Question

If negligibly small current is passed through a wire of length 15m and resistance of $5\Omega $, having uniform cross section of $6\times {{10}^{-7}}{{m}^{2}}$, then the coefficient of resistivity of material is,
$\begin{align}
  & \left( 1 \right)1\times {{10}^{-7}}\Omega m \\
 & \left( 2 \right)2\times {{10}^{-7}}\Omega m \\
 & \left( 3 \right)3\times {{10}^{-7}}\Omega m \\
 & \left( 4 \right)4\times {{10}^{-7}}\Omega m \\
\end{align}$

Answer 1

Hint: The resistance of a material is directly proportional to the length of the material and inversely proportional to the area of cross section of the material and the constant of proportionality is the coefficient of resistivity. By applying this concept, rearrange the equation for coefficient of resistivity. Thus on substituting the value of resistance, length of the material and cross section area of the material we will get the value of coefficient of resistivity.
Formula used:
$R=\dfrac{\rho l}{A}$
where, R is the resistance
$\rho $ is the coefficient of resistivity
l is the length of the material
A is the area of the cross section.

Complete step by step solution:
Let the coefficient of resistivity be $\rho $.
Then $R=\dfrac{\rho l}{A}$…………….(1)
Rearranging the equation (1) we get,
$\rho =\dfrac{RA}{l}$ ……………….(2)
Given that,
$R=5\Omega $
$A=6\times {{10}^{-7}}{{m}^{2}}$
l=15m
By substituting these values in equation (2) we get,
$\rho =\dfrac{5\times 6\times {{10}^{-7}}}{15}$
$\Rightarrow \rho =2\times {{10}^{-7}}\Omega m$

Therefore option (2) is correct.

Note:
The opposing while a current is flowing through a conductor is called resistance. Whereas the resistance per unit length for a unit cross section is called resistivity of a material. The SI unit of resistance is ohm and the unit of resistivity is ohm meter. With the increase in temperature the resistivity also increases. The insulators have a higher value of resistivity than that of conductors. Since the resistance of a material is directly proportional to the length of the material and inversely proportional to the area of cross section of the material and the constant of proportionality is the coefficient of resistivity. The resistance of material increases as the length of the wire increases and decreases with increase in cross sectional area and in case of resistivity its vice versa.