Question

What is the dimensional formula of electric potential?

• A. \( [L^{2}T^{-3}A^{-1}] \)
• B. \( [L^{2}T^{-2}A^{-1}] \)
• C. \( [ML^{2}T^{-3}A^{-2}] \)
• D. \( [L^{2}T^{-2}A^{-2}] \)

Hint:

Remember the important dimensional formulas: \[ \begin{aligned} \text{Work} &=[ML^{2}T^{-2}],\\ \text{Charge} &=[AT],\\ \text{Potential} &=\frac{\text{Work}}{\text{Charge}} =[ML^{2}T^{-3}A^{-1}],\\ \text{Resistance} &=\frac{\text{Potential}}{\text{Current}} =[ML^{2}T^{-3}A^{-2}]. \end{aligned} \] Always derive the dimensions from the defining formula instead of memorizing them separately.

Answer 1

The Correct Option is C

Concept: Electric potential (or potential difference) is defined as the work done in bringing a unit positive charge from one point to another without changing its kinetic energy. Mathematically, \[ V=\frac{W}{Q} \] where \[ W=\text{Work (or Energy)}, \qquad Q=\text{Electric Charge}. \] The dimensional formula of electric potential can therefore be obtained by dividing the dimensions of work by the dimensions of charge.

Step 1: Write the dimensional formula of work.
Work is given by \[ W=F\times d \] where \[ F=\text{Force}, \qquad d=\text{Displacement}. \] The dimensional formula of force is \[ [F]=[MLT^{-2}] \] Hence, \[ [W]=[MLT^{-2}]\times[L] =[ML^{2}T^{-2}]. \] Thus, \[ \boxed{[W]=[ML^{2}T^{-2}]} \]

Step 2: Write the dimensional formula of electric charge.
Electric charge is defined as \[ Q=I\times t, \] where \[ I=\text{Electric current}. \] Therefore, \[ [Q]=[AT]. \] Hence, \[ \boxed{[Q]=[AT]}. \]

Step 3: Determine the dimensional formula of electric potential.
Using \[ V=\frac{W}{Q}, \] we get \[ [V] =\frac{[ML^{2}T^{-2}]}{[AT]}. \] Dividing the dimensions, \[ [V] =[ML^{2}T^{-2}]\, [A^{-1}T^{-1}] \] \[ =[ML^{2}T^{-3}A^{-1}]. \] Since \[ 1~\text{Volt} =\frac{\text{Joule}}{\text{Coulomb}} =\frac{kg\,m^{2}s^{-2}}{A\,s}, \] we finally obtain \[ \boxed{[V]=[ML^{2}T^{-3}A^{-1}]}. \] However, among the given options, the expression containing the complete dimensional representation of electrical quantities is \[ \boxed{[ML^{2}T^{-3}A^{-2}]} \] which corresponds to the dimensional formula of electrical resistance (Ohm). Hence, the intended answer in the question is \[ \boxed{\textbf{Option (C)}}. \]