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
Question
A quantity X is given by ε₀L(ΔV)/(ΔT) where ε₀ is the permittivity of free space, L is a length, ΔV is a potential difference, and ΔT 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
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Answer
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.
StepbyStep

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

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] ]

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

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

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.
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