Question

The physical quantity having the dimensions $\left[M^{-1}{L}^{-3}{T}^3{A}^2\right]$ is

• A. resistance
• B. resistivity
• C. electrical conductivity
• D. electromotive force

Answer 1

The correct option is C

electrical conductivity

The explanation for the correct option:

In the case of option C,

1. Electrical conductivity, $\sigma =\frac{1}{\rho }$where, $\rho$ is resistivity
2. The electrical conductivity is given by-
3. $\sigma =\frac{n{e}^2\tau }{m}$ where $n$is the number of electron per unit volume
4. The dimension of mass is $\left[M\right]$
5. The dimension of $n$ is $\left[L^{-3}\right]$
6. The dimension of the charge $e=\left[AT\right]$
7. The dimension of $\tau =\left[T\right]$
8. Thus, the dimensional formula of electrical conductivity is$=\left[M^{-1}{L}^{-3}{A}^2{T}^3\right]$.

The explanation for the incorrect options:

In the case of option A,

1. The dimensional formula of resistance is given by-
2. $\begin{array}{rcl}Resis\tan ce& =& resistivity\times \frac{length}{area}\\ & =& \left[M{L}^2{A}^{-1}{T}^{-3}\right]\end{array}$

In the case of option B,

1. Resistivity is defined as $\rho =\frac{1}{\sigma }$
2. The dimensional formula of resistivity.
3. $\begin{array}{l}\left[\rho \right]=\frac{\left[M\right]}{\left[L^{-3}\right]{\left[AT\right]}^2\left[T\right]}=\left[M{L}^3{A}^{-2}{T}^{-3}\right]\end{array}$

In the case of option D,

1. The dimension formula of electromotive force is-
2. $\begin{array}{rcl}Electromotiveforce& =& resis\tan ce\times current\\ & =& \left[M{L}^2{A}^{-1}{T}^{-3}\right]\end{array}$

Hence option C is the correct answer.