Retarded potential importance and infinitesimal dipole

Retarded potential is the scalar or vector potential at a point P computed using the source value at the earlier (retarded) time t − R/v, because changes in a source take R/v seconds to reach P.

V = (1/4πε) ∫ ρ(t − R/v)/R dv
A = (μ/4π)  ∫ J(t − R/v)/R dv
Phasor form: ρ, J times e^(−jkR)

Importance

• Gives a simple way to find the fields of any current distribution: find A (and V), then H = ∇×A/μ, E = −jωA − ∇V.
• Explains finite propagation speed and the phase delay e^(−jkr) of radiated fields.
• Produces the 1/r radiation field term; static potentials give only 1/r² fields, which carry no net power.
• Basis for analysing dipoles, loops, arrays and all wire antennas.

Infinitesimal dipole

A very short wire, l ≪ λ (≤ λ/50), on the z-axis with uniform current I0.

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

Vector potential: Az = μ I0 l e^(−jkr)/(4πr).

Fields:

Hφ = j k I0 l sinθ e^(−jkr)/(4πr) [1 + 1/(jkr)]
Er = η I0 l cosθ e^(−jkr)/(2πr²) [1 + 1/(jkr)]
Eθ = jη k I0 l sinθ e^(−jkr)/(4πr) [1 + 1/(jkr) − 1/(kr)²]

Far field (kr ≫ 1):

Eθ ≈ jη k I0 l sinθ e^(−jkr)/(4πr),   Hφ = Eθ/η

Radiation parameters:

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

Field pattern

|Eθ| ∝ sinθ.

   E-plane (vertical)       H-plane (horizontal)
        z                        .---.
   .-.  |  .-.                 /       \
  (   ) | (   )               |    o    |
   '-'  |  '-'                 \       /
        |                        '---'

• Maximum at θ = 90°, nulls along the dipole axis.
• Omnidirectional in the H-plane; 3-D pattern is a doughnut.