Question

What is the diameter of a 1.00-m length of tungsten wire whose resistance is 0.47ohms

d= _________m

Answer 1

Concepts and reason

The concept used to solve this problem is resistance of the conductor.

Initially, use the expression in terms of resistance, resistivity, length of the wire, and area of cross section to find the area of cross section of the tungsten wire.

Then, use the value of area to find the diameter of the tungsten wire.

Fundamentals

The expression for resistance of the conducting wire is,

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

Here, $R$ is the resistance, $\rho$ is the resistivity, $L$ is the length of the wire, and $A$ is the area of cross section.

The expression for area of cross section of the wire is,

$A = \pi {r^2}$

Here, $r$ is the radius of the wire.

Expression for the diameter is,

$d = 2r$

Here, $d$ is the diameter.

The expression for resistance of the conducting wire is,

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

Rearrange the above expression for $A$.

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

Substitute $5.6 \times {10^{ - 8}}\,\Omega \cdot {\rm{m}}$ for $\rho$, $1.00\,{\rm{m}}$ for $L$, and $0.47\,\Omega$ for $R$.

$\begin{array}{c}\\A = \frac{{\left( {5.6 \times {{10}^{ - 8}}\,\Omega \cdot {\rm{m}}} \right)\left( {1.00\,{\rm{m}}} \right)}}{{\left( {0.47\,\Omega } \right)}}\\\\ = 1.19 \times {10^{ - 7}}\,{{\rm{m}}^2}\\\end{array}$

Therefore, the area of cross section of the wire is $1.19 \times {10^{ - 7}}\,{{\rm{m}}^2}$.

The expression for area of cross section of the wire is,

$A = \pi {r^2}$

Rearrange the above expression for $r$.

$r = \sqrt {\frac{A}{\pi }}$

Substitute $1.19 \times {10^{ - 7}}\,{{\rm{m}}^2}$ for $A$ and $3.14$ for $\pi$.

$\begin{array}{c}\\r = \sqrt {\frac{{1.19 \times {{10}^{ - 7}}\,{{\rm{m}}^2}}}{{3.14}}} \\\\ = 1.9467 \times {10^{ - 4}}\,{\rm{m}}\\\\ \approx 1.95 \times {10^{ - 4}}\,{\rm{m}}\\\end{array}$

Expression for the diameter is,

$d = 2r$

Substitute $1.95 \times {10^{ - 4}}\,{\rm{m}}$ for r.

$\begin{array}{c}\\d = 2\left( {1.95 \times {{10}^{ - 4}}\,{\rm{m}}} \right)\\\\ = 3.90 \times {10^{ - 4}}\,{\rm{m}}\\\end{array}$

Therefore, the diameter of the tungsten wire is $3.90 \times {10^{ - 4}}\,{\rm{m}}$.

Ans:

The diameter of the tungsten wire is $3.90 \times {10^{ - 4}}\,{\rm{m}}$.