Question

What will be the area of cross-section of a wire if its length is 1 m, its resistance is 23\[\Omega \] and the resistivity of material of the wire is \[1.84\times {{10}^{-6}}\Omega m\]?

Answer 1

Hint: We need to understand and apply the correct relations between the given physical parameters such as the resistance, resistivity of the material and length of the wire to find the unknown area of cross-section of the given wire to solve this problem.

Complete answer:
We know from Ohm's law that the resistance for a wire is the property which is constant for any range of potential drop and the corresponding current flow through the wire. From the Ohm’s law, we know that –
\[\begin{align}
  & V\propto I \\
 & \text{and,} \\
 & V=IR \\
\end{align}\]
Here, the resistance is the constant of proportionality that balances and connects the voltage drop across the wire and the current through it.
The resistance is defined as the opposing effect experienced by the electrons in the movement under an applied external potential difference. It is dependent on the length of the wire, the area of cross-section of the wire and the nature of the wire.
From experiments, we have derived that the resistance increases with the length of the conductor and decreases with increasing area of cross-section of the wire. We can mathematically write this as –
\[\begin{align}
  & R\propto \dfrac{l}{A} \\
 & \Rightarrow R=\rho \dfrac{l}{A} \\
\end{align}\]
The constant of proportionality is the resistivity of the material. It is a characteristic property of the material that doesn’t change with any physical changes except for temperature.
The area of cross-section of the given resistor can be found from this relation as –
\[\begin{align}
  & R=\rho \dfrac{l}{A} \\
 & \Rightarrow A=\dfrac{\rho l}{R} \\
 & \Rightarrow A=\dfrac{1.84\times {{10}^{-6}}\times 1}{23} \\
 & \therefore A=8\times {{10}^{-8}}{{m}^{2}} \\
\end{align}\]
The area of cross-section of the given wire is –
\[A=8\times {{10}^{-8}}{{m}^{2}}\]

Note:
The resistance of a conducting material is dependent on the physical dimensions such as the length of the wire and it’s area of cross-section, whereas the resistivity is a material property which can change only with a change in temperature.