Dimensional Representation of Power using Fundamental Quantities
Understanding Dimensions
The goal is to express the dimensions of Power (P) in terms of Kinetic Energy (K), Current (I), and Charge (Q) as the fundamental quantities.
Deriving Base Dimensions
- Kinetic Energy (K): The dimension of energy is derived from $K = \frac{1}{2}mv^2$. Thus, the dimension is $[K] = [M][L]^2[T]^{-2}$.
- Current (I): This is taken as a fundamental quantity, so its dimension is $[I]$.
- Charge (Q): Charge is related to current and time by $Q = It$. Thus, the dimension is $[Q] = [I][T]$.
- Power (P): Power is the rate of change of energy, $P = \frac{Energy}{Time}$. Thus, the dimension is $[P] = \frac{[K]}{[T]}$.
Expressing Power in Fundamental Quantities
We need to express $[P] = \frac{[K]}{[T]}$ using K, I, and Q. From the dimension of charge, $[Q] = [I][T]$, we can find the dimension of time:
- $[T] = \frac{[Q]}{[I]} = [Q][I]^{-1}$.
Now, substitute the dimension of Time into the dimension of Power:
- $[P] = \frac{[K]}{[T]} = \frac{[K]}{[Q][I]^{-1}}$
- $[P] = [K][Q]^{-1}[I]$
- $[P] = [K I Q^{-1}]$
Therefore, the dimensional representation of power using K, I, and Q as fundamental quantities is $[K I Q^{-1}]$.
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