Question

The physical quantity having the dimensions \[[{M^{ - 1}}{L^{ - 3}}{T^3}{A^2}]\]
A. Resistance
B. Resistivity
C. Electrical Conductivity
D. Electromotive force

Answer 1

Hint: In order to solve this question, we should know that every physical quantity has a dimensional formula that is determined by its mathematical expression, and here we will first determine the Dimensional formula of the given option and then matches it with the given dimensions.

Complete step by step solution:
Let us start with the dimensions of Electromotive force which is defined as the work done per unit charge so, its dimension is $[M{L^2}{T^{ - 3}}{A^{ - 1}}]$. and Dimensions of Resistance which is defined as the electromotive force per unit current so, its dimensions is $[M{L^2}{T^{ - 3}}{A^{ - 2}}]$.

Now As we know, the resistivity of a material is calculated as,
$\rho = \dfrac{m}{{n{e^2}t}}$
where, m is the mass having dimensions of $[M]$ ,n is the number of electrons per unit volume having dimensions of $[{L^{ - 3}}]$, e is the electronic charge having dimensions of $[AT]$ and t denotes for time having dimensions of $[T]$.

Now, putting all dimension into resistivity formula we get dimensions of Resistivity as,
$\rho = \dfrac{{[M]}}{{[{L^{ - 3}}{A^2}{T^3}]}} \\
\Rightarrow \rho = [M{L^3}{A^{ - 2}}{T^{ - 3}}] \\ $
Now, since we know that electrical conductivity is related to resistivity as,
$\sigma = \dfrac{1}{\rho }$
Hence, the dimensional formula for electrical conductivity will be the inverse of the dimensional formula for resistivity hence, we get;
$\sigma = \dfrac{1}{{[M{L^3}{A^{ - 2}}{T^{ - 3}}}}] \\
\Rightarrow \sigma = [{M^{ - 1}}{L^{ - 3}}{A^2}{T^3}] \\ $
And given dimensional formula is \[[{M^{ - 1}}{L^{ - 3}}{T^3}{A^2}]\] and it matches with the dimensional formula of electrical conductivity. The dimensional formula of resistance is $[M{L^2}{A^{ - 2}}{T^{ - 3}}]$ and the dimensional formula of electromotive force is $[M{L^2}{A^{ - 1}}{T^{ - 3}}]$, so only the option of electrical conductivity matches the given dimensional formula.

Hence, the correct option is C.

Note: It should be remembered that, $[A]$ denotes the dimensional formula of electric current and there are mainly seven fundamental physical quantities that can be used to determine any other physical quantity dimensions and these seven fundamental physical quantities are Mass, Length, Time, Current, Amount of Substance, Luminous Intensity, and Temperature.