{ d{Cu} } =\sqrt{\frac{p_{Fe} }{ p_{Cu} }}$

Explanation:

R=(ρL)/A

• R is resistance,
• L is length,
• A is area,
• ρ is resistivity
• d is diameter

from the question the two materials have the same resistance per unit length.

$\frac{R}{L}= \frac{p}{A}$

$\frac{R}{L}$   for iron = $\frac{R}{L}$  for copper

This means we can equate ρ/A for both materials.

$\frac{p_{Fe} }{A_{Fe} } =\frac{p_{Cu} }{A_{Cu} }$

re-arranging the equation we have,

$\frac{A_{Fe}}{A_{Cu} } =\frac{p_{Fe} }{ p_{Cu} }$

$A=\pi \frac{d^{2} }{4}$

$\frac{A_{Fe}}{A_{Cu} } =\frac{d^{2}{Fe} }{ d^{2}{Cu} }$

$\frac{d^{2}{Fe} }{ d^{2}{Cu} } =\frac{p_{Fe} }{ p_{Cu} }$

$\frac{d{Fe} }{ d{Cu} } =\sqrt{\frac{p_{Fe} }{ p_{Cu} }}$