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.

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.


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