Question

A wire 50cm long and $1m{{m}^{2}}$in cross section carries a current of 4A when connected to a 2V battery. The resistivity of the wire is:
$\begin{align}
  & \left( A \right)2\times {{10}^{-7}}\Omega m \\
 & \left( B \right)5\times {{10}^{-7}}\Omega m \\
 & \left( C \right)4\times {{10}^{-6}}\Omega m \\
 & \left( D \right)1\times {{10}^{-6}}\Omega m \\
\end{align}$

Answer 1

Hint: Apply Ohm’s law to find the resistance. 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:
According to Ohm’s law the voltage or potential difference and the current flowing through a conductor are directly proportional. And here the proportionality constant is the resistance.
$V=IR$
where, V is the voltage or potential difference.
I is the current flowing through a conductor.
and R is the resistance.
$R=\dfrac{\rho l}{A}$
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 answer:
$V=IR$
$\Rightarrow R=\dfrac{V}{I}$
$\begin{align}
  & \Rightarrow R=\dfrac{2}{4} \\
 & \therefore R=0.5\Omega \\
\end{align}$
Here l=50cm=0.5m
$A=1m{{m}^{2}}$ $={{10}^{-6}}{{m}^{2}}$
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)
$\Rightarrow \rho =\dfrac{0.5\times {{10}^{-6}}}{0.5}$
$\therefore \rho =1\times {{10}^{-6}}\Omega m$

So, the correct answer is “Option (D)”.

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 unit of resistivity is ohm meter. With the increase in temperature the resistivity also increases. The insulators have high 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.