Question

A quantity \( X \) is given by \[ X=\varepsilon_0 L \frac{\Delta V}{\Delta t}, \] where \( \varepsilon_0 \) is permittivity of free space, \( L \) is length, \( \Delta V \) is potential difference and \( \Delta t \) is time interval. The dimension of \( X \) is same as that of

• A. Resistance
• B. Charge
• C. Voltage
• D. Current

Hint:

Dimensional analysis: \begin{itemize} \item Convert everything into base units. \item Simplify stepwise. \end{itemize}

Answer 1

The Correct Option is D

Concept: Use dimensional analysis. Step 1: {\color{red}Dimensions of each quantity.} Permittivity: \[ [\varepsilon_0]=\frac{C^2}{N\cdot m^2} \] Potential difference: \[ [V]=\frac{J}{C} \] So: \[ \frac{\Delta V}{\Delta t} = \frac{J}{C\cdot s} \] Step 2: {\color{red}Combine terms.} \[ X=\varepsilon_0 L \frac{\Delta V}{\Delta t} \] Substitute: \[ \frac{C^2}{Nm^2}\cdot m \cdot \frac{J}{Cs} \] Using \( J=Nm \): \[ \frac{C^2}{Nm^2}\cdot m \cdot \frac{Nm}{Cs} =\frac{C}{s} \] Step 3: {\color{red}Final dimension.} \[ \frac{C}{s} = \text{Current} \]