Question

A metal wire of specific resistance $64 \times {10^{ - 6}}\,\Omega \,m$ and length 1.98 m has a resistance of $7\,\Omega$. Find its radius.

Answer 1

Use the formula for resistance of metal wire of length l and area of cross-section A. The area of cross-section of the wire of radius r is $\pi {r^2}$.

Formula used:

$R = \dfrac{{\rho l}}{A}$

Here, resistance of wire of length is l, cross-sectional area A and specific resistance $\rho$.

Complete step by step answer:
We know the resistance of wire of length l, cross-sectional area A and specific resistance $\rho$ is,$R = \dfrac{{\rho l}}{A}$

The cross-sectional area of the metal wire is, $A = \pi {r^2}$, where, r is the radius of wire. Therefore, the above equation becomes,
$R = \dfrac{{\rho l}}{{\pi {r^2}}}$

Rearrange the above equation for r as follows,
$r = \sqrt {\dfrac{{\rho l}}{{\pi R}}}$

We have given the specific resistance of wire is $64 \times {10^{ - 6}}\,\Omega \,m$, the length is 1.98 m and resistance is $7\,\Omega$. Therefore, the above equation becomes,
$r = \sqrt {\dfrac{{\left( {64 \times {{10}^{ - 6}}\,\Omega \,m} \right)\left( {1.98\,m} \right)}}{{3.14 \times 7\,\Omega }}}$
$\Rightarrow r = \sqrt {\dfrac{{1.2672 \times {{10}^{ - 4}}\,{m^2}}}{{21.98}}}$
$\Rightarrow r = \sqrt {5.765 \times {{10}^{ - 6}}\,{m^2}}$
$\Rightarrow r = 2.40 \times {10^{ - 3}}\,m$ or $r = 0.24\,cm$

Therefore, the radius of metal wire is 0.24 cm.

Additional information: The resistance of metal wire is, $R = \dfrac{{\rho l}}{A}$. The resistance of metal depends on its specific resistance as it is proportional to the resistance. The specific resistance varies metal to metal. The metal wire with low specific resistance is considered suitable wire for electric home appliances. From the above relation, it is clear that the resistance is also proportional to the length of wire, therefore, as the length of wire increases, the resistance also increases. Also, the resistance is inversely proportional to the square of radius of the metal wire.

Note: To solve such type of questions, students can use, $R = \dfrac{{\rho l}}{{\pi {r^2}}}$ directly without the terms cross-sectional area if the radius of the metal wire is given. If the resistance of the second wire of the same specific resistance but having different radius is asked to calculate, then students can take the ratio of resistance of two metal wires.