Question

Equal potentials are applied on an iron and a copper wire of same length. In order to have the same current flow in the two wires, the ratio r (iron) /r (copper) of their radii must be: [Give that: specific resistance of iron is \[1.0 \times {10^{ - 7}}\Omega m\] and specific resistance of copper is \[1.7 \times {10^{ - 8}}\Omega m\] ]
A. about 1.2
B. about 2.4
C. about 3.6
D. about 4.8

Answer 1

Hint:We know the relationship between the resistance, the resistivity and the radius of a conductor. For a given potential difference, the value of resistance is constant for a constant current. Therefore, we compare the radius of given iron and copper wires for the given situation and find the ratio of their radii.

Formula used:
Potential difference, $V = RI$
where $R$ is the resistance and $I$ is the current.
Resistance, $R = \dfrac{{\rho l}}{A}$
where $\rho $ is the resistivity of the conductor (here, wire), $l$ is the length of the conductor and $A$ is the area of the cross section of the wire.
Area of cross section, $A = \pi {r^2}$
where $r$ is the radius of the wire.

Complete step by step solution:
We know that, according to Ohm’s law, $V = RI$. This implies that the value of resistance is constant for a given potential difference and a value of current. Therefore, to find the ratio of radii of iron and copper wires when equal potentials are applied to them such that the same current flows through both the wires, we will find the expression of resistance for both the wires and compare them.

Now, we know that $R = \dfrac{{\rho l}}{A}$ where $A = \pi {r^2}$ .
This gives,
$R = \dfrac{{\rho l}}{{\pi {r^2}}}$
This implies that,
${r^2} = \dfrac{{\rho l}}{{\pi R}}$
Now, for a given value of length of the conductor (here, wire) and for the given value of resistance of the wire, ${r^2} \propto \rho $. That is,
$r \propto \sqrt \rho $

Comparing the radius of given iron $\left( {Fe} \right)$ and copper \[\left( {Cu} \right)\] wire, we get,
$\dfrac{{{r_{Fe}}}}{{{r_{Cu}}}} = \sqrt {\dfrac{{{\rho _{Fe}}}}{{{\rho _{Cu}}}}} $ ...(1)
Given: ${\rho _{Fe}} = 1.0 \times {10^{ - 7}}\Omega m$ and ${\rho _{Cu}} = 1.7 \times {10^{ - 8}}\Omega m$
Substituting these values in equation (1) we get,
$\dfrac{{{r_{Fe}}}}{{{r_{Cu}}}} = \sqrt {\dfrac{{1.0 \times {{10}^{ - 7}}}}{{1.7 \times {{10}^{ - 8}}}}} $
Thus, we get, $\dfrac{{{r_{Fe}}}}{{{r_{Cu}}}} = 2.4$

Hence, option B is the correct answer.

Note: In this question the two expressions of resistance obtained for iron and copper wires rather than establishing a proportionality between the radius of the wire and the resistivity of the material.