Question

Calculate the resistivity of the material of a wire 1 m long , 0.4 mm in diameter and having a resistance 2Ω.
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

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 the material measured in ohms
l is the length 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 the resistivity of material.

Complete Step-by-step solution:
As mentioned the known
$R=2\Omega$
$l=1m$
$d=0.4mm=0.0004m$
First, we will calculate the area of cross-section of material which is given by
$A=\dfrac{\pi {{d}^{2}}}{4}$--------- (1)
Substitute the value of d in equation (1) we get
$A=\dfrac{\pi \times {{(0.0004)}^{2}}}{4}$
$A=\dfrac{3.14\times 0.00000016}{4}$
$A=1.256\times {{10}^{-7}}{{m}^{2}}$
Now, we need to find the resistivity of the material.
We know, $\rho =R\dfrac{A}{l}$------------- (2)
Substituting the values in equation (2) ,we get
$\rho =\dfrac{2\times 1.256\times {{10}^{-7}}}{1}$
$\rho =2.514\times {{10}^{-7}}\Omega m$

**Hence, the resistivity of the material is $2.514\times {{10}^{-7}}\Omega m$
Correct Option is B **

Note:
A material's resistivity can also be defined in terms of the magnitude of the electric field through it that gives a certain density of current. An electrical resistivity formula can be designed which is generated 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.
The resistivity of perfect conductors is 0. There is no resistance at all in perfect conductors.
Resistivity is limitless for ideal insulators. As a consequence of resistance, there are so many barriers that current does not flow at all.
Resistance tells of the conductor's resistance as a whole, while resistivity tells of a given material's resistance.