Operation of infinitesimal dipole

An antenna is a transducer that converts guided electromagnetic energy on a transmission line into free-space radiated waves (and vice versa).

Infinitesimal (Hertzian) dipole

An infinitesimal dipole is a very short straight wire of length l ≪ λ (l ≤ λ/50) carrying a uniform current I(z) = I0 along its length, placed at the origin along the z-axis. Real antennas can be built as a sum of such elements.

        z
        |   P(r, θ, φ)
        |  /
       ┌┼┐/ r
     l │I0│ θ
       └┼┘
        |
        +----------- y
       /
      x

Vector potential

A = âz μ I0 l e^(−jkr) / (4π r)
Ar = Az cosθ,  Aθ = −Az sinθ

Field components (H = ∇×A/μ, E from Maxwell)

Hφ = j k I0 l sinθ e^(−jkr)/(4πr) · [1 + 1/(jkr)]
Er = η I0 l cosθ e^(−jkr)/(2πr²) · [1 + 1/(jkr)]
Eθ = jηkI0 l sinθ e^(−jkr)/(4πr)
     · [1 + 1/(jkr) − 1/(kr)²]
Hr = Hθ = Eφ = 0

Field regions:

• kr ≪ 1 (near/reactive field): 1/r² and 1/r³ terms dominate; energy is stored, not radiated.
• kr ≫ 1 (far field): only 1/r terms remain:

Eθ ≈ jη k I0 l sinθ e^(−jkr) / (4πr)
Hφ ≈ j k I0 l sinθ e^(−jkr) / (4πr)
Er ≈ 0,   Eθ / Hφ = η = 120π Ω

The far fields are transverse, in phase, and form a TEM wave.

Power, radiation resistance and directivity

P_rad = η (π/3) |I0 l / λ|²
Rr    = 2 P_rad / |I0|² = 80π² (l/λ)²  Ω
U(θ) ∝ sin²θ  →  D = 1.5 (1.76 dBi), HPBW = 90°

Example: l = λ/50 → Rr = 80π²/2500 ≈ 0.316 Ω (very small, so efficiency is low).

Field pattern

The far-field pattern is |Eθ| ∝ sinθ:

• E-plane (vertical plane): figure-of-eight, maximum at θ = 90°, nulls along the axis (θ = 0°, 180°).
• H-plane (horizontal plane): circle – omnidirectional.
• 3-D: doughnut shape.

   E-plane (sinθ)        H-plane
        z                   y
        |                 .---.
   .-.  |  .-.          /       \
  (   ) | (   )        |    o    |  x
   '-'  |  '-'          \       /
        |                 '---'
     null on axis       circle