Question

Find the resistance of 1000M 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 \[1\;{\text{sq}} {\text{mm}}\] 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 \[25\;{\text{sq}} {\text{mm}}\]. 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 \[{\text{1}}\;{\text{sq}} {\text{mm}}\].
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 \[{\text{1}}\;{\text{sq}} {\text{mm}}\].
\[ \Rightarrow \rho = \left( {\dfrac{1}{{58}}} \right)\dfrac{{\left( 1 \right)}}{{\left( {1000} \right)}}\]
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 \[25\;{\text{sq}} {\text{mm}}\].
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 \[2{\text{5}}\;{\text{sq}} {\text{mm}}\] is the cross section.

Note: From the above calculation we can conclude that the resistance of the wire depends on the length and the cross-sectional area of the wire that is the resistance is directly proportional to the length and inversely proportional to the area of cross-section of the wire.