Question

A wire of length 1m and radius 0.1mm has a resistance 100Ω. Find the resistivity of the material.

Answer 1

Hint: The resistivity of a material is given as resistance of a material per unit length and for unit cross sectional area. Here we have given the radius of the wire, its length and the resistance and we have to find resistivity. To calculate resistivity we will first calculate the cross section area of the wire.
Formula used:
\[\rho =R\dfrac{A}{l}\]

Complete answer:
Resistivity is the property of a material due to which it has a resistance which opposes the flow of current and it is given as the resistance of the material per unit length and for unit cross sectional area. Hence resistivity can be given as
\[\rho =R\dfrac{A}{l}\]
Where R is the resistance of the material, l is the length of the material and A is the cross-sectional area of the material.
Here we have to calculate the resistivity of the wire, and the cross sectional area of the wire is given as
\[A=\pi {{r}^{2}}\]
Where r is the radius of the wire.
Hence the resistivity of the wire is given as
\[\rho =\dfrac{\pi {{r}^{2}}R}{l}\]
We have given \[R=100\Omega ,\text{ }l=1m\text{ and }r=0.1mm=0.1\times {{10}^{-3}}m\]
Substituting these values in the above equation
\[\begin{align}
  & \rho =\dfrac{(3.14){{(0.1\times {{10}^{-3}})}^{2}}(100)}{1} \\
 & \rho =3.14\times {{10}^{-6}}\Omega m \\
\end{align}\]
Hence the resistivity of the wire is \[3.14\times {{10}^{-6}}\Omega m\]

Note:
Many people get confused in resistivity and resistance but they are two different quantities, we already discuss what resistivity is whereas resistance is the measure of voltage per unit area. And we can also see that the unit of resistance is ohm whereas the unit of resistivity is ohm-m. Resistivity depends on material and the temperature not on the length or cross sectional area of the material.