Question

A certain electrical conductor has a square cross-section, 2.0 mm on a side and is 12 m long. The resistance between its end is $0.072\Omega $. The resistivity of its material is equal to: A. $2.4 \times {10^{ - 6}}\Omega m$
B. $1.2 \times {10^{ - 5}}\Omega m$
C. $1.2 \times {10^{ - 8}}\Omega m$
D. $2.4 \times {10^{ - 8}}\Omega m$

Answer 1

Hint:The resistance offered by a conductor is dependent on the characteristic property of the conductor called resistivity, length and area of cross-section.
Resistance –
$R = \rho \dfrac{l}{A}$
Resistivity, $\rho = \dfrac{{RA}}{l}$
where A and l are an area of cross-section and length respectively.

Complete step-by-step answer:
The given conductor has a square cross-section of side 2 mm.
Hence, the area of cross-section, $A = {a^2}$
$
  Given, \\
  a = 2.0mm \\
  A = {\left( {2.0} \right)^2} = 4m{m^2} \\
  A = 4 \times {10^{ - 6}}{m^2} \\
 $
Given, $l = 12m\& R = 0.072\Omega $
Substituting in the formula, we get –
$
  \rho = \dfrac{{RA}}{l} \\
  \rho = \dfrac{{0.072 \times 4 \times {{10}^{ - 6}}}}{{12}} \\
  Solving, \\
  \rho = \dfrac{{{{0.072}}0.024 \times {4} \times {{10}^{ - 6}}}}{{{{12}}{3}}} \\
  \rho = 0.024 \times {10^{ - 6}}\Omega - m \\
  rearranging \\
  \rho = 2.4 \times {10^{ - 8}}\Omega - m \\
 $
Hence, the correct option is Option D

Note: Due to the formula, the students might sometimes get a wrong notion that resistivity can be calculated by resistance.
The resistivity is a characteristic property of the material which solely depends on the temperature. The resistivity is independent of the cross-sectional area while the resistance is dependent on the dimensions of the conductor.
The resistances, commercially used are of two types – Wirebound resistors and Carbon resistors
Wirebound resistors are made by winding metallic wires of alloys such as maganin, constantan, nichrome or similar. Mainly, their resistivities should not increase dramatically with the temperature and remain constant over a range of temperatures.
Carbon resistors are in the higher range and made from carbon. They are compact, small size and inexpensive making them best suited for usage in electronic circuits as opposed to the wirebound resistors. The values of the resistances of these carbon resistors are given by a color code along with a metallic colour at the end indicating the tolerance band.