The dimensional formula of Capacitance is
[ M-1 L-2 T2 A2 ]
where [M], [L], [T], and [A] are the fundamental quantities: Mass, Length, Time, and Ampere.
From the above formula, the dimensions of capacitance are (-1, -2, 2, 2).
How to calculate the dimensional formula of capacitance?
In physics, we can determine the amount of charge by the capacitance and the voltage difference across the capacitors. Mathematically, it means
Charge (Q) = Capacitance (C) * Voltage (V) Capacitance = Charge / Voltage
We know that,
- Charge = Current * time
- Voltage = Work / Charge
Now, let’s drive this formula in terms of fundamental quantities:
Voltage = Work / Charge
Work = Force * displacement, so the formula becomes
Voltage = (Force * displacement) / Charge
Replacing the value of Force by mass * acceleration, we get
Voltage = (mass * acceleration * displacement) / Charge
- acceleration = speed / time
- speed = distance / time
- acceleration = distance / time2
Voltage = (mass * distance / time2 * displacement) / Charge = (mass * distance * displacement) / (Charge * time2) = (mass * distance * displacement) / (Current * time * time2) = (mass * distance * displacement ) / (Current * time3) = (mass * distance * displacement * Current-1 * time-3)
Now, the formula of capacitance becomes
Capacitance = (Current * time) / (mass * distance * displacement * Current-1 * time-3) = (Current * time * Current * time3) / (mass * distance * displacement) = ( Current2 * time4 ) / (mass * distance * displacement) = (Current2 * time4 * mass-1 * distance-1 * displacement-1)
Dimensionally, we use
- [M] for mass
- [L] to represent both distance and displacement
- [T] for time
- and [A] for current
Now, the formula becomes
Capacitance = [A2] * [T4] * [M-1] * [L-2] = [M-1] * [L-2] * [T4] * [A2]
Hence, the dimensional equation of Capacitance is C = [ M-1 L-2 T4 A2 ].