Magnetic Dipole Radiation

The chapter's first radiation formula came from charges sloshing up and down a stick — the oscillating electric dipole, with its $\sin^2\theta$ donut and $\omega^4$ power law. This article asks the obvious next question: what does a current loop broadcast? A small loop with an alternating current is the elementary magnetic dipole, a tiny bar magnet whose poles trade places $\omega/2\pi$ times a second. It is electrically neutral everywhere — no charge separation ever develops — so the electric dipole moment is identically zero and the leading term of the radiation ladder simply is not there. The loop must radiate, if it radiates at all, through the next rung. Two rewards await. The first is structural: the calculation mirrors the electric dipole step for step, and at the end the two results are related by the cleanest symmetry in electrodynamics — the duality substitution $\mathbf{p}\to\mathbf{m}/c$ with the roles of $\mathbf{E}$ and $c\mathbf{B}$ exchanged. Same donut, same $\omega^4$ , polarization rotated by ninety degrees. The second is a number with teeth: for a source of size $d$ whose charges move at speed $v$ , the magnetic channel is weaker than the electric one by $(v/c)^2$ . That single suppression factor explains why loop antennas make superb receivers but hopeless transmitters, why "forbidden" spectral lines take megayears instead of nanoseconds, and why the 21 cm line that maps our galaxy exists at all. The article closes the pair EDRAD opened and sets up the one after it: magnetic dipole and electric quadrupole radiation arrive at exactly the same order of the multipole expansion, and honesty requires both. The $M1$ story here, the $E2$ story next, then the full ladder.

Key takeaways

A small loop carrying $I(t)=I_0\cos\omega t$ has an oscillating magnetic moment $\mathbf{m}(t)=m_0\cos\omega t\,\hat{\mathbf{z}}$ with $m_0=I_0\pi b^2$ , and it radiates even though its electric dipole moment is exactly zero.
The radiation fields follow from the retarded vector potential $\mathbf{A}=\tfrac{\mu_0}{4\pi}\nabla\times[\mathbf{m}(t_r)/r]$ : keeping the $1/r$ piece gives $\mathbf{B}_{\text{rad}}\parallel\hat{\boldsymbol\theta}$ and $\mathbf{E}_{\text{rad}}\parallel\hat{\boldsymbol\phi}$ — the electric dipole pattern with the two fields swapped.
Total power: $\langle P_{M1}\rangle=\dfrac{\mu_0 m_0^2\omega^4}{12\pi c^3}$ — the electric dipole formula under the duality map $p_0\to m_0/c$ . Same $\omega^4$ , same $\sin^2\theta$ donut.
The suppression is in the moments, not the frequency: $m\sim qvd$ against $p\sim qd$ gives $P_{M1}/P_{E1}\sim(v/c)^2$ — tiny for any nonrelativistic source.
Consequences priced in this article: a 10 cm loop antenna at 100 MHz has a radiation resistance of 24 m $\Omega$ ; hydrogen's 21 cm hyperfine line ( $M1$ ) has an 11-million-year lifetime; a pulsar's spin-down is a rotating magnetic dipole radiating at $10^{31}$ W.
$M1$ and $E2$ radiation enter at the same order of the multipole expansion — when the electric dipole vanishes by symmetry, both must be kept (next article).
Key idea: a magnetic dipole radiates exactly like an electric dipole under the substitution $\mathbf{p}\to\mathbf{m}/c$ with $\mathbf{E}$ and $c\mathbf{B}$ interchanged — same donut, same $\omega^4$ , polarization rotated $90°$ , power down by $(v/c)^2$ .

The core insight

A magnetic dipole radiates exactly like an electric dipole with $\mathbf{p}\to\mathbf{m}/c$ and the electric and magnetic fields interchanged; the radiated power $P=\mu_0 m_0^2\omega^4/(12\pi c^3)$ carries the same $\omega^4$ law but is down by $(v/c)^2$ from $E1$ .

Where this lives today

The $(v/c)^2$ suppression is the quiet architect of modern astrophysics' most important spectral line. Hydrogen's ground-state hyperfine flip at 21 cm is a magnetic dipole transition — the electron's spin moment reorients, no charge displaces — and its transition rate is so suppressed ( $A\approx2.9\times10^{-15}$ s⁻¹, a lifetime of eleven million years) that only the absurd column depths of interstellar space make the line visible. That feebleness is a feature: the line stays optically thin through entire galaxies, which is why 21 cm radio maps trace galactic rotation curves — the original dark-matter evidence — and why the coming generation of cosmology experiments (SKA, HERA) hunts the same $M1$ photon from the universe's first billion years. Meanwhile every "forbidden" line in a nebular spectrum — the auroral greens and nebular reds of [O III] and [O I] — is an $M1$ or $E2$ transition surviving because the gas is too dilute to collisionally quench a state that waits seconds instead of nanoseconds to decay. The engineering ledger is just as direct. NFC and RFID at 13.56 MHz are deliberate near-field magnetic dipole systems: with $kb\sim10^{-3}$ , equation (3) makes the radiated power vanishingly small, so the link is private by physics — inductive coupling dies as $1/r^6$ , and an eavesdropper meters away gets nothing. Implanted medical devices and mine-rescue radios exploit the same loophole in reverse: magnetic near fields penetrate conducting tissue and rock that would eat any radiating electric field. And the pulsar timing of worked example 2, run at nanosecond precision across an array of millisecond pulsars, is now a gravitational-wave detector the size of the galaxy — the magnetic dipole spin-down law is the clock-drift model underneath it all.

Origins · 1 minHeinrich Hertz and how this was discovered

Heinrich Hertz

1857–1894 · Karlsruhe — whose receiver was a loop: the first magnetic dipole antenna

Hertz's 1886–88 experiments are remembered for the transmitter — the spark-gap electric dipole that first proved Maxwell's waves real. But the detector on the other side of his lab was a simple loop of wire with a micrometer spark gap: a resonant magnetic dipole, reading the wave's oscillating $\mathbf{B}$ through Faraday induction. Hertz chose it deliberately: a loop responds to the magnetic field's circulation and can be rotated to read polarization — which is precisely how he mapped the field directions, verified transversality, and demonstrated reflection, refraction and standing waves with the full optical toolkit. The pairing was prophetic. To this day the electrically small loop is the receiver of choice where a whip antenna fails, from AM ferrite rods to magnetometer coils hunting gravitational-wave-adjacent physics — and every one of them is Hertz's resonator, miniaturized. He died at 36, before "radio" was a word; asked about applications of his waves, he reportedly answered "nothing, I guess." The loop in his hand said otherwise.

History

1820

— Ampère models magnetism as circulating currents: the magnetic moment is born as a current loop

1873

— Maxwell's Treatise contains the duality of E and B in free space, the symmetry behind the M1/E1 twinship

1883

— FitzGerald computes the power radiated by an alternating current loop, the first magnetic dipole radiation formula

1888

— Hertz detects electromagnetic waves with a resonant loop — the magnetic dipole as receiver

1898

— Liénard's moving-charge results seed the systematic multipole ordering of radiation

1930

s — Condon and Shortley codify M1/E2 selection rules; "forbidden" nebular lines identified as such by Bowen

1951

— Ewen and Purcell detect the 21 cm hyperfine M1 line of galactic hydrogen

1968

— the Crab pulsar's spin-down is matched to magnetic dipole radiation, weighing a teragauss field

The physical picture

Hold the two elementary broadcasters side by side. The electric dipole is two charges trading places along a line: charge sloshes, $\mathbf{p}$ oscillates, and the news of each reversal peels off at $c$ . The magnetic dipole is a closed loop of current whose circulation reverses: no charge ever accumulates anywhere — the current is divergence-free — but the loop's magnetic moment $\mathbf{m}=I\pi b^2\,\hat{\mathbf{z}}$ flips sign twice a cycle. Far away it is a bar magnet whose north and south poles swap $\omega/2\pi$ times per second, and that reversal is news too. The deep reason the two problems are twins is the symmetry of the source-free Maxwell equations under $\mathbf{E}\to c\mathbf{B}$ , $c\mathbf{B}\to-\mathbf{E}$ . In the radiation zone, where the waves have detached from their sources and propagate freely, a solution's dual is also a solution. The magnetic dipole's radiation is the dual of the electric dipole's: the same $\sin^2\theta$ donut of intensity, with the field roles exchanged. For the electric dipole, $\mathbf{E}$ lies in the meridian plane (along $\hat{\boldsymbol\theta}$ ) and $\mathbf{B}$ wraps the axis (along $\hat{\boldsymbol\phi}$ ); for the magnetic dipole, $\mathbf{B}$ takes the meridian and $\mathbf{E}$ wraps the axis. An antenna engineer would say the polarization is rotated by ninety degrees — and that rotation is measurable, the fingerprint that tells you which multipole you are hearing.

Figure

The two elementary broadcasters — dual sources, dual fields: same donut, polarization rotated 90 degrees

Why is the magnetic channel weak? Count the moments. A source of size $d$ built from charges $q$ moving at speed $v$ has electric dipole moment $p\sim qd$ but magnetic moment $m\sim I d^2\sim (qv/d)\,d^2 = qvd$ . The dimensionless ratio that duality puts into every field formula is $m/(cp)\sim v/c$ — one power of the charges' speed over the speed of light. Radiated power goes as the moment squared, so the magnetic channel is down by $(v/c)^2$ . For electrons in an atom $v/c\sim\alpha\approx1/137$ , so $M1$ rates are $\sim5\times10^{-5}$ of comparable $E1$ rates before frequency factors. For a mains transformer, $(v/c)^2$ of the drift velocity is beyond negligible. Magnetic dipole radiation is the unsung sibling: it matters precisely when symmetry silences the electric dipole — and then it is the loudest voice left.

The retarded potential of an oscillating moment

The machinery is the chapter's standard kit. For a localized current distribution the retarded vector potential, expanded for a source small compared to both $r$ and $\lambda$ , has as its leading neutral-loop term the dipole potential — the time-dependent upgrade of the magnetostatic result from the vector-potential article of the magnetostatics chapter. The only change, exactly as with the retarded potentials themselves, is the dating of the source:

\mathbf{A}(\mathbf{r},t)=\frac{\mu_0}{4\pi}\,\nabla\times\left[\frac{\mathbf{m}(t_r)}{r}\right],\qquad t_r=t-\frac{r}{c}. \qquad (1)

where: $\mathbf{m}(t)=m_0\cos\omega t\,\hat{\mathbf{z}}$ is the loop's moment ( $m_0=I_0\pi b^2$ for a loop of radius $b$ ), and $r$ is measured from the loop's center. Read (1) against its static ancestor $\mathbf{A}=\tfrac{\mu_0}{4\pi}\mathbf{m}\times\hat{\mathbf{r}}/r^2$ : the curl form packages the same content, but now the curl acts on a moment evaluated at the retarded time, and that extra dependence is where the radiation comes from. Since the loop is neutral and the current divergence-free, the scalar potential vanishes identically in this order — the entire broadcast is carried by $\mathbf{A}$ , which is why the magnetic dipole is the cleanest possible showcase of the vector potential doing physics on its own.

Derivation ARadiation fields of the magnetic dipole from the retarded vector potential

The goal is the far-zone field pair of the oscillating magnetic dipole, mirroring the electric dipole derivation of the previous article rung by rung. The strategy: evaluate the curl in (1) honestly — the derivative acts on the explicit $1/r$ and on the retarded time inside $\mathbf{m}$ — sort the two resulting terms by their range, keep the $1/r$ radiation term, then take one more curl for $\mathbf{B}$ . Watch for the same lesson the retarded-potentials article taught: the term the naive static calculation misses (the derivative through $t_r$ ) is precisely the term that survives at infinity. We work in the radiation zone $r\gg\lambda\gg b$ , so terms are kept or dropped by powers of $1/r$ at fixed $\omega$ .

Step 1. Evaluate the curl of $\mathbf{m}(t_r)/r$ . Using $\nabla\times(f\mathbf{V})=\nabla f\times\mathbf{V}+f\,\nabla\times\mathbf{V}$ with $f=1/r$ and $\mathbf{V}=\mathbf{m}(t_r)$ , and the chain rule $\nabla\times\mathbf{m}(t_r)=\nabla t_r\times\dot{\mathbf{m}}=-\tfrac{1}{c}\hat{\mathbf{r}}\times\dot{\mathbf{m}}$ ... assembling both pieces:

\mathbf{A}=\frac{\mu_0}{4\pi}\left[\frac{\hat{\mathbf{r}}\times\mathbf{m}(t_r)}{r^2}+\frac{\hat{\mathbf{r}}\times\dot{\mathbf{m}}(t_r)}{c\,r}\right].

Why two terms: the first is the static dipole potential wearing a retarded timestamp — it falls as $1/r^2$ and dies at infinity. The second exists only because moving the field point changes which moment of the loop's history you read; it has the longer $1/r$ range. This is the first scent of radiation, in exactly the place the retarded-potentials article said to look.

Step 2. Keep the radiation term and make it explicit. With $\mathbf{m}=m_0\cos(\omega t_r)\hat{\mathbf{z}}$ , $\dot{\mathbf{m}}=-m_0\omega\sin(\omega t_r)\hat{\mathbf{z}}$ , and $\hat{\mathbf{r}}\times\hat{\mathbf{z}}=-\sin\theta\,\hat{\boldsymbol\phi}$ :

\mathbf{A}_{\text{rad}}=\frac{\mu_0}{4\pi c}\,\frac{\hat{\mathbf{r}}\times\dot{\mathbf{m}}(t_r)}{r}=\frac{\mu_0 m_0\omega}{4\pi c}\,\frac{\sin\theta}{r}\,\sin\!\big(\omega(t-r/c)\big)\,\hat{\boldsymbol\phi}.

Why $\hat{\boldsymbol\phi}$ : the potential wraps the dipole axis, inheriting the geometry of the current it summarizes — the loop's circulation. Compare the electric dipole, whose radiation $\mathbf{A}$ points along $\ddot{\mathbf{p}}$ , i.e. along the axis. The two geometries are orthogonal from the start, and the fields will inherit that.

Step 3. One more curl for $\mathbf{B}$ , radiation-zone rules. In the far zone every derivative either hits $1/r$ (down one power, drop) or hits $t_r$ (brings $\omega/c$ , keep): effectively $\nabla\to-(\hat{\mathbf{r}}/c)\,\partial_t$ acting on the oscillation. Carrying it out on $\mathbf{A}_{\text{rad}}$ :

\mathbf{B}_{\text{rad}}=\nabla\times\mathbf{A}_{\text{rad}}=-\frac{\mu_0 m_0\omega^2}{4\pi c^2}\,\frac{\sin\theta}{r}\,\cos\!\big(\omega(t-r/c)\big)\,\hat{\boldsymbol\theta},

and the electric field follows from the radiation-zone triad relation $\mathbf{E}=c\,\mathbf{B}\times\hat{\mathbf{r}}$ :

\mathbf{E}_{\text{rad}}=\frac{\mu_0 m_0\omega^2}{4\pi c}\,\frac{\sin\theta}{r}\,\cos\!\big(\omega(t-r/c)\big)\,\hat{\boldsymbol\phi}.

Why the pattern was inevitable: two time derivatives of $\mathbf{m}$ appear (one from each honest curl), so the amplitude carries $\ddot m\propto m_0\omega^2$ — radiation is sourced by the second derivative of the moment, exactly as $E1$ radiation rides $\ddot{\mathbf{p}}$ . The $\sin\theta$ is the lever-arm geometry of $\hat{\mathbf{r}}\times\hat{\mathbf{z}}$ : no broadcast along the axis the moment points.

Step 4. Audit against duality. Take the electric dipole radiation fields from the previous article, substitute $p_0\to m_0/c$ , and exchange $\mathbf{E}\to c\mathbf{B}$ , $c\mathbf{B}\to-\mathbf{E}$ : the pair above appears verbatim, signs and all. The calculation and the symmetry agree — either one now certifies the other. Verified: matches Griffiths §11.1.3 (Eqs. 11.36–11.38) and Jackson §9.3; the static limit $\omega\to0$ kills both radiation fields and the $1/r^2$ term of Step 1 reduces to the magnetostatic dipole potential exactly.

Power, and the duality dividend

Energy bookkeeping is the Poynting machinery of the conservation chapter, applied to the fields just built. Both fields fall as $1/r$ , their product as $1/r^2$ , and the sphere's area grows as $r^2$ : a finite, radius-independent power escapes — the operational definition of radiation. The flux and its angular density:

\left\langle\frac{dP}{d\Omega}\right\rangle=\frac{\mu_0 m_0^2\omega^4}{32\pi^2c^3}\,\sin^2\theta. \qquad (2)

where: the time average has replaced $\cos^2$ by $\tfrac12$ , and $\theta$ is measured from the moment's axis. This is the same donut as the electric dipole's — maximum broadside, silent along the poles — because duality exchanges the fields without touching the energy density; only the polarization of the donut knows which dipole made it.

Go deeper · Derivation B · 1 minTotal radiated power and the suppression factor

Derivation BTotal radiated power and the suppression factor

The goal is the total power $\langle P_{M1}\rangle$ and its ratio to the electric dipole result — the number that organizes all the applications. Strategy: integrate (2) over the sphere using the standard angular integral, then run the duality map against $\langle P_{E1}\rangle$ and count powers of $v/c$ in the moments. The only real work is bookkeeping; the payoff is the cleanest suppression law in radiation theory.

Step 1. The angular integral. With $d\Omega=\sin\theta\,d\theta\,d\phi$ and $\int_0^\pi\sin^3\theta\,d\theta=\tfrac43$ :

\oint\sin^2\theta\,d\Omega=2\pi\cdot\frac{4}{3}=\frac{8\pi}{3}.

Step 2. Assemble the total power:

\langle P_{M1}\rangle=\frac{\mu_0 m_0^2\omega^4}{32\pi^2c^3}\cdot\frac{8\pi}{3}=\frac{\mu_0 m_0^2\omega^4}{12\pi c^3}.

Step 3. Run the duality audit. The electric dipole total was $\langle P_{E1}\rangle=\mu_0 p_0^2\omega^4/12\pi c$ . Substitute $p_0\to m_0/c$ : the two extra powers of $c$ appear in the denominator and Step 2's formula is reproduced exactly. Duality is not a mnemonic — it is a symmetry of the free-field equations, and the power formula respects it to the letter.

Step 4. Count the suppression. For one source of size $d$ , charges $q$ at characteristic speed $v$ : $p_0\sim qd$ , $m_0\sim I d^2\sim qvd$ , so

\frac{\langle P_{M1}\rangle}{\langle P_{E1}\rangle}=\frac{m_0^2}{c^2p_0^2}\sim\left(\frac{v}{c}\right)^2.

Why this is a statement about moments, not frequencies: both channels carry the identical $\omega^4$ . The magnetic amplitude simply starts one factor of $v/c$ smaller, because building a magnetic moment requires the charges to move around the loop, while an electric moment only requires them to be separated. Squaring the amplitude ratio gives the power ratio. *Verified: power matches Griffiths Eq. 11.40 and Jackson §9.3; the equivalent-source check below (worked example 1) confirms $P=I_0^2R_{\text{rad}}/2$ numerically to all displayed digits.*

The engineering repackaging of Step 2 is the radiation resistance of a small loop. Setting $\langle P\rangle=\tfrac12 I_0^2R_{\text{rad}}$ with $m_0=I_0\pi b^2$ and $k=\omega/c$ :

R_{\text{rad}}^{\text{loop}}=\frac{\pi Z_0}{6}\,(kb)^4\approx 197\,(kb)^4\ \Omega, \qquad Z_0=\mu_0c\approx376.7\ \Omega. \qquad (3)

where: the fourth power of $kb=2\pi b/\lambda$ is the loop's shame: halve the loop size and the resistance falls sixteenfold. Compare a short electric dipole antenna, whose $R_{\text{rad}}=20\pi^2(\ell/\lambda)^2$ falls only quadratically. The extra two powers of size-over-wavelength are exactly the $(v/c)^2$ suppression wearing engineering clothes — for an antenna, the charges' effective $v/c$ is set by $kd$ , since they must circulate the structure once per cycle.

Figure

The suppression ledger — one law, twenty-five orders of magnitude: the (v/c) squared penalty across real systems

Limiting cases

case	result	meaning
$\omega\to0$	$P\to0$ as $\omega^4$ ; $\mathbf{A}\to$ magnetostatic dipole potential	a steady moment broadcasts nothing; magnetostatics is the silent limit
duality map $p_0\to m_0/c$	$E1$ formulas $\to$ $M1$ formulas exactly	the free-field $\mathbf{E}\to c\mathbf{B}$ , $c\mathbf{B}\to-\mathbf{E}$ symmetry at work
same source, both moments	$P_{M1}/P_{E1}\sim(v/c)^2$	magnetic channel audible only when the electric one is silenced
$\mathbf{p}=0$ by symmetry	$M1$ and $E2$ lead, same order in $d/\lambda$	forbidden-line territory; next article completes the pair
loop antenna, $kb\ll1$	$R_{\text{rad}}=(\pi Z_0/6)(kb)^4$	the $\propto(b/\lambda)^4$ transmit penalty; receivers unaffected
rotating (not oscillating) $\mathbf{m}$ , angle $\alpha$	$P=\mu_0 m^2\omega^4\sin^2\alpha/6\pi c^3$	two orthogonal oscillating components — twice the single-axis power; the pulsar case

Common trapSame omega-to-the-fourth — the suppression is not a frequency effect, and the polarization is not optional

Two errors account for most lost credit on this topic. First: reading " $M1$ is weaker than $E1$ " as a statement about frequency scaling. Both powers scale as $\omega^4$ ; the suppression lives entirely in the ratio of moments, $m/(cp)\sim v/c$ , and no amount of frequency helps a magnetic dipole catch up with an electric dipole of the same source. (The channel that genuinely carries different frequency scaling is the electric quadrupole, $\omega^6$ — next article.) Second: forgetting that the polarization swaps. The magnetic dipole's electric field wraps the axis along $\hat{\boldsymbol\phi}$ — azimuthal, horizontal for a vertical loop axis — opposite to the electric dipole's meridional $\hat{\boldsymbol\theta}$ . Get this backwards and every antenna-orientation and line-polarization deduction that follows is backwards too. The donut shape is common property; the field directions are the channel's signature.

Worked examples

1 · The small loop antenna — why your car radio's loop cannot transmit (confidence builder)

Setup. A circular loop of radius $b=5$ cm drives $I_0=1$ A at $f=100$ MHz (FM band, $\lambda=3.00$ m). Find $R_{\text{rad}}$ and the radiated power; check the power two ways.

Step 1. The size parameter: $kb=\dfrac{2\pi b}{\lambda}=\dfrac{2\pi(0.05)}{3.00}=0.105$ — comfortably in the small-loop regime $kb\ll1$ , so (3) applies.

Step 2. Radiation resistance: $R_{\text{rad}}=\dfrac{\pi Z_0}{6}(0.105)^4=197.4\times1.20\times10^{-4}=\mathbf{23.8\ m\Omega}$ . For comparison, a short electric dipole of the same 10 cm span has $R_{\text{rad}}=20\pi^2(0.1/3.0)^2=0.22\ \Omega$ — nine times larger even at this tiny size, and the gap widens as $(kb)^{-2}$ for smaller structures.

Step 3. Power, route one: $\langle P\rangle=\tfrac12I_0^2R_{\text{rad}}=\tfrac12(1)^2(0.0238)=\mathbf{11.9\ mW}$ .

Step 4. Power, route two — the physics formula. $m_0=I_0\pi b^2=\pi(0.05)^2=7.85\times10^{-3}$ A·m², $\omega=6.28\times10^8$ s⁻¹:

\langle P\rangle=\frac{\mu_0 m_0^2\omega^4}{12\pi c^3}=\frac{(4\pi\times10^{-7})(7.85\times10^{-3})^2(6.28\times10^8)^4}{12\pi(3.00\times10^8)^3}=\mathbf{11.9\ mW}.

The two routes agree to all displayed digits — the resistance formula is the power formula.

The lesson: a milliwatt-scale radiator burning an ampere is a terrible transmitter (any ohm of wire resistance swamps 24 m $\Omega$ , so efficiency is under 3%) — but a superb receiver, since reception cares about induced EMF, not radiated power. AM ferrite loop antennas ( $kb\sim2\times10^{-4}$ , $R_{\text{rad}}\sim10^{-13}\ \Omega$ !) live entirely on this asymmetry.

2 · A pulsar as a magnetic dipole transmitter (exam level)

Setup. The Crab pulsar spins with period $P_{\text{rot}}=33.4$ ms, slowing at $\dot P=4.21\times10^{-13}$ s/s; its moment of inertia is $I_\star\approx10^{38}$ kg·m² and radius $R_\star\approx10$ km. Model the spin-down as magnetic dipole radiation from a moment inclined at angle $\alpha$ to the spin axis and infer the surface field.

Step 1. The observed power budget. Rotational energy $E=\tfrac12I_\star\Omega^2$ drains at

\dot E=I_\star\Omega\dot\Omega,\qquad \Omega=\frac{2\pi}{P_{\text{rot}}}=188.1\ \text{s}^{-1},\qquad \dot\Omega=\frac{2\pi\dot P}{P_{\text{rot}}^2}=2.37\times10^{-9}\ \text{s}^{-2},

giving $\dot E=(10^{38})(188.1)(2.37\times10^{-9})=\mathbf{4.5\times10^{31}\ W}$ — a hundred thousand Suns, from a 20-km-wide object.

Step 2. The radiating moment. A moment $\mathbf{m}$ inclined at $\alpha$ and rotating at $\Omega$ is two perpendicular oscillating dipoles of amplitude $m\sin\alpha$ in quadrature, so $P=\dfrac{\mu_0(m\sin\alpha)^2\Omega^4}{6\pi c^3}$ — twice the single-axis formula. Setting $P=\dot E$ and solving:

m\sin\alpha=\sqrt{\frac{6\pi c^3\dot E}{\mu_0\Omega^4}}=\sqrt{\frac{6\pi(2.70\times10^{25})(4.5\times10^{31})}{(1.26\times10^{-6})(188.1)^4}}=\mathbf{3.8\times10^{27}\ A\cdot m^2}.

Step 3. The surface field. With the polar-field convention $B_p=\mu_0 m/(2\pi R_\star^3)$ and $\sin\alpha\sim1$ :

B_p=\frac{(1.26\times10^{-6})(3.8\times10^{27})}{2\pi(10^4)^3}=\mathbf{7.6\times10^{8}\ T}\approx7.6\times10^{12}\ \text{G},

(the equatorial-field convention quotes half this, the familiar $3.8\times10^{12}$ G). Either way: a teragauss magnet, deduced from a stopwatch.

Step 4. Audit the scalings. $P\propto m^2\Omega^4$ : as the pulsar slows, the broadcast dies as the fourth power of the spin — young pulsars scream, old pulsars whisper, exactly the observed population. And note the frequency: $\Omega/2\pi\approx30$ Hz. This "light" has a 10,000 km wavelength; the star ( $R_\star\ll\lambda$ ) is the ultimate small-loop antenna, $kR_\star\sim6\times10^{-3}$ .

The lesson: the same $\omega^4 m^2$ law prices a 24 m $\Omega$ hobby loop and a $10^{31}$ W neutron star. That is what a good radiation formula is for.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §11.1.3 · Jackson, Classical Electrodynamics, 3rd ed., §9.2–9.3 · Zangwill, Modern Electrodynamics, Ch. 20 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11

Provenance. Field expressions and power formula verified against Griffiths Introduction to Electrodynamics 4th ed. §11.1.3 (Eqs. 11.36–11.40) and Jackson Classical Electrodynamics 3rd ed. §9.2–9.3. A verifier should recompute: (1) the curl of the retarded dipole vector potential keeping the $1/r$ term; (2) the angular integral $\int\sin^2\theta\,d\Omega=8\pi/3$ ; (3) the final power $\mu_0m_0^2\omega^4/(12\pi c^3)$ ; and (4) the duality map $p_0\to m_0/c$ against the electric dipole power $\mu_0p_0^2\omega^4/(12\pi c)$ . All symbols in SI units. Addendum for this edition: per the chapter's division of labor, the three radiation-zone approximations ( $d\ll r$ , $d\ll\lambda$ , $r\gg\lambda$ ) and the $E1$ fields recalled here are owned by Electric Dipole Radiation and never re-derived; the systematic same-order pairing with the electric quadrupole is stated here and prosecuted in Electric Quadrupole Radiation; the exact (all- $kr$ ) magnetic multipole fields belong to the E11 treatment Magnetic Dipole and Electric Quadrupole Radiation. Worked numbers for this edition (loop $R_{\text{rad}}=23.8$ m $\Omega$ at $kb=0.105$ with the two-route power check $11.9$ mW; Crab $\dot E=4.5\times10^{31}$ W, $m\sin\alpha=3.8\times10^{27}$ A·m², $B_p=7.6\times10^8$ T; 21 cm lifetime $1.1\times10^7$ yr from $A=2.9\times10^{-15}$ s⁻¹) recomputed for this edition.

Builds on: Electric Dipole Radiation · Retarded Potentials · Lorenz Gauge · Scalar and Vector Potentials Leads to: Electric Quadrupole Radiation · Jefimenko's Equations

If the loop is neutral, what is actually waving in the radiated electric field?

The vector potential — and behind it, the current. A neutral, divergence-free current produces no scalar potential at this order, but the radiation electric field does not need one: $\mathbf{E}=-\partial\mathbf{A}/\partial t$ , and $\mathbf{A}$ carries the retarded news of the changing circulation. This is the cleanest demonstration in the chapter that the vector potential is not bookkeeping: here it is the only potential on stage, and differentiating it produces a perfectly real, energy-carrying azimuthal electric field. If you prefer field language: the changing $\mathbf{B}$ of the reversing moment induces a curling $\mathbf{E}$ (Faraday), that changing $\mathbf{E}$ regenerates $\mathbf{B}$ (displacement current), and the pair bootstraps outward at $c$ exactly as in the wave chapter — the loop merely sets the pattern spinning.

Is the duality substitution a rigorous argument or a plausibility trick?

Rigorous, with one honest caveat. In source-free regions Maxwell's equations are exactly invariant under $\mathbf{E}\to c\mathbf{B}$ , $c\mathbf{B}\to-\mathbf{E}$ , so the dual of any free-field solution is a solution. The radiation zone is source-free, and matching the dual solution onto a magnetic dipole source instead of an electric one is precisely the substitution $\mathbf{p}\to\mathbf{m}/c$ . The caveat: duality is a symmetry of the fields, not of the sources — nature supplies electric charges but (as far as anyone has measured) no magnetic monopoles, so the two problems are physically distinct at the source even though their radiation fields are mirror images. That is why this article also runs the honest retarded-potential derivation: duality certifies it, but the potential calculation is what grounds it in the actual current loop.

Why does building a magnetic moment cost a factor of v/c when building an electric moment does not?

Because the two moments are different moments of the same charge distribution. The electric dipole moment $\mathbf{p}=\int\rho\,\mathbf{r}\,d^3r$ is a configuration property: separate charges by $d$ and you have $p\sim qd$ , whether or not anything moves. The magnetic moment $\mathbf{m}=\tfrac12\int\mathbf{r}\times\mathbf{J}\,d^3r$ is a motion property: it requires current, and $J\sim qv/d^2$ over a region $d$ gives $m\sim qvd$ . The comparison that duality forces on us is $m/c$ against $p$ , i.e. $qvd/c$ against $qd$ — one factor of $v/c$ . Deeper version, for when you reach the relativistic chapter: $\mathbf{p}$ and $\mathbf{m}/c$ sit in the same relativistic object (the dipole-moment analogue of how $\mathbf{E}$ and $c\mathbf{B}$ share the field tensor), and boosting a static electric dipole generates a magnetic moment at order $v/c$ — the counting is a Lorentz-structure fact, not an accident.

Do M1 and E2 radiation always come together, and can they interfere?

They arrive at the same order in $d/\lambda$ — both stem from the first retardation correction across the source, as the next article shows explicitly — so a careful next-to-leading calculation must evaluate both. But "same order" does not mean "both present": symmetry decides. A perfect current loop has $M1$ but zero electric quadrupole; two back-to-back electric dipoles have $E2$ but no net magnetic moment; a bent molecule may carry both. When both are present they can interfere in the angular distribution (their fields add coherently at each direction), though not in the total power once you integrate over all angles with the full multipole machinery — the channels are orthogonal on the sphere, a theorem the multipole-expansion article proves. Selection rules in atoms use exactly this taxonomy: a transition is labeled M1, E2, or a mixture, and the measured angular/polarization pattern identifies which.

The formulas here assume an oscillating moment along a fixed axis. What about a rotating magnet, like a pulsar?

Decompose it. A moment of fixed magnitude $m$ rotating at $\Omega$ about an axis at inclination $\alpha$ is a constant component $m\cos\alpha$ along the spin axis (silent — it never changes) plus two perpendicular components of amplitude $m\sin\alpha$ oscillating at $\Omega$ , ninety degrees out of phase. Each radiates the standard $\mu_0(m\sin\alpha)^2\Omega^4/12\pi c^3$ , and because their fields are in quadrature the cross term time-averages to zero: total $P=\mu_0m^2\Omega^4\sin^2\alpha/6\pi c^3$ , exactly twice the single-axis formula. The radiation is circularly polarized along the spin axis and linearly polarized in the equatorial plane — and an aligned rotator ( $\alpha=0$ ) is silent in this model, which is why real pulsar spin-down (which persists even for nearly aligned pulsars, via magnetospheric currents) is measured against this vacuum-dipole baseline rather than explained entirely by it.

Why is the magnetic dipole a good receiver if it is such a bad transmitter — does reciprocity not forbid that?

Reciprocity is not violated; the comparison changes. As a transmitter, what matters is radiated power per unit drive current: $R_{\text{rad}}\propto(kb)^4$ makes the loop feeble, and ohmic resistance devours the efficiency. As a receiver, what matters is signal-to-noise into your amplifier, and in the regimes where loops rule (AM broadcast, NFC, magnetometry) the limiting noise is external — atmospheric or ambient interference — not the antenna's own losses. A small loop delivers an EMF $\mathcal{E}=\omega B_0\cdot N\pi b^2$ (boostable with $N$ turns and a ferrite core) and picks up the magnetic field, which penetrates conductors and shrugs off the electric-field noise that plagues short whips. Reciprocity still holds antenna-for-antenna in identical conditions; the loop simply plays a different game — near-field, magnetically coupled, externally noise-limited — where its transmit penalty never enters the score.

At what loop size does the small-dipole treatment fail, and what happens then?

The expansion behind equation (1) requires $kb\ll1$ — the current must be effectively uniform around the loop and the source's internal retardation negligible. As $kb\to1$ the loop becomes a resonant antenna: the current distribution develops standing-wave structure around the circumference, higher multipoles ( $E2$ , $M2$ , ...) enter with comparable strength, and the neat $\sin^2\theta$ donut deforms and eventually splits into multiple lobes. A full-wave loop ( $2\pi b=\lambda$ , i.e. $kb=1$ ) radiates strongly — radiation resistance of order 100 Ω, no longer any suppression — because the charges now traverse the structure in one period: effectively $v/c\sim1$ in the suppression counting. The systematic tool for that regime is the full multipole expansion two articles ahead, or numerically, the method-of-moments machinery mentioned in the retarded-potentials article. The $(kb)^4$ law is the small-loop asymptote, not a property of loops as such.