
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
162.6k+ views
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.
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.
Recently Updated Pages
How To Find Mean Deviation For Ungrouped Data

Difference Between Molecule and Compound: JEE Main 2024

Ammonium Hydroxide Formula - Chemical, Molecular Formula and Uses

Difference Between Area and Surface Area: JEE Main 2024

Difference Between Work and Power: JEE Main 2024

Difference Between Acetic Acid and Glacial Acetic Acid: JEE Main 2024

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Charging and Discharging of Capacitor

Wheatstone Bridge for JEE Main Physics 2025

Formula for number of images formed by two plane mirrors class 12 physics JEE_Main

In which of the following forms the energy is stored class 12 physics JEE_Main
