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At a distance of 20 m from a hypothetical isotropic antenna is radiating in free space, the total electric field (Eθ) is measured to be 1.5 V/m. Find the(a)    power density (Wrad) and(b)    power radiated (Prad)

Question

At a distance of 20 m from a hypothetical isotropic antenna is radiating in free space, the total electric field (Eθ) is measured to be 1.5 V/m. Find the(a)    power density (Wrad) and(b)    power radiated (Prad)

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Solution

(a) Power Density (Wrad)

The power density (Wrad) in watts per square meter (W/m²) at a distance (r) from an isotropic antenna radiating in free space can be calculated using the formula:

Wrad = |Eθ|^2 / (2 * η)

where:

  • |Eθ| is the magnitude of the electric field in volts per meter (V/m), and
  • η is the intrinsic impedance of free space, approximately 377 ohms.

Given that |Eθ| = 1.5 V/m, we can substitute these values into the formula to find Wrad:

Wrad = (1.5 V/m)² / (2 * 377 Ω) = 2.98 * 10^-3 W/m²

(b) Power Radiated (Prad)

The total power radiated (Prad) by an isotropic antenna in free space can be calculated using the formula:

Prad = Wrad * Area

where Area is the surface area of a sphere with radius r. The surface area of a sphere can be calculated using the formula 4πr².

Given that r = 20 m, we can calculate the surface area:

Area = 4π * (20 m)² = 5026.55 m²

We can then substitute the values for Wrad and Area into the formula to find Prad:

Prad = 2.98 * 10^-3 W/m² * 5026.55 m² = 14.98 W

So, the power density at a distance of 20 m from the antenna is approximately 2.98 * 10^-3 W/m², and the total power radiated by the antenna is approximately 14.98 W.

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