A radio signal of length L, emits a signal of wavelength λ and power P. These equation quantities are related by P = K I² [L/λ]² where I is current. Determine the S.I unit of K

• $P$ (Power) has the SI unit of Watt (W).
• $I$ (Current) has the SI unit of Ampere (A).
• $L$ (Length) has the SI unit of meter (m).
• $\lambda$ (Wavelength) has the SI unit of meter (m).

We want to find the unit of $K$. Let $[X]$ denote the unit of quantity $X$.

Substitute the known units:

The term $\left[\frac{\text{m}}{\text{m}}\right]^2$ simplifies to $(1)^2 = 1$, as it is a ratio of two lengths, making it dimensionless.
So the equation becomes:

Divide both sides by $\text{A}^2$:

The SI unit of $K$ is $\boxed{\frac{\text{W}}{\text{A}^2}}$.

Alternatively, we can express this in terms of base SI units:
Since $1 \text{ W} = 1 \frac{\text{J}}{\text{s}}$ and $1 \text{ J} = 1 \text{ N} \cdot \text{m} = 1 \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \cdot \text{m} = 1 \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}$,
Then $1 \text{ W} = 1 \frac{\text{kg} \cdot \text{m}^2}{\text{s}^3}$.
So, the unit of $K$ can also be expressed as: