Question

What length of a copper wire of cross-sectional area $0.01{\text{ }}m{m^2}$will be needed to prepare a resistance of $1{\text{ }}k\Omega $? Resistivity of copper$ = 1.7 \times {10^{ - 8}}{\text{ }}\Omega m$.

Answer 1

Hint: First of all, we have to find the relation between the resistance and the resistivity of the given material in the given question. Then, by considering the unknown value as a variable and substituting other values in SI units in the relation we will find the answer.

Complete step by step answer:
Resistance of a wire is defined as the obstruction or blockage in the flow of electric current or charges in a conductor. Specific resistance of a wire is defined as the resistance of the wire of unit cross-sectional area and unit length.The relation between specific resistance and resistance is as follows,
$R = \rho \dfrac{l}{A}$
where $R = $ resistance of the wire, $\rho = $ specific resistance of the wire, $l = $ length of the given wire and $A = $ area of cross-section of the given wire.

The values assigned in the given question are-
$R = 1{\text{ }}k\Omega {\text{ }} = 1000{\text{ }}\Omega $
$\Rightarrow \rho = 1.7 \times {10^{ - 8}}{\text{ }}\Omega m$
$\Rightarrow A = $$0.01{\text{ }}m{m^2} = {10^{ - 8}}{\text{ }}{m^2}$
Let the length of the wire be $l$.
Substituting all the values in the given equation we get,
$1000 = 1.7 \times {10^{ - 8}} \times \dfrac{l}{{{{10}^{ - 8}}}}$
Simplifying the equation we get,
$\therefore l = 0.58 \times {10^3}{\text{ }}m \approx 0.6{\text{ }}km$

So, $0.6{\text{ }}km$ length of a copper wire of cross-sectional area $0.01{\text{ }}m{m^2}$ will be needed to prepare a resistance of $1{\text{ }}k\Omega $.

Note: Specific resistance is a materialistic property. It does not alter the similar material. Different material corresponds to different specific resistance whereas resistance is not a materialistic property. Hence when we change the other factors of a material then, its resistance changes but not resistivity. Conversion of all units in the form of SI units is mandatory for solving this type of question as different values are assigned with different units.