Question

Find the resistance of \[1000\;{\text{meters}}\] of copper wire \[25\;{\text{sq}} \cdot {\text{mm}}\] in cross section. The resistance of copper is \[1/58\;{\text{ohm}}\] per meter length and \[1\;{\text{sq}} \cdot {\text{mm}}\] cross section.

Answer 1

Hint:In this question use the concept of the resistivity, that is it is the material property, it does not depend on the resistance, length, and the cross-section area of the wire. It depends on the material, that is it will be constant for the copper wire.

Complete step by step answer:
In the question, it is given that the copper wire has the length of \[1000\;{\text{meters}}\]and the area of cross section is \[25\;{\text{sq}} \cdot {\text{mm}}\]. And we are also given that the resistance of copper wire is \[1/5{\text{8}}\;{\text{ohm}}\] per meter length and the area of cross section is \[{\text{1}}\;{\text{sq}} \cdot {\text{mm}}\].

For calculating the resistance of the copper wire, we use the resistivity formula,
\[\rho = R\dfrac{A}{l}\]
Here, \[\rho \] is the resistivity of the material in ohm meter, and \[l\] is the length of copper wire, and \[A\] is the area of the cross section of the copper wire.

We substitute the values for the resistivity of the copper if the resistance of copper wire is \[1/5{\text{8}}\;{\text{ohm}}\] per meter or $1000\;{\text{mm}}$ length and the area of cross section is \[{\text{1}}\;{\text{sq}} \cdot {\text{mm}}\].
\[ \Rightarrow \rho = \left( {\dfrac{1}{{58}}} \right)\dfrac{{\left( 1 \right)}}{{\left( {1000} \right)}}\]
After simplification we get,
$ \Rightarrow \rho = \dfrac{1}{{58000}}\;\Omega \cdot {\text{mm}}$

Now, we calculate the resistance for the copper wire has the length of \[1000\;{\text{meters}}\]and the area of cross section is \[25\;{\text{sq}} \cdot {\text{mm}}\].
For calculating the resistance of the copper wire, we use the resistivity formula,
\[\rho = R\dfrac{A}{l}\]
Now we rearrange the above formula
$ \Rightarrow R = \dfrac{{\rho l}}{A}$
Substitute the values in the above equation
\[R = \left( {\dfrac{1}{{58000}}} \right)\left( {\dfrac{{1000 \times {{10}^3}}}{{25}}} \right)\]

After simplification we get,
\[\therefore R = \dfrac{{20}}{{29}}\;\Omega \]
Therefore, the resistance of copper wire is \[\dfrac{{20}}{{29}}\Omega \] and the length of wire is \[1000\;{\text{meters}}\]and \[2{\text{5}}\;{\text{sq}} \cdot {\text{mm}}\] is the cross section.

Note:We know that the resistivity of the material depends on the property of the material, temperature, pressure, and the microstructure of the material. It does not depend on the resistance, length, and the cross-section area of the wire.