Electric Dipole Radiation and Radiation Patterns

Core Idea

Electric dipole radiation from time-varying dipole moment p(t) dominates for non-relativistic sources. Radiated power and angular distribution depend on dipole acceleration magnitude and direction. Maximum radiation perpendicular to acceleration; zero along it. Dipole antennas exploit this pattern.

Explainer

In your study of multipole expansions and the Larmor formula, you found that accelerating charges radiate electromagnetic energy, and that the leading-order term in the multipole expansion of the radiation field comes from the electric dipole moment p⃗(t) of the source. For any localized charge distribution oscillating at frequency ω with size much smaller than the radiation wavelength (the non-relativistic, long-wavelength limit), the dipole term dominates all higher multipole contributions by factors of (kd) ≪ 1, where d is the source size. This is why dipole radiation is the first and most important radiation mechanism to master.

The electric dipole moment p⃗ = Σ qᵢrᵢ captures the overall charge separation in the source. For a sinusoidally oscillating dipole p(t) = p₀ cos(ωt), the second time derivative p̈ = −ω²p₀ cos(ωt) enters the radiation fields. The radiated power follows from the Larmor formula generalized to dipoles: P = p̈²/(6πε₀c³) (in SI). The key insight is that it is the acceleration of the dipole moment, not just its existence, that produces radiation. A static dipole radiates nothing; a uniformly moving dipole radiates nothing; only a changing p̈ (equivalently, changing current distribution) produces radiation.

The angular radiation pattern is one of the most beautiful results in classical electrodynamics. The power radiated per unit solid angle varies as dP/dΩ ∝ sin²θ, where θ is the angle measured from the direction of p̈. This means: maximum radiation is emitted perpendicular to the oscillation direction (θ = 90°, a band around the "equator" of the dipole), and zero radiation is emitted along the dipole axis (θ = 0°, the "poles"). Visualize a toroidal or donut-shaped pattern, with the hole aligned along the dipole. The radiation is also polarized: the electric field in the far zone lies in the plane containing the observation direction and the dipole axis.

This pattern directly explains antenna design. A half-wave dipole antenna is a conducting rod driven to oscillate current back and forth along its length. The pattern broadcasts strongest broadside (perpendicular to the rod) and nothing off the ends — exactly the sin²θ shape. Engineers orient antennas accordingly. The same physics governs how atoms radiate light: an excited atom's oscillating electron distribution can be approximated as an oscillating dipole, and dipole selection rules (which transitions are allowed) govern which spectral lines appear bright. The sin²θ pattern, the ω⁴ power dependence on frequency, and the dominance of perpendicular emission are features you will encounter repeatedly in radiation physics, optics, and antenna engineering.