Electric Dipole Radiation

Derivation

Electric Dipole Radiation

D-171 Home PU-204 Threads waves · light · energy Depends on Retarded Potentials and Jefimenko's Equations, Larmor Formula for Radiated Power

Statement

For an oscillating point electric dipole \( \vec p(t) = p_0 \cos(\omega t)\,\hat{\mathbf z} \) in vacuum, we derive the radiation-zone fields, the time-averaged angular power distribution \( \dfrac{d\langle P\rangle}{d\Omega} = \dfrac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\,\sin^2\theta \), and the total time-averaged radiated power \( \langle P\rangle = \dfrac{\mu_0 p_0^2 \omega^4}{12\pi c} \), starting from the retarded potentials and working consistently to leading order in \( d/\lambda \), \( d/r \), and \( \lambda/r \).

Why it matters

Dipole radiation is the elementary act of light emission in classical electrodynamics: it is the leading term in the multipole expansion of any localized oscillating source, so almost every emitter you meet — a radio antenna shorter than its wavelength, a molecule scattering sunlight, an atom modelled as a bound electron — radiates, to first approximation, with the \( \sin^2\theta \) pattern and the \( \omega^4 \) power law derived here. The \( \omega^4 \) dependence alone explains why the sky is blue and sunsets are red, and the total-power formula fixes the radiation resistance of short antennas and the classical radiative lifetime of atoms.

Structurally, this is the canonical worked example of how retardation creates radiation. The static dipole field falls off as \( 1/r^3 \) and carries no energy to infinity; only by keeping the retarded time \( t - r/c \) and extracting the terms that fall off as \( 1/r \) does one obtain fields whose Poynting flux through a large sphere is independent of the radius. Everything in antenna theory, scattering theory, and spontaneous emission builds on this calculation.

Assumptions

Point-dipole limit, \( d \ll \lambda \).If the source size is comparable to the wavelength (e.g. a half-wave antenna, \( d = \lambda/2 \)), the current distribution along the source matters: contributions from different parts interfere, the angular pattern sharpens, and the power formula changes (a half-wave dipole has radiation resistance \( \approx 73\,\Omega \), not the short-dipole value derived below).

Far field / dipole geometry, \( d \ll r \).Close to the source the two charges are not seen as a single dipole; the fields are those of the individual charges and the multipole expansion in \( d/r \) fails.

Radiation zone, \( r \gg \lambda \).At \( r \lesssim \lambda \) the discarded \( 1/r^2 \) and \( 1/r^3 \) (induction and quasi-static) terms dominate; the fields are reactive, \( \vec E \) and \( \vec B \) are out of phase, and the Poynting vector is not purely radial.

Harmonic time dependence \( p(t) = p_0\cos\omega t \).For a general \( \vec p(t) \) the derivation still goes through with \( -\omega^2 p_0 \cos[\omega(t-r/c)] \to \ddot p(t - r/c) \); the harmonic case is recovered mode by mode via Fourier superposition (Maxwell's equations are linear).

Vacuum, no boundaries.In a medium of refractive index \( n \) the phase velocity, impedance, and hence the radiated power all change; near conducting boundaries image currents modify the pattern (this is why real antennas over ground differ from free-space dipoles).

Nonrelativistic source motion, \( v \ll c \).This is implied by \( d \ll \lambda \): charge speeds are \( \sim \omega d \ll \omega\lambda \sim c \). If dropped, retardation across the source can no longer be linearized, radiation beams forward, and one needs the Liénard generalization of the Larmor formula.

Lorenz gauge for the potentials.The retarded solutions \( V, \vec A \) used below are the Lorenz-gauge solutions of the inhomogeneous wave equations. In another gauge the intermediate potentials differ, but \( \vec E \) and \( \vec B \) — and hence every physical result — are unchanged; consistency of the approximated \( V \) and \( \vec A \) with \( \nabla\!\cdot\!\vec A = -\mu_0\varepsilon_0\,\partial V/\partial t \) is a useful check on the order counting.

Derivation

\[ \vec p(t) = q(t)\,d\,\hat{\mathbf z} = p_0 \cos(\omega t)\,\hat{\mathbf z}, \qquad q(t) = q_0\cos\omega t,\qquad I(t) = \dot q = -q_0\,\omega\sin\omega t \]

Model the dipole physically: charges \( \pm q(t) \) at \( z = \pm d/2 \) connected by a wire carrying the current \( I \) that shuttles the charge; \( p_0 = q_0 d \). Charge conservation is built in. A

\[ \vec A(\vec r, t) = \frac{\mu_0}{4\pi} \int_{-d/2}^{d/2} \frac{I\!\left(t - \left|\vec r - z'\hat{\mathbf z}\right|/c\right)}{\left|\vec r - z'\hat{\mathbf z}\right|}\, \hat{\mathbf z}\, dz' \]

Write the retarded vector potential (assumed prior result: retarded potentials solve the Lorenz-gauge wave equations with the source evaluated at the retarded time). The current is confined to the wire along \( \hat{\mathbf z} \). A

\[ \vec A(\vec r, t) \simeq \frac{\mu_0 d}{4\pi r}\, I\!\left(t - \tfrac{r}{c}\right) \hat{\mathbf z} = -\,\frac{\mu_0 p_0 \omega}{4\pi r}\, \sin\!\left[\omega\!\left(t - \tfrac{r}{c}\right)\right] \hat{\mathbf z} \]

Point-dipole limit: \( d \ll r \) lets us replace \( |\vec r - z'\hat{\mathbf z}| \to r \) in the denominator, and \( d \ll \lambda \) lets us do the same in the retarded time, since the phase error \( \omega z'/c \lesssim \pi d/\lambda \ll 1 \). The integrand is then constant and the integral gives a factor \( d \). Note \( q_0\omega d = p_0\omega \). B

\[ V(\vec r, t) = \frac{1}{4\pi\varepsilon_0} \left[ \frac{q\!\left(t - R_+/c\right)}{R_+} - \frac{q\!\left(t - R_-/c\right)}{R_-} \right], \qquad R_\pm = \left|\vec r \mp \tfrac{d}{2}\hat{\mathbf z}\right| \]

Retarded scalar potential of the two point charges; each charge is evaluated at its own retarded time. No approximation yet. A

\[ R_\pm \simeq r \mp \frac{d}{2}\cos\theta, \qquad \frac{1}{R_\pm} \simeq \frac{1}{r}\left(1 \pm \frac{d\cos\theta}{2r}\right), \] \[ q\!\left(t - \tfrac{R_\pm}{c}\right) \simeq q_0\left\{ \cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \mp \frac{\omega d \cos\theta}{2c}\, \sin\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \right\} \]

Expand to first order in the two independent small parameters \( d/r \) and \( \omega d/c = 2\pi d/\lambda \). The second expansion is the crucial one: retardation differs between the two charges by \( \Delta t = (d\cos\theta)/c \), and it is this differential retardation that will survive at large \( r \). Dropping it (i.e. using a common retarded time) kills the radiation term entirely. C

\[ V(\vec r, t) = \frac{p_0\cos\theta}{4\pi\varepsilon_0} \left\{ \frac{1}{r^2}\cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] - \frac{\omega}{c\,r}\, \sin\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \right\} \]

Insert the expansions of Step 5 into Step 4 and keep first-order terms; the zeroth-order monopole pieces cancel exactly (the system is neutral), and \( q_0 d = p_0 \). The \( 1/r^2 \) term is the retarded quasi-static dipole potential; the \( 1/r \) term exists only because of differential retardation. One can verify \( \nabla\!\cdot\!\vec A = -\mu_0\varepsilon_0\,\partial V/\partial t \) holds to this order — the approximations respect the Lorenz gauge. C

\[ V(\vec r, t) \simeq -\,\frac{p_0\,\omega}{4\pi\varepsilon_0 c}\, \frac{\cos\theta}{r}\, \sin\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \qquad (r \gg \lambda) \]

Radiation zone: the ratio of the discarded to the kept term is \( c/(\omega r) = \lambda/(2\pi r) \ll 1 \). We now hunt only for fields falling as \( 1/r \), since only they deliver energy to infinity (\( S \sim E^2 \sim 1/r^2 \) against sphere area \( 4\pi r^2 \)). B

\[ -\frac{\partial \vec A}{\partial t} = \frac{\mu_0 p_0 \omega^2}{4\pi r} \cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \left( \cos\theta\,\hat{\mathbf r} - \sin\theta\,\hat{\boldsymbol\theta} \right) \]

Differentiate Step 3 in time and resolve \( \hat{\mathbf z} = \cos\theta\,\hat{\mathbf r} - \sin\theta\,\hat{\boldsymbol\theta} \) onto the spherical basis at the field point. A

\[ -\nabla V \Big|_{1/r} = -\,\frac{\mu_0 p_0 \omega^2}{4\pi}\, \frac{\cos\theta}{r}\, \cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \hat{\mathbf r} \]

Take the gradient of Step 7 keeping only \( 1/r \) terms: the radial derivative acting on the phase \( \omega(t - r/c) \) brings down \( \omega/c \) without costing a power of \( r \), whereas \( \partial_r(1/r) \) and \( \tfrac{1}{r}\partial_\theta \) both produce \( 1/r^2 \) and are dropped. Used \( 1/(\varepsilon_0 c^2) = \mu_0 \). This selective differentiation — "differentiate the phase, not the envelope" — is the systematic radiation-zone rule. C

\[ \vec E = -\nabla V - \frac{\partial \vec A}{\partial t} = -\,\frac{\mu_0 p_0 \omega^2}{4\pi} \left( \frac{\sin\theta}{r} \right) \cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \hat{\boldsymbol\theta} \]

Add Steps 8 and 9: the radial components cancel identically, leaving a purely transverse field. This cancellation is not luck — a radiation field must be transverse, and it happens automatically because the approximated potentials still satisfy the Lorenz gauge (Step 6). B

\[ \vec B = \nabla \times \vec A \Big|_{1/r} = -\,\frac{\mu_0 p_0 \omega^2}{4\pi c} \left( \frac{\sin\theta}{r} \right) \cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \hat{\boldsymbol\varphi} = \frac{1}{c}\, \hat{\mathbf r} \times \vec E \]

Curl of Step 3 in spherical coordinates: \( B_\varphi = \tfrac{1}{r}\left[\partial_r (r A_\theta) - \partial_\theta A_r\right] \); again only the \( \partial_r \) acting on the phase survives at order \( 1/r \). The result is a locally plane wave: \( \vec E \perp \vec B \perp \hat{\mathbf r} \) with \( |\vec E| = c|\vec B| \), in phase. B

\[ \vec S = \frac{1}{\mu_0}\,\vec E \times \vec B = \frac{\mu_0 p_0^2 \omega^4}{16\pi^2 c}\, \frac{\sin^2\theta}{r^2}\, \cos^2\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right] \hat{\mathbf r} \quad\Rightarrow\quad \langle \vec S \rangle = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\, \frac{\sin^2\theta}{r^2}\, \hat{\mathbf r} \]

Form the Poynting vector from Steps 10 and 11 (\( \hat{\boldsymbol\theta} \times \hat{\boldsymbol\varphi} = \hat{\mathbf r} \)) and average over one period: \( \langle \cos^2 \rangle = \tfrac12 \). The flux is radially outward and falls as exactly \( 1/r^2 \). A

\[ \frac{d\langle P\rangle}{d\Omega} = r^2\, \langle \vec S \rangle \cdot \hat{\mathbf r} = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\, \sin^2\theta \]

Power per unit solid angle through a distant sphere: the \( r^2 \) from the area element \( dA = r^2\, d\Omega \) cancels the \( 1/r^2 \) of the flux, so the same power crosses every sphere — the signature of true radiation. A

\[ \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c} \int_0^{2\pi}\! d\varphi \int_0^{\pi} \sin^2\theta\, \sin\theta\, d\theta = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\cdot 2\pi \cdot \frac{4}{3} = \frac{\mu_0 p_0^2 \omega^4}{12\pi c} \]

Integrate over solid angle with the measure \( d\Omega = \sin\theta\, d\theta\, d\varphi \); \( \int_0^\pi \sin^3\theta\, d\theta = 4/3 \). A

\[ P_{\text{inst}} = \frac{\mu_0\, \ddot p^{\,2}(t-r/c)}{6\pi c}, \qquad \left\langle \ddot p^{\,2} \right\rangle = \tfrac12\, \omega^4 p_0^2 \;\Rightarrow\; \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{12\pi c}\;\checkmark \]

Cross-check against the assumed Larmor result: for a point charge, \( P = \mu_0 q^2 a^2 / 6\pi c \); the dipole's charges give \( q\,a \to \ddot p \). Time-averaging the harmonic \( \ddot p = -\omega^2 p_0 \cos[\omega(t-r/c)] \) reproduces Step 14 exactly. The general (non-harmonic) formula is \( P = \mu_0 \ddot p^{\,2}/6\pi c \). C

Result

\[ \vec E = -\frac{\mu_0 p_0 \omega^2}{4\pi}\frac{\sin\theta}{r}\cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right]\hat{\boldsymbol\theta}, \qquad \vec B = \frac{\hat{\mathbf r}\times\vec E}{c}, \] \[ \frac{d\langle P\rangle}{d\Omega} = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\sin^2\theta, \qquad \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{12\pi c} = \frac{p_0^2 \omega^4}{12\pi \varepsilon_0 c^3} \]

Reading. An oscillating dipole launches an outgoing spherical wave whose electric field lies along \( \hat{\boldsymbol\theta} \) (in the plane containing the dipole axis and the line of sight), with magnitude falling as \( 1/r \) and modulated by \( \sin\theta \): nothing is radiated along the dipole axis, and emission peaks in the equatorial plane. The radiated power grows as the fourth power of the frequency and the square of the dipole amplitude, and is the same through every distant sphere — energy genuinely leaves the source and never comes back.

Units check. \( [\mu_0 p_0^2 \omega^4 / c] = \left(\mathrm{kg\,m\,A^{-2}\,s^{-2}}\right)\left(\mathrm{A^2\,s^2\,m^2}\right)\left(\mathrm{s^{-4}}\right)\left(\mathrm{s\,m^{-1}}\right) = \mathrm{kg\,m^2\,s^{-3}} = \mathrm{W} \). ✓ For the field: \( [\mu_0 p_0 \omega^2/r] = \left(\mathrm{kg\,m\,A^{-2}\,s^{-2}}\right)\left(\mathrm{A\,s\,m}\right)\left(\mathrm{s^{-2}}\right)\left(\mathrm{m^{-1}}\right) = \mathrm{kg\,m\,A^{-1}\,s^{-3}} = \mathrm{V\,m^{-1}} \). ✓

Limiting cases

\( \omega \to 0 \): \( \langle P\rangle \to 0 \) as \( \omega^4 \) — a static dipole does not radiate; the fields smoothly reduce to electrostatics.
\( \theta \to 0, \pi \): \( d\langle P\rangle/d\Omega \to 0 \) — no radiation along the oscillation axis, because the transverse projection of the acceleration vanishes there.
Single accelerated charge: writing \( \ddot p = q a \), Step 15 reduces to the Larmor formula \( P = \mu_0 q^2 a^2 / 6\pi c \), as it must.
Near zone \( \omega r/c \ll 1 \): restoring the discarded terms, the fields reduce to the instantaneous (quasi-static) dipole field \( \vec E \propto (2\cos\theta\,\hat{\mathbf r} + \sin\theta\,\hat{\boldsymbol\theta})/r^3 \) evaluated with \( p(t) \) — retardation becomes invisible.
Scattering regime: for a bound electron driven below resonance, \( p_0 \propto E_0 \) independent of \( \omega \), so scattered power \( \propto \omega^4 \propto \lambda^{-4} \) — Rayleigh's law and the blue sky.

Breaks when

Source comparable to the wavelength (\( d \gtrsim \lambda \)). The current distribution can no longer be collapsed to a point dipole; interference between source elements reshapes the pattern and higher multipoles contribute at full strength. A half-wave antenna must be treated by integrating the actual current profile (radiation resistance \( \approx 73\,\Omega \), pattern \( \propto \cos^2\!\left(\tfrac{\pi}{2}\cos\theta\right)/\sin^2\theta \)), not by these formulas.
Inside the near zone (\( r \lesssim \lambda \)). The \( 1/r^2 \) and \( 1/r^3 \) terms discarded in Steps 7 and 9 dominate; \( \vec E \) and \( \vec B \) are out of phase, energy sloshes back and forth (reactive fields), and \( \langle \vec S\rangle \) is not given by the radiation formula. Antenna near-field measurements and RFID coupling live in this regime.
Relativistic source speeds. If charge velocities approach \( c \) (violating \( d \ll \lambda \) for the internal motion), retardation across the source cannot be linearized; radiation beams into a forward cone of half-angle \( \sim 1/\gamma \) and the power must come from the Liénard formula (synchrotron radiation is the standard example).
Quantum regime. When the emitted quantum \( \hbar\omega \) is comparable to the emitter's level spacings, emission is a discrete transition: the rate follows from the quantum dipole matrix element \( \langle f|\hat{\vec d}|i\rangle \) (Einstein \( A \) coefficient), and the classical formula survives only as the correspondence-principle limit.

Failure modes

Killing the radiation by dropping retardation. Using a common retarded time for both charges (or none at all) removes the \( 1/r \) term in \( V \) (Step 5) and yields zero radiated power — the single most instructive wrong answer in the subject.
Inconsistent order counting. Keeping the \( 1/r^2 \) piece of \( V \) but only the \( 1/r \) piece of \( \vec A \) (or vice versa) produces fields that violate the transversality cancellation of Step 10 and a spurious radial \( \vec E \).
Differentiating the envelope instead of the phase. In Step 9, taking \( \partial_r \) of \( 1/r \) but forgetting the \( \omega/c \) from the phase (or the reverse) misses or double-counts the radiation field.
\( \langle\cos^2\rangle = 1 \) instead of \( \tfrac12 \). Forgetting the time average overstates the power by a factor of 2; equivalently, confusing peak dipole moment \( p_0 \) with an RMS value.
Missing the Jacobian. Integrating \( \sin^2\theta\, d\theta \) instead of \( \sin^2\theta\,\sin\theta\, d\theta \) in Step 14 gives \( \pi/2 \) instead of \( 4/3 \) and a wrong total power.
Assuming radiation is strongest along the dipole axis. The pattern is exactly zero on-axis; the maximum is broadside (\( \theta = \pi/2 \)). Antennas are oriented accordingly.

Discussion

The deepest structural point is the origin of the \( 1/r \) fields. The retarded quasi-static field of the dipole falls as \( 1/r^3 \) (potential \( \sim 1/r^2 \)); no rearrangement of static terms can carry energy through a sphere whose area grows as \( r^2 \). Radiation appears only through differential retardation: the two charges are seen at slightly different retarded times, and differentiating that time offset trades a power of \( r \) for a factor \( \omega/c \). Each such trade costs one factor of \( \omega/c \), which is why the radiation field carries \( \omega^2 \) (two time derivatives of \( p \)) and the power carries \( \omega^4 \).

The \( \sin^2\theta \) pattern encodes the transversality of light. An observer sees radiation proportional to the component of \( \ddot{\vec p} \) perpendicular to the line of sight; along the axis that projection vanishes. The polarization follows the same projection: \( \vec E \parallel \hat{\boldsymbol\theta} \), i.e. along the projection of the dipole axis onto the sky. This is directly observable — skylight scattered through \( 90^\circ \) is strongly linearly polarized, because the scattering molecules are driven transverse to the sunlight and the observer at \( 90^\circ \) sees only one surviving dipole component.

In the multipole hierarchy, this result is the leading term of a systematic expansion of the retarded potentials in powers of \( d/\lambda \). Magnetic dipole and electric quadrupole radiation enter at the next order, suppressed in amplitude by \( \sim d/\lambda \) and in power by \( (d/\lambda)^2 \) — which is why "electric-dipole allowed" transitions dominate atomic spectroscopy, and why "forbidden" lines (M1, E2) are so weak that they are seen mainly in dilute astrophysical plasmas where collisions cannot de-excite the atoms first.

The same calculation runs in reverse as the classical skeleton of spontaneous emission. Writing \( \langle P\rangle = \hbar\omega\, \Gamma \) with \( p_0 \to 2|\langle f|q\hat{\vec r}|i\rangle| \) reproduces, up to the correct factor emerging from proper QED treatment of vacuum fluctuations plus radiation reaction, the Einstein coefficient \( A = \omega^3 |\langle f|q\hat{\vec r}|i\rangle|^2 / (3\pi\varepsilon_0 \hbar c^3) \). Likewise, radiation reaction on the source is the back-action required by the energy flux derived here: expanding the retarded self-field of the source one finds the Abraham–Lorentz force \( \vec F = \mu_0 q^2 \dot{\vec a}/6\pi c \), whose work accounts, on average, for exactly \( \langle P\rangle \). The pathologies of that force (pre-acceleration, runaways) mark the boundary where classical point-charge electrodynamics stops being self-consistent.

Common misconceptions. (i) "Any moving charge radiates" — uniform motion does not; the fields of a uniformly moving charge carry no energy to infinity (they can be obtained by boosting a static Coulomb field). Acceleration, here \( \ddot p \neq 0 \), is essential. (ii) "The near field is the radiation seen up close" — the reactive near field is a distinct, non-propagating energy store; the radiated power is only the part that survives to \( r \to \infty \). (iii) "Radiated intensity depends on how far you are in an essential way" — the \( 1/r^2 \) of intensity is pure geometry; the power through any distant closed surface is the same.

Worked examples

Example 1 — Short linear antenna: radiated power and radiation resistance. A center-fed wire antenna of length \( d = 5.0\ \mathrm{cm} \) carries an (idealized) uniform current of amplitude \( I_0 = 1.0\ \mathrm{A} \) at \( f = 100\ \mathrm{MHz} \). Find the time-averaged radiated power and the radiation resistance.

\[ \lambda = \frac{c}{f} = \frac{2.998\times 10^8\ \mathrm{m\,s^{-1}}}{1.00\times 10^8\ \mathrm{s^{-1}}} = 3.00\ \mathrm{m}, \qquad \frac{d}{\lambda} = \frac{0.050}{3.00} = 1.7\times 10^{-2} \ll 1 \]

Check the point-dipole condition before using the formula. A

\[ I_0 = q_0\,\omega \;\Rightarrow\; p_0 = q_0 d = \frac{I_0 d}{\omega}, \qquad \omega = 2\pi f = 6.283\times 10^{8}\ \mathrm{rad\,s^{-1}} \]

Relate the antenna current amplitude to the equivalent dipole amplitude (from Step 1 of the derivation, \( |I|_{\max} = q_0\omega \)). Symbols first. B

\[ p_0 = \frac{(1.0\ \mathrm{A})(0.050\ \mathrm{m})}{6.283\times 10^{8}\ \mathrm{s^{-1}}} = 7.96\times 10^{-11}\ \mathrm{C\,m} \]

Insert numbers with units. A

\[ \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{12\pi c} = \frac{\mu_0 \omega^2 I_0^2 d^2}{12\pi c} = \frac{(4\pi\times 10^{-7})(6.283\times 10^{8})^2 (1.0)^2 (0.050)^2}{12\pi\,(2.998\times 10^8)}\ \mathrm{W} = 0.11\ \mathrm{W} \]

Substitute \( p_0 = I_0 d/\omega \) symbolically first — two powers of \( \omega \) cancel — then evaluate. A

\[ \langle P\rangle = \tfrac12 I_0^2 R_{\mathrm{rad}} \;\Rightarrow\; R_{\mathrm{rad}} = \frac{\mu_0 \omega^2 d^2}{6\pi c} = \frac{2\pi}{3}\,\mu_0 c \left(\frac{d}{\lambda}\right)^2 \approx 789\left(\frac{d}{\lambda}\right)^2 \Omega \]

Define radiation resistance by analogy with Joule dissipation for a sinusoidal current; \( \mu_0 c = 376.7\ \Omega \) is the impedance of free space. B

\[ \langle P\rangle \approx 0.11\ \mathrm{W}, \qquad R_{\mathrm{rad}} = \frac{2\langle P\rangle}{I_0^2} \approx 0.22\ \Omega \]

Reading. A 5 cm wire at 100 MHz radiates only ~0.1 W per ampere-squared: its radiation resistance (0.22 Ω) is tiny compared with typical ohmic and feed impedances, which is precisely why electrically short antennas are inefficient and why practical antennas approach \( \lambda/2 \).

Units check. \( [\mu_0 \omega^2 d^2 / c] = (\mathrm{kg\,m\,A^{-2}\,s^{-2}})(\mathrm{s^{-2}})(\mathrm{m^2})(\mathrm{s\,m^{-1}}) = \mathrm{kg\,m^2\,A^{-2}\,s^{-3}} = \Omega \). ✓

Example 2 — Classical electron oscillator at optical frequency. Model an electron in a molecule as oscillating with amplitude \( x_0 = 1.0\times 10^{-10}\ \mathrm{m} \) at \( f = 5.0\times 10^{14}\ \mathrm{Hz} \) (orange light, \( \lambda \approx 600\ \mathrm{nm} \)). Find the time-averaged radiated power and the equivalent photon emission rate.

\[ p_0 = e\,x_0 = (1.602\times 10^{-19}\ \mathrm{C})(1.0\times 10^{-10}\ \mathrm{m}) = 1.60\times 10^{-29}\ \mathrm{C\,m} \]

The oscillating electron against the fixed ion core is an oscillating dipole of amplitude \( e x_0 \). Note \( x_0/\lambda \sim 10^{-4} \ll 1 \): point-dipole limit is excellent for atoms and light. A

\[ \omega = 2\pi f = 3.14\times 10^{15}\ \mathrm{rad\,s^{-1}}, \qquad \omega^4 = 9.74\times 10^{61}\ \mathrm{s^{-4}} \]

Angular frequency first; the fourth power is the dominant sensitivity in the problem. A

\[ \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{12\pi c} = \frac{(4\pi\times 10^{-7})(1.60\times 10^{-29})^2 (9.74\times 10^{61})}{12\pi\,(2.998\times 10^{8})}\ \mathrm{W} = 2.8\times 10^{-12}\ \mathrm{W} \]

Direct substitution into the main result. A

\[ \frac{\langle P\rangle}{\hbar\omega} = \frac{2.8\times 10^{-12}\ \mathrm{W}}{(1.055\times 10^{-34}\ \mathrm{J\,s})(3.14\times 10^{15}\ \mathrm{s^{-1}})} \approx 8\times 10^{6}\ \mathrm{s^{-1}} \]

Divide the classical power by the photon energy \( \hbar\omega = 3.3\times 10^{-19}\ \mathrm{J} \) to estimate an emission rate — a classical stand-in for the spontaneous-emission rate. B

\[ \langle P\rangle \approx 2.8\ \mathrm{pW}, \qquad \text{rate} \approx 8\times 10^{6}\ \text{photons s}^{-1} \;\Rightarrow\; \tau \sim 10^{-7}\ \mathrm{s} \]

Reading. A single Ångström-scale electron oscillation radiates picowatts — minuscule in absolute terms, yet the implied decay time of order \( 10^{-7}\,\mathrm{s} \) is within an order of magnitude of measured excited-state lifetimes (typically 1–100 ns): the classical dipole formula captures the correct scale of atomic radiative decay.

Units check. \( \mathrm{W}/(\mathrm{J\,s}\cdot\mathrm{s^{-1}}) = \mathrm{W}/\mathrm{J} = \mathrm{s^{-1}} \). ✓

Problems

Starting from \( \dfrac{d\langle P\rangle}{d\Omega} = \dfrac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\sin^2\theta \), perform the solid-angle integral explicitly and confirm \( \langle P\rangle = \mu_0 p_0^2 \omega^4 / 12\pi c \).
Solution
\[ \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c} \int_0^{2\pi} d\varphi \int_0^\pi \sin^2\theta\, \sin\theta\, d\theta. \] The azimuthal integral gives \( 2\pi \). With \( u = \cos\theta \), \( \int_0^\pi \sin^3\theta\, d\theta = \int_{-1}^{1}(1-u^2)\,du = 2 - \tfrac{2}{3} = \tfrac{4}{3} \). Hence \[ \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\cdot 2\pi \cdot \frac{4}{3} = \frac{\mu_0 p_0^2 \omega^4}{12\pi c}. \] The common error is integrating \( \sin^2\theta\,d\theta = \pi/2 \) without the Jacobian, which would give \( \langle P\rangle \) too small by a factor \( 3\pi/8 \approx 1.18 \).
What fraction of the total radiated power is emitted within \( 30^\circ \) of the equatorial plane, i.e. for \( 60^\circ \le \theta \le 120^\circ \)?
Solution
The fraction is \[ F = \frac{\int_{60^\circ}^{120^\circ} \sin^3\theta\, d\theta}{\int_0^{\pi} \sin^3\theta\, d\theta}. \] Using \( \int \sin^3\theta\, d\theta = -\cos\theta + \tfrac13\cos^3\theta \): at \( 120^\circ \), \( \cos\theta = -\tfrac12 \), the antiderivative is \( \tfrac12 - \tfrac{1}{24} = \tfrac{11}{24} \); at \( 60^\circ \), \( \cos\theta = \tfrac12 \), it is \( -\tfrac12 + \tfrac{1}{24} = -\tfrac{11}{24} \). The numerator is \( \tfrac{22}{24} = \tfrac{11}{12} \) and the denominator is \( \tfrac{4}{3} \), so \[ F = \frac{11/12}{4/3} = \frac{11}{16} = 0.6875. \] Nearly 69% of the power leaves through the middle third of the polar range — the dipole is a strongly broadside emitter.
An AM broadcast tower operating at \( f = 1.0\ \mathrm{MHz} \) radiates \( \langle P\rangle = 50\ \mathrm{kW} \). Treating it as an ideal oscillating dipole, find (a) the required dipole amplitude \( p_0 \) and (b) the peak electric-field amplitude at \( r = 10\ \mathrm{km} \) in the equatorial plane. (Verify \( r \gg \lambda \) first.)
Solution
\( \lambda = c/f = 300\ \mathrm{m} \), and \( r = 10^4\ \mathrm{m} = 33\lambda \gg \lambda \): radiation-zone formulas apply. \( \omega = 2\pi f = 6.283\times 10^6\ \mathrm{rad\,s^{-1}} \), \( \omega^2 = 3.948\times 10^{13}\ \mathrm{s^{-2}} \). (a) Invert the power formula: \[ p_0 = \frac{1}{\omega^2}\sqrt{\frac{12\pi c \langle P\rangle}{\mu_0}} = \frac{1}{3.948\times 10^{13}}\sqrt{\frac{12\pi (2.998\times 10^8)(5.0\times 10^4)}{4\pi\times 10^{-7}}}\ \mathrm{C\,m}. \] Inside the root: \( 3 \times 2.998\times 10^8 \times 5.0\times 10^4 \times 10^{7} = 4.50\times 10^{20} \), so the root is \( 2.12\times 10^{10} \) and \[ p_0 = \frac{2.12\times 10^{10}}{3.948\times 10^{13}} = 5.4\times 10^{-4}\ \mathrm{C\,m}. \] (b) At \( \theta = 90^\circ \): \[ E_0 = \frac{\mu_0 p_0 \omega^2}{4\pi r} = \frac{(10^{-7})(5.4\times 10^{-4})(3.948\times 10^{13})}{10^{4}}\ \mathrm{V\,m^{-1}} \approx 0.21\ \mathrm{V\,m^{-1}}, \] a realistic strong-signal field strength for AM reception near a transmitter.
Sunlight is scattered by air molecules whose induced dipole moment is \( p_0 = \alpha E_0 \) with polarizability \( \alpha \) essentially frequency-independent across the visible. Compute the ratio of scattered power at \( \lambda_b = 450\ \mathrm{nm} \) (blue) to that at \( \lambda_r = 650\ \mathrm{nm} \) (red) for equal incident field amplitudes, and state the observational consequence.
Solution
With \( p_0 \) fixed by \( \alpha E_0 \) (independent of frequency) and \( \omega = 2\pi c/\lambda \), \[ \frac{\langle P_b\rangle}{\langle P_r\rangle} = \left(\frac{\omega_b}{\omega_r}\right)^4 = \left(\frac{\lambda_r}{\lambda_b}\right)^4 = \left(\frac{650}{450}\right)^4 = (1.444)^4 \approx 4.35. \] Blue light is scattered about 4.4 times more strongly than red. Sky light — sunlight scattered toward the observer — is therefore blue-weighted, while direct sunlight near the horizon, having lost blue preferentially along a long air path, appears red. (Violet is scattered even more but is suppressed by the solar spectrum and the eye's response.)
A classical electron oscillator radiates away its own energy. Its mechanical energy is \( U = \tfrac12 m_e \omega^2 x_0^2 \) and its time-averaged radiated power is given by the dipole formula with \( p_0 = e x_0 \). (a) Show the fractional energy-loss rate is \( \gamma = \langle P\rangle / U = \dfrac{\mu_0 e^2 \omega^2}{6\pi m_e c} \), independent of amplitude. (b) Evaluate the classical radiative lifetime \( \tau = 1/\gamma \) at \( \lambda = 600\ \mathrm{nm} \) and compare with typical atomic excited-state lifetimes.
Solution
(a) \[ \gamma = \frac{\langle P\rangle}{U} = \frac{\mu_0 e^2 x_0^2 \omega^4 / 12\pi c}{\tfrac12 m_e \omega^2 x_0^2} = \frac{\mu_0 e^2 \omega^2}{6\pi m_e c}. \] The amplitude cancels: energy decays exponentially, \( U(t) = U(0)e^{-\gamma t} \), with a rate set only by \( \omega \) and fundamental constants. (b) \( \omega = 2\pi c/\lambda = 2\pi (2.998\times 10^8)/(6.00\times 10^{-7}) = 3.14\times 10^{15}\ \mathrm{rad\,s^{-1}} \), \( \omega^2 = 9.86\times 10^{30}\ \mathrm{s^{-2}} \). \[ \gamma = \frac{(1.257\times 10^{-6})(1.602\times 10^{-19})^2 (9.86\times 10^{30})}{6\pi (9.109\times 10^{-31})(2.998\times 10^{8})}. \] Numerator: \( 1.257\times 10^{-6} \times 2.566\times 10^{-38} \times 9.86\times 10^{30} = 3.18\times 10^{-13} \). Denominator: \( 18.85 \times 9.109\times 10^{-31} \times 2.998\times 10^{8} = 5.15\times 10^{-21} \). Hence \[ \gamma \approx 6.2\times 10^{7}\ \mathrm{s^{-1}}, \qquad \tau = \frac{1}{\gamma} \approx 1.6\times 10^{-8}\ \mathrm{s} = 16\ \mathrm{ns}. \] Measured allowed optical transitions have lifetimes of a few to a few tens of nanoseconds (e.g. sodium 3p: 16 ns) — the classical dipole result lands remarkably close, which is why the "classical electron oscillator" with an oscillator-strength correction was a workable model of spectral lines before quantum mechanics.