Two dipoles that agree everywhere but inside

The path a charge takes is not its field line concerned a common misreading of the lines drawn in a field diagram, and the diagram most often drawn in that discussion is a dipole. Nearly every picture of a dipole shows the same pattern from any reasonable distance: lines leaving one pole, curving round through space and returning to the other. The electric field of a pair of opposite charges has this shape, and so does the magnetic field of a loop of current, or a bar magnet. Physicists use one formula for both, and at the distances where anyone normally stands to look, the formula is correct.

That sameness is a large-distance fact, and it conceals a real physical difference in the middle. Field lines of the electric dipole begin on one charge and end on the other, so between the charges they run backward against the dipole’s moment. Field lines of the magnetic dipole never end, and so inside the loop they run forward, with the moment. Near enough to the source the two fields are opposite in direction. Common sense suggests that this should stop mattering in the limit where the dipole becomes very small. In the very case where people first needed to know, it does not. The hyperfine structure of hydrogen depends entirely on this interior term, and in that way it answers a question that sounds as if it could not possibly be asked of a single proton.

Two pictures, one far field #

Take two charges $\pm q$ a distance $2s$ apart along the $z$ axis, so the dipole moment $p = 2qs$ points from negative to positive. Take also a circular loop of radius $a$ carrying a current $I$, with moment $m = \pi a^2 I$ along the same axis, by the right-hand rule. The loop that behaves like a needle showed that the second of these acts on a compass the way a magnetic needle would, and at a distance that equivalence is exact for all practical purposes.

The electric field is the sum of two Coulomb fields, and it has a simple closed form. The magnetic field of the loop can be written with complete elliptic integrals, which is an exact expression and is calculated directly here. Drawing the two side by side, in units where each has the same moment, makes the point at once. Over most of the picture the arrows point in nearly the same directions. Near the centre, those of the charge pair run from the positive charge to the negative, downward and against the moment. Those through the loop run upward and with it.

The difference is due to the divergence of one field and the curl of the other. An electric field can have sources, and a charge pair has them right there in the picture. A magnetic field has none, as the field with no ends established, so each of its lines must close. A line that comes down the outside of a loop must go back up through the middle. There is nowhere else for it to go. An electric line coming down the outside of a charge pair does not need to go back up, because it finishes at the negative charge.

On the axis, where the numbers are simple #

The field along the axis shows the same result as a measurement.

In units where the moment is 1 and the size of the dipole is 1, the far field along the axis is $2/z^3$ for each. By $z = 2$ the two curves and the ideal dipole are close together, and at $z = 4$ they are indistinguishable by eye. Getting closer, the pair’s field passes through a singularity at the charge itself, where it swings from large and positive to large and negative. Inside, between the charges, it is negative. It is $-1$ at the centre. The loop’s curve simply rises smoothly and reaches $+2$ at the centre, in the same units, with no singularity on the axis at all.

The numbers $-1$ and $+2$ are not incidental, and they do not depend on the particular size chosen. Shrink each dipole in size while holding its moment fixed. The charge separation $2s$ shrinks and the charges increase in proportion; the centre field grows as $1/s^3$. The field outside at a fixed distance does not change. The interior term therefore becomes stronger but confined to a smaller volume, and a useful way to see what it comes to in the limit is to ask for its total, which requires an average.

The mean field inside a sphere #

Surround the dipole with a sphere of radius $R$, large enough to contain it, and average the field over the volume of that sphere. A theorem of electrostatics says that the mean electric field inside a sphere from charges inside it depends only on their total dipole moment, and is exactly

$$\langle \mathbf{E} \rangle = -\frac{1}{4\pi\varepsilon_0}\frac{\mathbf{p}}{R^3}.$$

The result does not rely on the shape of the charge arrangement or on where it sits inside. The corresponding theorem for currents gives

$$\langle \mathbf{B} \rangle = +\frac{\mu_0}{4\pi}\frac{2\mathbf{m}}{R^3}.$$

The signs differ, and the sizes differ by a factor of two. That is a remarkable statement, and a calculation is useful for checking that it has been read correctly.

The loop’s mean is integrated directly from its elliptic-integral field over the sphere, on a grid, for radii from just outside the loop to six times its radius. It gives values between 1.99 and 2.03, the small scatter being the grid’s handling of the rim where the current runs. Between two nested spheres, moreover, the field of a dipole averages to precisely zero, since the dipole’s angular pattern has no spherical average. So the entire mean is contributed by the region near the source, however small that may be, and it is fixed once the moment is. In the limit of a point dipole, the mean over any sphere stays finite while the volume where it arises goes to nothing. That is a delta function:

$$\mathbf{E} = \mathbf{E}_{\text{dipole}} - \frac{\mathbf{p}}{3\varepsilon_0}\delta^3(\mathbf{r}), \qquad \mathbf{B} = \mathbf{B}_{\text{dipole}} + \frac{2\mu_0\mathbf{m}}{3}\delta^3(\mathbf{r}).$$

Every textbook’s dipole formula is therefore missing a term that sits at the origin, and it is a different term for the two kinds. This is what the phrase “point dipole” has been hiding.

Why it would matter only if something could get inside #

For almost all of physics, that term is irrelevant. To feel something located at a point, a probe has to be there, and a compass needle or a test charge cannot sit in the middle of a dipole. A bar magnet or an electric-dipole molecule at any distance acts just as the formula says. The term sounds as if it might be a mathematician’s nicety.

There is one thing in the universe that does get inside. An electron in an atomic s state has a finite probability density at the nucleus, in the ordinary sense: its wave function does not vanish there. A nucleus such as a proton has a size of about a femtometre, and over that volume the electron cloud, some fifty thousand times wider, is uniform. So the electron’s magnetic moment samples the field inside the proton, and it does so with a uniform weighting. That average is exactly what the theorem computes.

Everything else cancels. For a spherical electron cloud, the ordinary dipole interaction between electron and proton averages to zero, because it has the angular pattern of a dipole field, which averages to nothing over a sphere. That leaves only the origin term, as Enrico Fermi saw in 1930 when he worked out the hyperfine structure of the alkali atoms. The energy of an s electron in the field of a nuclear moment is called the Fermi contact interaction, and for a proton it comes out at

$$\Delta E = \frac{4}{3} g_p \frac{m_e}{m_p}\alpha^4 m_e c^2$$

between the states with parallel and opposed spins, using nothing but the proton’s measured $g$-factor and constants of nature. With a reduced-mass correction and the electron’s own slightly anomalous moment included, this comes to 1420.5 MHz.

The measured frequency is 1420.4 MHz, the famous hydrogen line at a wavelength of 21 cm that the line that takes eleven million years described in its astronomical setting. The agreement is good evidence that the calculation is right. It also answers a question nobody set out to ask of it.

Which kind of dipole is a proton? #

A proton’s magnetism might in principle have been produced in two different ways. It might be a tiny current, a loop of circulating charge, which is roughly what its quark content suggests. Or it might be a tiny pair of north and south magnetic charges, a magnetic analogue of an electric dipole. For many years, physicists discussed magnetic matter both ways, and the Gilbert picture of magnetic poles is still common in engineering since it gives correct answers for forces between distant magnets. No magnetic charge has ever been found, but that alone would not settle what an elementary moment is made of.

The interior term settles it. With a pair of magnetic charges, the term at the origin would have the same form as the electric dipole’s: the mean interior field would point against the moment, and it would be half the size of the current loop’s. The energy order would be reversed, with the opposed-spin state above. The figure draws both cases. A pair of magnetic charges predicts hydrogen’s 21 cm transition with the opposite sign and a frequency near 710 MHz. It is the measured size and order of the line that show a nucleon’s moment acts like a current. J. D. Jackson’s textbook makes a point of this argument, attributing the original reasoning to Fermi; a beam of neutrons sent through magnetised iron in the early 1950s provided the same answer by a different route, since scattering from the magnetic field inside the iron depends on $\mathbf{B}$ and not on $\mathbf{H}$, and the difference between those two quantities in a magnet comes from exactly the same term. The field the matter takes away dealt with the electric version of that averaging inside bulk matter.

It is unusual for a physical question like this one to have such a direct test. Where the moment comes from is a question about something a billion times smaller than any microscope can see, and the evidence is a frequency in the radio band, which radio astronomers routinely use to map galaxies.

Only an s electron is inside #

If the term is real, its effect should scale with the electron density at the nucleus and with nothing else about the state. Hydrogen’s s states have $|\psi(0)|^2 \propto 1/n^3$. So the 2s splitting should be precisely an eighth of the 1s, or 177.56 MHz. The measured value, 177.557 MHz, differs by a part in ten thousand, and that small difference is itself a precision test of quantum electrodynamics. The 2p state has a node at the nucleus, and so no contact term at all. Its much smaller hyperfine structure comes from the ordinary dipole interaction at a distance, since a 2p cloud is not spherical and so does not average it to zero.

The same scaling explains why the contact term dominates in heavy atoms. A heavy nucleus pulls its s electrons much closer, raising the density at the nucleus by a large factor. The caesium ground-state hyperfine splitting has been defined as exactly 9,192,631,770 Hz since 1967, and that splitting is the basis of the second. A transition in the caesium atom depends on the spin of the outermost electron, in the s state, and so the definition of the second relies on the same delta function drawn above. The wait that sets the precision shows how a clock interrogates this transition; the transition itself is an interior term. Nuclear magnetic resonance also shows the contact term in everyday chemistry: the shift of a nucleus’s resonance due to an unpaired electron, the Knight shift in metals, and the through-bond couplings of spectra that the image made of frequencies depends on are largely contact effects, passed through the s character of a chemical bond.

How the interior term looks from the other direction #

An alternative argument reaches the same answer from Maxwell’s equations, without any sphere. Imagine a uniformly magnetised ball, with magnetisation $M$ throughout. Its magnetic field inside is uniform, $\tfrac23\mu_0 M$, and points along $M$. Imagine alternatively a uniformly polarised ball with polarisation $P$; its electric field inside is $-P/3\varepsilon_0$ and points against $P$. These are standard exact results. They were developed in connection with the field that points against the magnet it is in, which reaches an uncomfortable-sounding conclusion about demagnetising fields only because it looks at $\mathbf{H}$ rather than $\mathbf{B}$.

Multiply each by the ball’s volume, $\tfrac43\pi R^3$, and the total moment is $M$ or $P$ times that volume. The interior fields give the mean over the ball, and they are just the theorem’s two results with signs and factors unchanged. Since the outside of a uniformly magnetised ball is an exact dipole, the ball is the ideal case: it shows one object with a dipole field outside and a specific uniform field inside, and those two fit together to give the same factor of $-1$ or $+2$. A point dipole is the limit as $R$ goes to zero at fixed moment, and the delta function is nothing else than that uniform interior field, with its fixed integral, shrunk onto a point. That provides a complete check on the signs.

This also makes clear what is geometry and what is physics here. The factor of two and the sign come from Gauss’s law and from the absence of magnetic sources. If magnetic monopoles existed, then the argument would require a magnetic dipole to be classified by what it is made from and not by what it does at a distance, and either result could occur, with experiments distinguishing them.

The limits of what the drawings show #

Every field drawn here is classical. A proton is not literally a loop of current with a radius, nor is it a static object; its moment is a property of quark spins and orbital motion that lattice calculations now get roughly right. The theorem still applies, because all it asks for is a current distribution with no sources. An electron has an interior term of its own but no known internal structure at all, down to the smallest distances measured, and treating it as a current loop with a finite size runs into trouble elsewhere, as the angular momentum that is not a rotation discussed. The two-dimensional cut through the loop in the first figure shows only the fields in one plane, and the loop’s rim is a real singularity in that plane.

The treatment of hydrogen’s hyperfine splitting also makes approximations. It uses non-relativistic wave functions and a proton that is effectively a point. Corrections due to the proton’s actual size and internal structure move the answer by a few tens of parts per million. They are large compared with the precision to which the frequency has been measured, one of the most exact measurements in all of physics, so a theoretical calculation currently stops being useful well before the measurement does.

Still open: what the proton’s interior term says about its structure #

Hydrogen’s hyperfine frequency is known to thirteen significant figures, yet theory cannot match it beyond about six. The limitation is exactly the term discussed here. A proton is not a point: the electron’s cloud sees a spatial spread of charge and magnetism, summed up in a quantity called the Zemach radius, which together with effects of the proton’s excited states sets the leading uncertainty. Measurement of the same structure in muonic hydrogen, in which a muon sits two hundred times closer and feels the proton’s interior correspondingly more strongly, has been pursued in recent years. Where the electron probably is tells the story of how a muonic atom overturned the proton’s charge radius. A comparable measurement of the proton’s magnetic radius, by laser spectroscopy of the muonic hyperfine line, is now in progress, with several collaborations still working to detect the extremely faint resonance. If successful, this will measure how the interior term behaves when the probe is big enough to sense the proton’s actual extent.

A charge pair and a current loop share a field at any distance worth measuring, and differ on the inside by a sign and a factor of two: $-1$ against $+2$, at every radius of averaging. That difference is a term concentrated at the origin, felt only by what can get inside. An s electron can, so hydrogen’s 21 cm line is that term, and its size and level order say that the proton’s magnetism is a current.