Question

Calculate the resistivity of the material of a wire 1 m long , 0.4 mm in diameter and having a resistance 2$\Omega $.
A) $300\Omega m$
B) $2.514 \times {10^{ - 7}}\Omega m$
C) $2 \times {10^7}\Omega m$
D) $1 \times {10^{ - 15}}\Omega m$

Answer 1

Hint: Resistivity also known as specific electrical resistance is defined as the measure of the resistance of a given size of a particular material to electrical conduction.
Electrical resistivity is the electrical resistance per unit length and per unit of cross-sectional area at a specified temperature and is given by-
$\rho = R\dfrac{A}{l}$ where,
R is the electrical resistance of a uniform specimen of the material measured in ohms
l is the length of the piece of material measured in metres, m
A is the cross-sectional area of the specimen measured in square metres, ${m^2}$
In the given question, we will first find the area of cross-section and then put all the values in formula to find resistivity of the material.

Complete step-by-step answer:
Given, $R = 2\Omega $
$l = 1m$
$d = 0.4mm = 0.0004m$
First, we will find out the area of cross-section of material given by
⇒$A = \dfrac{{\pi {d^2}}}{4}$
⇒$A = \dfrac{{\pi \times {{(0.0004)}^2}}}{4}$
⇒$A = \dfrac{{3.14 \times 0.00000016}}{4}$
⇒$A = 0.0000001256$
⇒$A = 1.256 \times {10^{ - 7}}{m^2}$
Now, we need to find the resistivity of the material.
We know, $\rho = R\dfrac{A}{l}$
Putting in the values in formula, we get
$\rho = \dfrac{{2\Omega \times 1.256 \times {{10}^{ - 7}}{m^2}}}{{1m}}$
⇒$\rho = 2.514 \times {10^{ - 7}}\Omega m$
Hence, the resistivity of the material is $2.514 \times {10^{ - 7}}\Omega m$

Option B is correct.

Note: The resistivity of a material can also be defined in terms of the magnitude of the electric field across it that gives a certain current density. It is possible to devise an electrical resistivity formula which is given by-
$\rho = \dfrac{E}{J}$ where,
ρ is the resistivity of the material.
E is the magnitude of the electric field.
J is the magnitude of the current density.