Electrodynamics

Electromagnetic radiation from a dipole

Accelerating charge radiates; in the far zone an electric dipole has a sin²θ angular pattern and average power proportional to ω⁴p₀².

Consider a short electric dipole p(t) = p₀ cos(ωt) along z. In the radiation zone r ≫ λ, transverse fields fall as 1/r and carry energy outward.

Interpretation and consequences

The total average power is P = μ₀ω⁴p₀²/(12πc). The ω⁴ factor explains stronger radiation at higher oscillation frequency; near fields also contain 1/r² and 1/r³ terms.

Integrating over solid angle uses $\int\sin^2\theta\,d\Omega=8\pi/3$, giving $\langle P\rangle=\mu_0\omega^4p_0^2/(12\pi c)$. The pattern has two lobes and no radiation along the dipole axis. The short-dipole approximation requires the source size to be much smaller than the wavelength; otherwise phase varies across the source, and higher multipoles can alter both total power and angular pattern.

The angular pattern reflects radiation polarization: the far electric field is perpendicular to both the observation direction $\hat r$ and dipole moment $\hat z$. Flux through a sphere of radius $r$ is $r^2\langle S_r\rangle$, so the field falls as $1/r$ while total power is independent of distance. Near the source, $1/r^2$ and $1/r^3$ terms store energy cyclically rather than carrying it steadily to infinity.

The far-zone condition is $kr\gg1$; only the radiative $1/r$ terms contribute a finite outward power through a sphere in the distant limit. Antenna design therefore depends strongly on operating frequency.

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