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

To determine the SI unit of $K$, we need to analyze the given equation and the SI units of each variable.

The given equation is: $P = KI^2 \left[\frac{L}{\lambda}\right]^2$

Step 1: Identify the SI units of the known quantities. • $P$ (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).

Step 2: Substitute the units into the equation. We want to find the unit of $K$. Let $[X]$ denote the unit of quantity $X$. $[P] = [K] [I]^2 \left[\frac{L}{\lambda}\right]^2$ Substitute the known units: $\text{W} = [K] (A)^2 \left[\frac{m}{m}\right]^2$

Step 3: Simplify the units. The term $\left[\frac{m}{m}\right]^2$ simplifies to $(1)^2 = 1$, as it is a ratio of two lengths, making it dimensionless. So the equation becomes: $\text{W} = [K] \cdot A^2 \cdot 1$ $\text{W} = [K] \cdot A^2$

Step 4: Solve for the unit of $K$. Divide both sides by $\text{A}^2$: $[K] = \frac{W}{A^2}$

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

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

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