A quantity X is given by e_0*L(Delta V)/(Delta T) where e_0 is the permittivity of free space, L is a length, Delta V is a potential difference, and Delta T is a time interval. The dimensional formula for X is the same as that of: A. Resistance B. Electric charge C. Voltage D. Electric current

To determine the dimensional formula of the quantity (X) given by:

[ X = \frac{\varepsilon_0 L (\Delta V)}{\Delta T} ]

where:

• (\varepsilon_0) is the permittivity of free space,
• (L) is a length,
• (\Delta V) is a potential difference (voltage), and
• (\Delta T) is a time interval.

We need to compare its dimensional formula with those of the given options: Resistance, Electric charge, Voltage, and Electric current.

Step-by-Step

1. Given Formula: [ X = \frac{\varepsilon_0 L (\Delta V)}{\Delta T} ]

2. Capacitance Relation: The capacitance $C$ is related to these quantities: [ C = \varepsilon_0 \frac{A}{d} ] where $A$ is the area (with dimensional formula $[L^2]$) and $d$ is the separation (with dimensional formula $[L]$). Thus: [ C \sim \frac{[L^2] \varepsilon_0}{[L]} = [L \varepsilon_0] ]

3. Simplifying (X): Given $X = \frac{\varepsilon_0 L (\Delta V)}{\Delta T}$: [ X \sim C \cdot \frac{\Delta V}{\Delta T} ]

4. Understanding Charge (Q): The charge $Q$ on a capacitor is given by: [ Q = C \Delta V ] Hence, [ X \sim \frac{Q}{\Delta T} ]

5. Identifying the Dimensional Formula: The quantity $\frac{Q}{\Delta T}$ represents the electric current $I$, where: [ I = \frac{Q}{t} ]

Therefore, (X) has the same dimensional formula as the electric current (I).

Conclusion

[ \boxed{D} ] The dimensional formula for (X) matches that of electric current.