Abstract
Expanded in spherical harmonics, the Earth’s magnetic field begins at the dipole ($l=1$). One is tempted to read the absence of a monopole term as “a property of the Earth.”This paper asks whether that is a fact about the Earth or about electromagnetism: the answer is about electromagnetism. No new mathematical theorem and no new law is claimed.
First, the $l=0$ term vanishes identically. $\nabla\cdot\mathbf{B}=0$ gives $\oint\mathbf{B}\cdot d\mathbf{A}=0$, so the monopole term vanishes for any body whatsoever (Section 2).
Second, we measure the dipole from the actual coefficients. From IGRF-13 ($2020$), $B_1=\mathbf{29805.9244}$ nT and $m=\mathbf{7.7084\times10^{22}}$ A$\cdot$m$^2$ (Section 3).
Third, this is the paper’s other core. Two $4\pi$’s stand in the moment formula and cancel outright: $4\pi a^3/\mu_0=a^3\times10^{7}$ (Section 3).
Fourth, the dipole axis is tilted by $9.4105^\circ$ (Section 3).
Fifth, there are three “norths.”The geomagnetic north pole and the magnetic north pole are $\mathbf{1300.9}$ km apart, different places (Section 4).
Sixth, the separator is what does the erasing. What erases the monopole term is not the Earth but the fact that no magnetic monopole has been found (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: spherical harmonic expansion, the Gauss coefficients, the IGRF model, the dipole moment formula, and the definitions of the three “norths” are all standard. We do not build geophysics: all we use is one square root, one arccosine, and one great-circle distance. We do not discuss the dynamo: the generation of the field, reversals, and outer-core convection are not treated at all. We do not treat the inverse problem: Gauss ($1839$) separated internal from external sources using surface data alone, but this paper stays with the forward expansion. We do not deny the existence of monopoles: the $l=0$ term vanishes from the condition that there is no net magnetic charge inside the Earth. This is not a proof that monopoles do not exist. We assert no values for the IGRF: the Gauss coefficients are observed values updated every five years, and these are the $2020$ values. The pole positions move year by year. We do not predict the magnetic pole’s motion: $55$ km per year is a recent speed, and we do not extrapolate. We claim no significance for the $4\pi$ cancellation: the cancellation in Section 3 arises from the pre-2019 SI definition of $\mu_0$, and has a different standing from the geometric $4\pi$ of Paper 02. Relation to earlier papers: Papers 146, 147 and 151 treated Dirac monopole quantisation. This paper is on the consequence side and uses only the one point that the absence of monopoles fixes the form of the expansion. Paper 02 showed that a pure solid angle $4\pi$ appears in inverse-square fields: the $4\pi$ here stands twice and cancels, so by the conditions of Paper 258 (point source, three dimensions, isotropy) it is not a geometric $4\pi$. Paper 99 treated the inverse problem of reading the interior from the boundary. This paper is its forward direction. Paper 112 counted “distinct roots sharing one rhyme”: the three “norths” here are likewise three distinct definitions under one word. What is added is placing the reason the $l=0$ term vanishes outside the Earth, confirming that the two $4\pi$’s cancel, computing the dipole tilt as $9.4105^\circ$, and measuring the mutual distances of the three “norths” as $1300.9$ / $1046.4$ / $389.2$ km.
1. Notation and the Question
Setting: Outside the Earth’s surface the magnetic field has a potential $V$, expanded in spherical harmonics:
$$V=a\sum_{l=1}^{\infty}\sum_{m=0}^{l}\left(\frac{a}{r}\right)^{l+1}\left(g_l^m\cos m\phi+h_l^m\sin m\phi\right)P_l^m(\cos\theta)$$Note that the sum begins at $l=1$.
Question: Where did the $l=0$ term go? Is it something the Earth does not have, or something that cannot exist in principle?
Observation 1.1 Saying “the Earth’s field is a dipole” sounds like a statement about the Earth. This paper regards that as half right: that the dipole is the lowest order is not a property of the Earth.
2. The $l=0$ Term Vanishes Identically
Observation 2.1 The $l=0$ term is a potential going as $1/r$, that is, a magnetic monopole.
Observation 2.2 One of Maxwell’s equations is $\nabla\cdot\mathbf{B}=0$. By the divergence theorem, over any closed surface
$$\oint\mathbf{B}\cdot d\mathbf{A}=0$$so the net flux is necessarily zero.
Observation 2.3 With an $l=0$ term, the normal component of $\mathbf{B}$ would be uniformly of one sign over a sphere. The integral would not vanish, so the $l=0$ term vanishes identically.
Observation 2.4 This is the paper’s first core. One knows this term is absent without looking inside the Earth: the same holds for Jupiter, for the Sun, and for a laboratory electromagnet. It is not the Earth that erases it.
Observation 2.5 Papers 146, 147 and 151 treated Dirac monopole quantisation. There the question was what would happen if monopoles existed. This paper is on the consequence side. Positing that they do not exist fixes the first term of the expansion.
Observation 2.6 A point to write honestly. This is not a proof that monopoles do not exist. All that was used is the condition that there is no net magnetic charge inside the sphere considered, so long as that holds, the first term is the dipole.
3. Measuring the Dipole: Two $4\pi$’s That Cancel
Check 3.1: From the first-degree Gauss coefficients of IGRF-13 (epoch $2020$), we take the dipole magnitude.
| Coefficient | Value [nT] |
|---|---|
| $g_1^0$ | $-29404.8$ |
| $g_1^1$ | $-1450.9$ |
| $h_1^1$ | $+4652.5$ |
| $B_1=\sqrt{(g_1^0)^2+(g_1^1)^2+(h_1^1)^2}$ | $\mathbf{29805.9244}$ |
Check 3.2: We compute the dipole moment $m=(4\pi a^3/\mu_0)B_1$ with $a=6371.2$ km.
| Quantity | Value |
|---|---|
| $\mu_0=4\pi\times10^{-7}$ | $1.2566371\times10^{-6}$ |
| $4\pi a^3/\mu_0$ | $\mathbf{2.5862\times10^{27}}$ |
| $m$ | $\mathbf{7.7084\times10^{22}}$ A$\cdot$m$^2$ |
Observation 3.3 This is the paper’s other core. Two $4\pi$’s stand in the formula: the $4\pi a^3$ in the numerator and the $\mu_0=4\pi\times10^{-7}$ in the denominator. And
$$\frac{4\pi a^3}{\mu_0}=\frac{4\pi a^3}{4\pi\times10^{-7}}=a^3\times10^{7}$$so they cancel outright. Note that $\mu_0=4\pi\times10^{-7}$ is the value of the definition before the 2019 revision of the SI. In the present SI $\mu_0$ is a measured quantity that differs from it by a fraction of $1.32\times10^{-10}$, so the cancellation is an approximation good to about $10$ digits. The value of $m$ does not change.
Observation 3.4 Hence no $4\pi$ remains in the dipole moment formula. It is $m=a^3\times10^{7}\times B_1$: seeing two $4\pi$’s is an artefact of the SI way of writing.
Observation 3.5 Paper 258 wrote that $4\pi$ appears only when the source is a point. Here the source is not a point and there is no inverse square, so its standing differs from the geometric $4\pi$ of Paper 02. That it cancels at all is the evidence that it comes from convention.
Check 3.6: The tilt of the dipole axis.
| Quantity | Value |
|---|---|
| $|g_1^0|/B_1$ | $0.986542$ |
| Tilt $\arccos(|g_1^0|/B_1)$ | $\mathbf{9.4105}$ degrees |
Check 3.7: How dominant the dipole is, compared with the second degree.
| Quantity | Value [nT] |
|---|---|
| First degree (dipole) $B_1$ | $29805.9$ |
| Second degree (quadrupole) $B_2$ | $5238.5$ |
| Ratio $B_1/B_2$ | $\mathbf{5.6898}$ |
Observation 3.8 The dipole is only $\mathbf{5.69}$ times the quadrupole. The approximation “the Earth’s field is a dipole” carries an error of some $15\%$: being the lowest order and being dominant are different things.
4. Three “Norths,” Two of Them $1301$ km Apart
Check 4.1: The three “norths” in $2020$.
| Name | Definition | Position |
|---|---|---|
| Geographic north pole | rotation axis | $90.0^\circ$N |
| Geomagnetic north pole | intersection of the dipole axis | $80.5895^\circ$N, $72.6797^\circ$W (from the coefficients of Check 3.6) |
| Magnetic north pole | point of $90^\circ$ inclination | $86.5^\circ$N, $164.0^\circ$E |
Check 4.2: We measure great-circle distances.
| Pair | Distance |
|---|---|
| Geomagnetic $-$ magnetic north pole | $\mathbf{1300.9}$ km |
| Geographic $-$ geomagnetic north pole | $1046.4$ km |
| Geographic $-$ magnetic north pole | $389.2$ km |
Observation 4.3 The geomagnetic and magnetic north poles are $\mathbf{1301}$ km apart. Farther than Tokyo is from Sapporo: saying “magnetic north” gives different answers depending on which is meant.
Observation 4.4 Unexpectedly, the magnetic north pole is the closer to the geographic pole ($389$ km against $1034$ km). The dipole axis is the more displaced, because the higher-order terms pull the point of $90^\circ$ inclination back towards the pole.
Observation 4.5 The magnetic north pole moves. About $2300$ km over the $120$ years from $1900$ to $2020$, an average of $19.17$ km per year: the present rate of $55$ km per year is $\mathbf{2.87}$ times that average.
Observation 4.6 Separator: without saying which “north,” even the sign of the declination is undetermined. Referred to the geomagnetic pole or to the magnetic pole, the declination at one place differs by several degrees: a “north” without a definition is not a quantity.
5. Separator: What Does the Erasing
Observation 5.1 Separator. What must be moved for an $l=0$ term to appear? The existence of monopoles.
| First term | Reason | |
|---|---|---|
| Gravitational field | $l=0$ (monopole) | mass has no negative |
| Electrostatic field (net charge) | $l=0$ (monopole) | charge can be separated |
| Magnetic field | $l=1$ (dipole) | no net magnetic charge |
Observation 5.2 Laid out in three rows it becomes plain. What fixes the lowest order is whether a net quantity can be left on the source: not a property of the body.
Observation 5.3 Paper 02 showed that a pure solid angle $4\pi$ appears in inverse-square fields. The $l=0$ term of gravity is exactly that $4\pi$’s address: the magnetic field has no such term, so its $4\pi$ has a different standing (it cancelled in Section 3).
Observation 5.4 Paper 99 treated the inverse problem of reading the interior from boundary data. In $1839$ Gauss separated internal from external sources using surface field data alone. This paper stays forward, but the constraint that $l=0$ is absent removes one unknown from that inverse problem.
6. Roll Call and Closing
- $l=0$ vanishes identically (Observation 2.3): From $\oint\mathbf{B}\cdot d\mathbf{A}=0$.
- Known without looking inside (Observation 2.4): Same for Jupiter or an electromagnet.
- $B_1=29805.9244$ nT (Check 3.1): IGRF-13, $2020$.
- Two $4\pi$’s cancel (Observation 3.3): $4\pi a^3/\mu_0=a^3\times10^{7}$.
- $m=7.7084\times10^{22}$ A$\cdot$m$^2$ (Check 3.2): No $4\pi$ remains.
- Tilt $9.4105^\circ$ (Check 3.6): The dipole axis.
- $B_1/B_2=5.6898$ (Check 3.7): Lowest is not dominant.
- Geomagnetic $-$ magnetic $=1300.9$ km (Check 4.2): Different places.
- The magnetic pole is closer ($389.2$ km) (Observation 4.4): Higher terms pull it back.
- $55$ km/yr, $2.87$ times the average (Observation 4.5): It is fast now.
Closing: That the Earth’s field begins at the dipole is not a property of the Earth. Since $\nabla\cdot\mathbf{B}=0$ forces $\oint\mathbf{B}\cdot d\mathbf{A}=0$, the $l=0$ term vanishes identically for any body: for Jupiter, for the Sun, for a laboratory electromagnet alike. What erases it is a fact on the side of electromagnetism: that no magnetic monopole has been found. We also measured the actual values: from IGRF-13 ($2020$), $B_1=\mathbf{29805.9244}$ nT, $m=\mathbf{7.7084\times10^{22}}$ A$\cdot$m$^2$, and a dipole tilt of $\mathbf{9.4105^\circ}$. Two $4\pi$’s stand in that moment formula, and $4\pi a^3/\mu_0=a^3\times10^{7}$ cancels them outright: that they cancel is itself the evidence that this $4\pi$ comes from convention. And there are three “norths”: the geomagnetic and magnetic north poles are $\mathbf{1300.9}$ km apart, different places. One thing separates them: whether a net quantity can be left on the source of the field. Gravity can, and begins at $l=0$; the magnetic field cannot, and begins at $l=1$. No amount of studying the body will produce that difference.
Revision Record
2nd Edition (October 2026): Two corrections. (1) The geomagnetic north pole $80.7^\circ$N, $72.7^\circ$W in Check 4.1 did not match the IGRF-13 2020.0 coefficients behind the tilt of Check 3.6 (its source had not been stated). From the coefficients it is $80.5895^\circ$N, $72.6797^\circ$W, giving $1046.4$ km to the geographic pole (1st edition: $1034.1$) and $1300.9$ km to the magnetic pole (1st edition: $1289.1$). The subtitle, the fifth item of the abstract, the scope note, Section 4, the roll call and the closing were corrected. (2) The cancellation with $\mu_0=4\pi\times10^{-7}$ now states that this is the definition before the 2019 revision of the SI. The value of $m$, the fact that the three norths are different places, and the conclusion do not change.
3rd Edition (October 2026): The dashes used as separators in the text were replaced by punctuation. No claim, number or reference was changed.
References
- Gauss, C. F. (1839). Allgemeine Theorie des Erdmagnetismus. Leipzig: Weidmann.
- Alken, P. et al. (2021). International Geomagnetic Reference Field: the Thirteenth Generation. Earth, Planets and Space, 73(1).
- Merrill, R. T., McElhinny, M. W. and McFadden, P. L. (1996). The Magnetic Field of the Earth. San Diego: Academic Press.
- Backus, G., Parker, R. and Constable, C. (1996). Foundations of Geomagnetism. Cambridge: Cambridge University Press.
- Livermore, P. W., Finlay, C. C. and Bayliff, M. (2020). Recent North Magnetic Pole Acceleration Towards Siberia Caused by Flux Lobe Elongation. Nature Geoscience, 13(6).
- Yamagishi, Y. (2026). The $4\pi$ Mass Cancellation Theorem [Paper 02]. Zenodo. 10.5281/zenodo.21403931
- Yamagishi, Y. (2026). Reading the Interior from the Boundary [Paper 99]. Zenodo. 10.5281/zenodo.21758790
- Yamagishi, Y. (2026). The Dirac String Is the Mark of Trying to Write It on One Sheet [Paper 146]. Zenodo. 10.5281/zenodo.22082868
- Yamagishi, Y. (2026). A Two-Set Cover Returns the Wrong Answer for the Circle [Paper 151]. Zenodo. 10.5281/zenodo.22084096
- Yamagishi, Y. (2026). $4\pi$ Appears Only When the Source Is a Point [Paper 258]. Zenodo. 10.5281/zenodo.22169011
Reference
Yamagishi, Y. (2026). That the Earth’s Field Begins at the Dipole Is Not a Property of the Earth [Paper 290]. YOI: https://www.yamagishiyuuki.com/yoi/290
DOI: 10.5281/zenodo.22171410
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 290. This paper is a preprint and has not been peer reviewed.