Question

When the resistance of copper wire is $0.1\;\Omega$ and the radius is $1\;mm$, then the length of the wire is (specific resistance of copper is $3.14 \times 10^{-8}\;\Omega m$):
A.10cm
B.10m
C.100m
D.100cm

Answer 1

Try and think on the lines of the factors that the resistance of the wire depends upon. In other words, we know that the resistance of a wire is directly proportional to its length and inversely proportional to its area. To make it an equality we remove the proportionality of the relation by introducing the specific resistance of the wire since it remains constant throughout the wire. Using this relation that you obtain, plug in the values and find the length of the wire.

Formula Used:
Resistance of a wire: $R = \rho\dfrac{l}{A}$, where $\rho$ is the resistivity or specific resistance of the wire, l is the length of the wire, and A is the cross-sectional area of the wire.

Complete answer:
We know that resistance is the opposition that a wire employs against the flow of current through it. Now, the resistance of a wire depends on the resistance of the material that the wire is made of. This characteristic resistance of the material of the wire is called resistivity or specific resistance and remains constant throughout the bulk of the wire.
In addition to the material of the wire, its magnitude of resistance also depends on the length and area of cross-section of the wire. More length implies more resistance as the electrons flowing through the wire face more opportunities to collide with the atoms (ions) of the wire. However, more area is indicative of less resistance to current as the electrons flowing through are much more spaced out and are less likely to undergo as many collisions as they did with a lesser area.
From the above discussion we can deduce a relationship between resistance R, length l and area A of the wire as: $R \propto \dfrac{l}{A} \Rightarrow R= \rho\dfrac{l}{A}$, where $\rho$ is the specific resistance of the material of the wire that is introduced as a constant of proportionality since it remains the same for any wire made of this specific material.
In the context of our question, we can apply the above relations and the given values to obtain the length of the wire.
We have $R = \rho\dfrac{l}{A} \Rightarrow l = \dfrac{RA}{\rho}$
Given that $R = 0.1\;\Omega$, $\rho = 3.14 \times 10^{-8}\;\Omega m$
$r=1\;mm = 10^{-3}\;m \Rightarrow A = \pi r^2 = 3.14 \times (10^{-3})^2 = 3.14 \times 10^{-6}\;m^2$
Substituting these values back in the relation:
$l = \dfrac{0.1 \times 3.14 \times 10^{-6}}{3.14 \times 10^{-8}} = 0.1 \times 10^2 = 10\;m$

Therefore, the correct choice would be B. 10m.

Note:
-Do not get confused between resistance and resistivity.
-Resistance is a measure of the opposition that an object offers to a flow of current, whereas resistivity is a characteristic property of the material making up the object which resists the flow of current through it.
-Resistance of a wire is dependent on the length, cross-sectional area and temperature of the wire, whereas its resistivity is dependent only on the material and temperature of the wire. Thus the resistivity of a wire remains the same with a change in the length or area of the wire but changes with variations in temperature.