Step 1 : Understanding the Question:
The topic of this question is Dimensional Analysis within the context of Electrostatics. Every physical quantity in physics can be expressed in terms of fundamental dimensions: Mass [M], Length [L], Time [T], and Electric Current [A]. Capacitance is the ability of a system to store an electric charge per unit of electric potential. To find its dimensional formula, we must break down its definition into these base fundamental units through a series of step-by-step derivations involving charge, work, and potential.
Step 2 : Key Formulas and approach:
We utilize the following fundamental relationships to derive the dimensions:
1. Capacitance: $C = \frac{Q}{V}$ (Charge divided by Potential)
2. Electric Potential: $V = \frac{W}{Q}$ (Work done per unit Charge)
3. Substituting $V$ in $C$: $C = \frac{Q^2}{W}$
4. Basic Units: Charge ($Q$) = Current ($I$) $\times$ Time ($t$) and Work ($W$) = Force $\times$ Displacement.
Step 3 : Detailed Explanation:
Step 1: Find dimensions of Charge [Q]. Since $Q = I \times t$, the dimensional formula is $[A^1 T^1]$.
Step 2: Find dimensions of Work [W]. Work is Force ($M L T^{-2}$) multiplied by Displacement ($L$). Thus, $[W] = [M^1 L^2 T^{-2}]$.
Step 3: Derive dimensions of Potential [V]. Since $V = \frac{W}{Q}$, we have $[V] = \frac{[M^1 L^2 T^{-2}]}{[A^1 T^1]} = [M^1 L^2 T^{-3} A^{-1}]$.
Step 4: Calculate Capacitance [C] using the formula $C = \frac{Q}{V}$.
Step 5: Substitute the dimensions: $[C] = \frac{[A^1 T^1]}{[M^1 L^2 T^{-3} A^{-1}]}$.
Step 6: Move all terms to the numerator by changing the signs of the exponents: $[M^{-1} L^{-2} T^{1 - (-3)} A^{1 - (-1)}]$.
Step 7: Simplifying the exponents for Time and Current, we get $T^4$ and $A^2$.
The final resulting formula is $[M^{-1} L^{-2} T^4 A^2]$, which shows capacitance has an inverse relationship with mass and length but a direct relationship with the square of current and the fourth power of time.
Step 4 : Final Answer:
The derived dimensional formula for capacitance is $[M^{-1} L^{-2} T^4 A^2]$, which corresponds to option (A).