Radiation & Antennas
Radiation & Antennas
Accelerating charges radiate electromagnetic energy. Larmor's formula quantifies this power, and the analysis of oscillating dipoles explains antenna operation, molecular emission spectra, and synchrotron radiation.
Definition
Larmor formula (non-relativistic): radiated power \(P = q^2 a^2/(6\pi\epsilon_0 c^3)\) where \(a\) is the acceleration. Radiation is emitted in a characteristic donut pattern around the acceleration axis.
Key Result
An oscillating electric dipole \(p(t) = p_0\cos(\omega t)\) radiates power \(P = \omega^4 p_0^2/(12\pi\epsilon_0 c^3)\) — the \(\omega^4\) dependence explains why blue light scatters more than red (Rayleigh scattering).
Example 1
A half-wave dipole antenna of length \(\lambda/2\) fed at its center has input impedance \(\approx 73\,\Omega\) and radiates with gain 1.64 relative to an isotropic antenna.
Example 2
Synchrotron radiation: relativistic electrons in a circular orbit emit radiation in a narrow forward cone. Peak frequency \(\omega_{c} = 3\gamma^3 c/(2R)\) extends far into X-rays for GeV-scale electrons.
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Practice
- Derive the radiation resistance of a short dipole antenna.
- Why does an accelerating charge radiate but a uniformly moving charge does not?
- What is the far-field approximation, and when does it apply?
- Estimate the power radiated by an electron in a 1 T magnetic field.
Show Answer Key
1. Short dipole (length $l \ll \lambda$): radiated power $P = \frac{\pi}{3}\eta_0(I_0 l/\lambda)^2$ where $\eta_0 = 120\pi\,\Omega$. Radiation resistance: $R_r = P/(I_0^2/2) = \frac{2\pi}{3}\eta_0(l/\lambda)^2 = 80\pi^2(l/\lambda)^2\,\Omega$. For $l/\lambda = 0.01$: $R_r \approx 0.08\,\Omega$ (very small, hence short antennas are inefficient).
2. Radiation requires acceleration. By the Larmor formula: $P = \frac{q^2 a^2}{6\pi\epsilon_0 c^3}$. A uniformly moving charge has $a=0$ in its rest frame → no radiation (the fields are static Coulomb/Biot-Savart in that frame, just Lorentz-boosted). An accelerating charge's field has a $1/r$ radiation component that carries energy to infinity.
3. Far-field: observation distance $r \gg \lambda$ and $r \gg$ antenna size $D$. Fields fall as $1/r$, are transverse, and $E/H = \eta_0$. The angular pattern depends only on direction, not distance. Near-field ($r < \lambda$): reactive fields dominate ($1/r^2$, $1/r^3$), storing energy rather than radiating. The transition occurs at $r \sim \lambda/(2\pi)$.
4. Cyclotron radiation: $P = \frac{q^2\omega_c^2 v_\perp^2}{6\pi\epsilon_0 c^3}$ where $\omega_c = eB/m$. For $B=1$ T, electron with $v_\perp \sim 0.1c$: $\omega_c = 1.76\times10^{11}$ rad/s. $P \approx \frac{(1.6\times10^{-19})^2(1.76\times10^{11})^2(3\times10^7)^2}{6\pi(8.85\times10^{-12})(3\times10^8)^3} \approx 6.4\times10^{-12}$ W per electron.