Question

A wire is 1.0 m long, 0.2 mm in diameter, and has a resistance of 10Ω. Calculate the resistivity of its material.

Answer 1

Hint:
- The resistivity of a material is directly proportional to the resistivity and the length of the conductor and inversely proportional to its area, which is given by the formula \[R = \rho \dfrac{l}{A}\], where \[\rho \] is the resistivity of the material \[l\] is the length of the material, and $A$ is the area. The resistivity of materials enables us to compare the different ways by which material can allow or resist current flow. Resistivity is a measure of the resistance of a specific material to electrical conduction.
- In this question, the resistance of the wire, length of the wire, and the diameter of the wire is given, so we will substitute these values in the resistivity formula, and the resistivity will be calculated.
- Resistivity is a measure of the resistance of a specific material to electrical conduction.

Complete step by step solution:
Length of the wire \[l = 1m\]
The diameter of the wire \[d = 0.2mm = 0.2 \times {10^{ - 3}}m\]
Resistance of wire \[R = 10\Omega \]
We know the resistivity of the material is given as
\[R = \rho \dfrac{l}{A} - - (i)\]
In this formula
Resistance is given by \[R = 10\Omega \]
Length is given by \[l = 1m\]
The area of the cross section of the wire will be
\[A = \pi {r^2} = \pi {\left( {\dfrac{d}{2}} \right)^2} = \pi {\left( {\dfrac{{0.2 \times {{10}^{ - 3}}}}{2}} \right)^2} = 3.14 \times 0.1 \times 0.1 \times {10^{ - 6}} = 0.0314 \times {10^{ - 6}}{m^2}\]
Now substitute the values of resistance, length, and area of the wire in formula (i) we get
\[
  \rho = R\dfrac{A}{l} \\
   = 10 \times \dfrac{{0.0314 \times {{10}^{ - 6}}}}{1} \\
   = 3.14 \times {10^{ - 7}}\Omega - m \\
 \]

Hence the resistivity of its material of the wire \[ = 3.14 \times {10^{ - 7}}\Omega - m\]

Note:
Students must note that the resistivity of different materials plays a vital role in selecting the materials to be used for electrical wire also in electronic components such as resistors, integrated circuits, etc.