Question

The dimensional formula of electrical conductivity is
A. $$[M^{(-1)}{L}^{(-3)}{T}^3{A}^2]$$
B. $$[M{L}^3{T}^3{A}^2]$$
C. $$[M^2{T}^3{M}^{(-3)}{T}^2]$$
D. $$[M^1{L}^3{T}^{(-3)}{A}^{(-2)}]$$

Answer 1

Hint: To find electrical conductivity, first find the resistivity through resistance which could be obtained from the ohm’s law. To use the ohm’s law, first find the dimensions of voltage, which can be obtained through the electric field obtained by the coulomb’s law, and then apply the ohm’s law by using the dimensions of current.

Complete step by step answer:
The electrical conductivity of a substance is the inverse of its electrical resistivity. Resistivity can be obtained through resistance, which we can find out through voltage and current. We know the standard dimensional formula of current which is$$A^1$$. The relationship between current and resistance is given by the ohm’s law, which is written as follows:
 $$V=I\cdot R$$
Here, V is the voltage, while I and R are current and resistance respectively.
Voltage can be obtained from the formula , where E is the electric field and d is the distance. The standard unit of distance is meter. Hence the dimensional formula of d is$$L^1$$.
Electric field is obtained from the following formula:
$$E={\displaystyle \frac{F}{q}}$$
Here, F is the Coulomb’s force. Force has a unit of Newton, hence it has the dimensional formula o$$[ML{T}^{(-2)}]$$ and q is the charge which has the dimension $$q=[AT]$$ . Thus the dimensional formula of electric field is:
$$\begin{gathered} E={\displaystyle \frac{[ML{T}^{(-2)}]}{[AT]}} \\ ⟹E=[ML{T}^{(-3)}{A}^{(-1)}] \end{gathered}$$
Thus, from this, we get the dimensional formula of voltage as shown below:
 $$\begin{gathered} V=E\cdot d \\ ⟹V=[ML{T}^{(-3)}{A}^{(-1)}]\cdot [L] \\ ⟹V=[M{L}^2{T}^{(-3)}{A}^{(-1)}] \end{gathered}$$
 Now, the dimensional formula of resistance would be
$$\begin{gathered} R={\displaystyle \frac{V}{I}} \\ ⟹R={\displaystyle \frac{[M{L}^2{T}^{(-3)}{A}^{(-1)}]}{[A]}} \\ ⟹R=[M{L}^2{T}^{(-3)}{A}^{(-2)}] \end{gathered}$$
Now, we can obtain resistivity through resistance. It can be obtained from the formula below:
 $$\rho ={\displaystyle \frac{R\cdot A}{l}}$$
Here, A has the dimensions of area, while l has the dimensions of length.
Thus, the dimensions of resistivity are as follows:
$$\begin{gathered} \rho ={\displaystyle \frac{[M{L}^2{T}^{(-3)}{A}^{(-2)}][L^2]}{[L]}} \\ ⟹\rho =[M{L}^3{T}^{(-3)}{A}^{(-2)}] \end{gathered}$$
The electrical conductivity is the inverse of resistivity. Hence,
$$\sigma =[M^{(-1)}{L}^{(-3)}{T}^3{A}^2]$$

So, the correct answer is “Option A”.

Note:
A common place where a mistake can be made in such questions is that they write incorrect standard dimensions. For example, students mistake the dimension of charge as $$A^1{T}^{(-1)}$$ instead of$$A^1{T}^1$$. The potential V can also be founded from the formula E=qV, where E is the energy and q is the charge.