In short

Earth's magnetic field, to a first approximation, is the field of a giant bar magnet buried at the centre, tilted roughly 11^\circ away from the rotation axis. Near the surface its strength is about 25–65\ \mu\text{T} — tiny compared with a fridge magnet, but enough to align every compass needle on the planet.

At any point on Earth the field \vec{B} has three numbers that fully specify it:

  • Declination \delta_d — the angle (in the horizontal plane) between magnetic north and true geographic north. It tells you how much a compass lies.
  • Dip (inclination) \delta — the angle the field makes with the horizontal. At the magnetic equator, \delta = 0 (field is horizontal). At the magnetic poles, \delta = 90^\circ (field is vertical).
  • Horizontal component B_H — the part of \vec{B} that lies in the horizontal plane; this is what a compass needle feels.

The geometry gives the three core relations:

\boxed{\;B_H = B\cos\delta, \quad B_V = B\sin\delta, \quad \tan\delta = \frac{B_V}{B_H}\;}

Maps of declination (isogonic lines) and dip (isoclinic lines) cover the globe. India's national geomagnetic observatory at Alibag, run by the Indian Institute of Geomagnetism, has measured B, \delta, and \delta_d continuously for over a hundred years. In April 2026 at Alibag, B \approx 46\ \mu\text{T}, \delta \approx 27^\circ, \delta_d \approx -1^\circ (just west of true north).

Why it matters. Every compass, every migratory bird, every satellite orbit correction (ISRO's GSAT series live in this field), and every power-grid disturbance during a solar storm is governed by the same three angles.

A trekker in the Thar desert near Jaisalmer pulls out a brass-cased compass. The needle wobbles, settles, and points almost north — but not quite. It leans about one degree west of the North Star she can see rising above the dunes. A climber on the Siachen glacier does the same thing and finds the needle leans east of true north. A geologist in Chennai sees almost no tilt at all. Same planet, same instrument, three different answers.

None of them is broken. The compass is telling the truth — just not the truth about geographic north. It is telling the truth about where Earth's magnetic north is, and magnetic north sits in the Canadian Arctic, more than a thousand kilometres from the geographic pole. The angle between where the compass points and where true north actually is changes from place to place, and that angle has a name: declination.

This article is about the three numbers you need to fully describe Earth's magnetic field at any point — declination, dip, and the horizontal component — and the simple geometry that ties them together. By the end you will know what a geomagnetic observatory measures, why a compass needle in Srinagar tips downward but a compass needle in Bengaluru lies nearly flat, and how the Chandrayaan-2 orbiter's magnetometer readings fit the same dipole picture.

Earth is approximately a bar magnet

Drop iron filings around a bar magnet and you see the classic pattern — field lines emerging from the north pole, curving through space, re-entering at the south pole. Now imagine that bar magnet shrunk and placed at the centre of a planet. Extend its field lines out through the crust into space, and you get — to a very good first approximation — Earth's magnetic field.

This picture is called the centred dipole model. It captures about 90% of Earth's real field. The remaining 10% comes from more complicated sources (crustal rocks, ocean currents, the ionosphere) — but for a first understanding, the dipole is everything.

Two facts about this imaginary internal magnet matter:

  1. It is tilted. The axis of the dipole is not aligned with Earth's rotation axis. It is tipped by about 11^\circ. The dipole's "south" pole sticks out near northern Canada, and its "north" pole near Antarctica.

  2. Its south pole is near geographic north. This is the confusing sign convention. A compass needle's north end points toward Earth's geographic north — and the reason is that the compass's north end is attracted to whatever pole of Earth is up there. That pole must therefore be a south magnetic pole (opposite poles attract). So what geographers call the North Magnetic Pole is physically a magnetic south pole. Nobody renames it, because the convention is older than the physics.

Earth as a tilted magnetic dipole A sphere representing Earth with the rotation axis vertical and the magnetic dipole axis tilted 11 degrees from it. Field lines emerge from the southern magnetic pole, curve around the globe, and re-enter at the northern magnetic pole. Geographic and magnetic north poles are labelled separately. rotation axis magnetic axis (11° tilt) true N mag. N (phys. S) mag. S (phys. N) field lines (southward)
The imaginary bar magnet at Earth's centre, tilted about $11^\circ$ from the rotation axis. Field lines emerge from the southern end of the dipole (near the south geographic pole) and curve back in at the northern end (near the north geographic pole). A compass needle's north tip is attracted to the northern magnetic pole — which means that pole is physically a south pole.

How strong is this field? Between 25\ \mu\text{T} near the magnetic equator and about 65\ \mu\text{T} near the poles. Compare that with a typical refrigerator magnet (\sim 5\ \text{mT}) or the field in a hospital MRI scanner (1.5\ \text{T} to 3\ \text{T}). Earth's field is a hundred times weaker than a kitchen magnet and fifty thousand times weaker than an MRI. The fact that it can still line up a steel needle across an iron ore desert is a testament to how sensitive a well-balanced compass is.

Geographic north vs magnetic north — the distinction in one sentence

Geographic north is where Earth's rotation axis meets the surface — the point around which stars appear to rotate. Magnetic north is where a compass needle's north tip would point if you were standing directly on top of it (and it would point straight down). These two points are currently about 500 km apart — and magnetic north is drifting across the Arctic at roughly 50 km per year, fast enough that aviation charts are redrawn every five years.

Geographic poles are fixed by rotation. Magnetic poles are set by complicated fluid motion in Earth's iron core and drift over time. This is why every old ship's chart carries a date — the declination it printed was right for that year and slightly wrong the year after.

Splitting \vec{B} into components

Stand at any point on Earth's surface. Set up a local coordinate system with three axes:

This is the standard geodetic frame. Now look at Earth's magnetic field \vec{B} at this point. It has a total magnitude B, and it points in some direction in 3D. The question is: what is that direction, and how do we describe it?

There are two angles you need. First, the direction the field makes in the horizontal plane. Second, the angle it tips up or down from horizontal.

The horizontal component B_H and the vertical component B_V

Project \vec{B} onto the horizontal plane. Call that projection \vec{B}_H, and its magnitude B_H. Then project \vec{B} onto the vertical axis, magnitude B_V. By the Pythagorean theorem:

B^2 = B_H^2 + B_V^2 \qquad \Rightarrow \qquad B = \sqrt{B_H^2 + B_V^2}

Why: \vec{B}_H and \vec{B}_V are perpendicular by construction (one horizontal, one vertical), so the magnitude of the full vector is the square root of the sum of the squares of the components.

The dip angle \delta (sometimes I for inclination) is the angle \vec{B} makes with the horizontal plane:

\tan\delta = \frac{B_V}{B_H}

From the right triangle formed by B, B_H, and B_V:

\boxed{\;B_H = B\cos\delta, \qquad B_V = B\sin\delta\;}

Why: in the vertical plane containing \vec{B}, you have a right triangle with hypotenuse B and angle \delta measured from the horizontal. The side adjacent to \delta (horizontal) is B\cos\delta; the side opposite (vertical) is B\sin\delta. Standard trigonometry.

At the magnetic equator, the field is entirely horizontal — \delta = 0, B_V = 0, B_H = B. At the magnetic poles, the field is entirely vertical — \delta = 90^\circ, B_H = 0, B_V = B. Everywhere in between, \delta varies smoothly between these limits.

Decomposing Earth's field into horizontal and vertical components A 3D perspective showing the local frame at a point on Earth's surface. North, east, and down axes are drawn. The magnetic field vector B is shown in red tipping downward by the dip angle delta from horizontal, and its horizontal projection B_H is shown. The horizontal projection is then tilted by the declination angle delta_d from true north. geographic N up east B_H (horizontal) geodetic horizontal B (total field) B_V δ δ_d δ = dip (vertical tilt) δ_d = declination (horizontal tilt from N)
At any point on Earth, the full field $\vec{B}$ (red) tips downward from horizontal by the dip angle $\delta$. Its horizontal projection $\vec{B}_H$ (dashed) is itself rotated by the declination $\delta_d$ away from geographic north. Together, $B$, $\delta$, and $\delta_d$ specify the field completely.

Declination \delta_d — the compass-vs-map difference

The horizontal projection \vec{B}_H has its own direction in the horizontal plane. In general it does not point along geographic north. The angle between \vec{B}_H and true north, measured in the horizontal plane, is the declination \delta_d (sometimes written D).

A compass needle aligns itself along \vec{B}_H (because that is the horizontal torque it feels — see the dipole torque derivation). The needle does not care about B_V: the vertical component can pull one end down slightly, but the horizontal rotation is set by \vec{B}_H alone.

So the compass points along \vec{B}_H, not along geographic north. The angle between the two is the declination. A 5° declination east means the compass reads "north" when you are actually facing 5° east of true north.

A worked mental check: the three angles at Alibag

In 2026, the Indian Institute of Geomagnetism's observatory at Alibag (south of Mumbai) records:

From the formulas:

B_H = B\cos\delta = 46 \times \cos 27^\circ = 46 \times 0.891 \approx 41\ \mu\text{T}
B_V = B\sin\delta = 46 \times \sin 27^\circ = 46 \times 0.454 \approx 21\ \mu\text{T}

These are the raw numbers a torsion magnetometer or fluxgate at Alibag reads. The horizontal part (41\ \mu\text{T}) is what the compass needle in Mumbai actually feels. The vertical part (21\ \mu\text{T}) would be felt by a dip needle — a compass needle mounted to rotate in the vertical plane rather than the horizontal one.

Why the dip changes with latitude

If Earth were a pure centred dipole, the dip at magnetic latitude \lambda would obey an exact and beautiful formula:

\boxed{\;\tan\delta = 2\tan\lambda\;}

This is a standard result worth deriving. It shows why the needle stands vertical at the poles and lies horizontal at the equator, and how smoothly it transitions between them.

Step 1. Model Earth as a magnetic dipole of moment \vec{m} at the centre, pointing along the magnetic axis (from southern to northern magnetic pole). At a point on the surface at magnetic latitude \lambda (the angle from the magnetic equator), the position vector \vec{r} makes angle \theta = 90^\circ - \lambda with \vec{m}.

Step 2. The magnetic field of a dipole at distance r has two components in spherical coordinates (with \theta measured from the dipole axis):

B_r = \frac{\mu_0}{4\pi}\,\frac{2m\cos\theta}{r^3}, \qquad B_\theta = \frac{\mu_0}{4\pi}\,\frac{m\sin\theta}{r^3}

Why: these are the standard dipole field formulas (derived in magnetic-dipole-moment by taking the gradient of the dipole potential). B_r points radially (along \hat{r}, outward); B_\theta points along \hat{\theta}, in the direction of increasing \theta — i.e. tangent to a great circle toward the equator.

Step 3. At the surface, the vertical direction is radial: \hat{z} (straight down, into the ground) points along -\hat{r}. The horizontal direction (toward magnetic north, toward the near pole) points along -\hat{\theta}. So:

B_V = -B_r = -\frac{\mu_0}{4\pi}\,\frac{2m\cos\theta}{r^3}
B_H = -B_\theta = -\frac{\mu_0}{4\pi}\,\frac{m\sin\theta}{r^3}

Signs: in the northern magnetic hemisphere (\theta < 90^\circ, \cos\theta > 0), B_r is positive — the radial component points outward — but the field points inward (into the ground), so B_V > 0 means downward. The bookkeeping works.

Step 4. Take the ratio:

\frac{B_V}{B_H} = \frac{2m\cos\theta / r^3}{m\sin\theta / r^3} = \frac{2\cos\theta}{\sin\theta} = 2\cot\theta

Step 5. Substitute \theta = 90^\circ - \lambda, so \cot\theta = \tan\lambda:

\tan\delta = \frac{B_V}{B_H} = 2\tan\lambda

Why: the factor of 2 comes from the dipole field itself — the radial component is twice the tangential component (per unit of \cos\theta vs \sin\theta). This factor is not an accident; it is the same "2" that appears in Gauss's law for a magnetic dipole and in the ratio of axial to equatorial fields of a bar magnet.

Let us tabulate this to see the geography:

Location Magnetic latitude \lambda Dip \delta = \arctan(2\tan\lambda)
Magnetic equator (Trivandrum-ish) 0^\circ 0^\circ — needle flat
Chennai / Bengaluru \sim 3^\circ N mag \sim 6^\circ
Mumbai / Alibag \sim 13^\circ N mag \sim 25^\circ (measured: 27^\circ)
Delhi \sim 22^\circ N mag \sim 39^\circ
Srinagar / Leh \sim 26^\circ N mag \sim 45^\circ
Magnetic pole 90^\circ 90^\circ — needle straight down

This table is why a dip needle in Srinagar tips sharply into the ground while a dip needle in Bengaluru stays almost flat. The dipole physics is doing the work.

The dipole relation $\tan\delta = 2\tan\lambda$. At the magnetic equator the dip is zero; at the poles it reaches $90^\circ$. The factor of 2 makes the curve rise faster than the straight line $\delta = \lambda$ would.

Worked examples

Example 1: A compass in Delhi

A geological survey team in Delhi measures B_H = 32\ \mu\text{T} and B_V = 26\ \mu\text{T} at their site. Find the total intensity B and the dip \delta.

Vector diagram for Example 1 A right triangle with horizontal side labelled B_H equals 32 microteslas, vertical side labelled B_V equals 26 microteslas, and hypotenuse labelled B. The angle at the horizontal vertex is labelled delta. B_H = 32 μT (horizontal, toward mag. N) B_V = 26 μT (down) B (total) δ
The horizontal and vertical components are the legs of a right triangle; the total field is the hypotenuse, and the dip is the angle at the horizontal vertex.

Step 1. Compute the total intensity.

B = \sqrt{B_H^2 + B_V^2} = \sqrt{32^2 + 26^2} = \sqrt{1024 + 676} = \sqrt{1700} \approx 41.2\ \mu\text{T}

Why: B_H and B_V are perpendicular, so the full magnitude is their Pythagorean sum.

Step 2. Compute the dip.

\tan\delta = \frac{B_V}{B_H} = \frac{26}{32} = 0.8125
\delta = \arctan(0.8125) \approx 39.1^\circ

Why: the dip is the angle the total vector makes with the horizontal. Its tangent is the ratio of vertical to horizontal components — the standard definition from the right triangle.

Step 3. Sanity-check with the dipole formula.

Delhi's magnetic latitude is about 22^\circ. The dipole model predicts

\tan\delta = 2\tan 22^\circ = 2 \times 0.404 = 0.808, \quad \delta \approx 38.9^\circ.

The measured 39.1^\circ agrees with the pure-dipole prediction to within 0.2^\circ — a striking confirmation that the centred dipole captures almost all of the real field.

Result: B \approx 41.2\ \mu\text{T}, \delta \approx 39^\circ.

What this shows: two numbers a geologist's fluxgate instrument reads (the components) convert directly to the two angles the dipole model predicts. Observation and theory meet.

Example 2: Flight-plan declination over Alibag

A small aircraft departs Mumbai's Juhu airstrip to fly due magnetic north — the pilot holds the compass reading on 0^\circ throughout. At Alibag the current declination is \delta_d = -1^\circ (compass points 1^\circ west of true north). The cruise is 200\ \text{km}. How far west of the intended true-north endpoint does the plane actually land?

Compass vs true-north heading for Example 2 A plane starts at the lower left. Two vectors extend from the start: one labelled true north going straight up, and one labelled magnetic north tilted slightly to the left by 1 degree. After 200 km the magnetic-north vector ends to the west of the true-north vector by about 3.5 km. start true N (intended) mag. N (flown) ≈ 3.5 km west 1°
Over 200 km at $1^\circ$ declination west, the sideways drift is about $200 \sin 1^\circ \approx 3.5$ km.

Step 1. Translate the declination. The plane's compass reads 0^\circ (magnetic north), but magnetic north is tilted 1^\circ west of true north. So the plane's actual heading is 1^\circ west of true north throughout the flight.

Step 2. Decompose the flight path into true-N and true-W components after L = 200\ \text{km}:

\text{true-north distance} = L\cos\delta_d = 200\cos 1^\circ \approx 199.97\ \text{km}
\text{true-west distance} = L\sin\delta_d = 200\sin 1^\circ \approx 3.49\ \text{km}

Why: the compass holds the plane along \vec{B}_H, which is the magnetic-north direction. Geometrically, this is a straight line tilted 1^\circ west of true north. Project it onto the true frame and you get the two components above.

Step 3. Interpret.

After 200\ \text{km}, the aircraft lands about 3.5 km west of the point it would have reached flying due true north. For a regional flight this is a minor drift; for a trans-Atlantic flight (where declinations can exceed 20^\circ) ignoring the correction would put you in the wrong country.

Result: horizontal drift \approx 3.5\ \text{km} west; along-track distance essentially unchanged.

What this shows: declination is not an abstract footnote — it is the difference between where your compass says you are going and where you actually end up. Every aeronautical chart in the world carries declination contours (see below) for exactly this reason.

Example 3: A dip circle near the magnetic equator

A dip-circle demonstration at IIT Madras (magnetic latitude \approx 3^\circ N) gives \delta = 6.2^\circ. The same instrument in Srinagar (magnetic latitude \approx 26^\circ N) gives \delta = 44.3^\circ. Verify that both readings are consistent with the centred-dipole model.

Dip needles at low and high magnetic latitude Left: a dip needle near the magnetic equator tilted 6 degrees below horizontal. Right: a dip needle at latitude 26 degrees tilted 44 degrees below horizontal. horizon (Chennai) δ ≈ 6° horizon (Srinagar) δ ≈ 44°
Same instrument, same physics, wildly different dip angles. The dipole formula $\tan\delta = 2\tan\lambda$ explains both readings.

Step 1. Apply \tan\delta = 2\tan\lambda at Chennai.

\tan\delta = 2\tan 3^\circ = 2 \times 0.0524 = 0.1048
\delta = \arctan(0.1048) \approx 5.99^\circ

Measured: 6.2^\circ. Agreement: within 0.2^\circ.

Step 2. Apply the same formula at Srinagar.

\tan\delta = 2\tan 26^\circ = 2 \times 0.4877 = 0.9754
\delta = \arctan(0.9754) \approx 44.3^\circ

Measured: 44.3^\circ. Agreement: exact to three figures.

Why: the centred-dipole formula assumes a perfect dipole at Earth's centre and nothing else. The fact that two locations thousands of kilometres apart agree to within a fraction of a degree is not luck — it is the dipole actually being a very good model.

Step 3. Interpret the residuals.

Real Earth has about 10% non-dipole field (crustal anomalies, core flux patches). That 10% shows up as small corrections at the level of a degree or so in dip — consistent with what we see. A big residual would flag an unusual iron-ore deposit under the instrument.

Result: both measurements match the dipole prediction to \sim 0.2^\circ.

What this shows: a single formula with a single parameter (magnetic latitude) predicts compass behaviour across the whole subcontinent. This is what physicists mean when they call the dipole "a good first approximation" — it is very good.

Mapping Earth's field — isogonic and isoclinic lines

Every point on Earth's surface has its own (B, \delta, \delta_d). Mapmakers connect points with the same value of each:

These maps are compiled every five years into the World Magnetic Model (WMM) and the International Geomagnetic Reference Field (IGRF), both updated in 2025 with data from the ESA Swarm satellites and from ground observatories — including Alibag, one of the oldest continuously-running geomagnetic stations in the world (recording since 1904). Other Indian observatories contributing to the IGRF: Hyderabad, Pondicherry, Nagpur, Shillong, Tirunelveli — a chain of stations operated by the Indian Institute of Geomagnetism.

Isoclinic lines over India A stylized map of India with dashed lines showing constant dip angles. The magnetic equator passes south of Cape Comorin. Lines of 10 degrees, 20 degrees, 30 degrees, 40 degrees and 50 degrees dip sweep upward across the subcontinent. Observatory locations for Alibag, Hyderabad, Shillong, Nagpur are marked. India (approximate outline) δ = 10° 20° 30° 40° 50° Alibag Hyderabad Nagpur Shillong Delhi
A schematic isoclinic map over India. Dashed lines connect points of equal dip. The dip increases smoothly from about $10^\circ$ near the south coast to about $50^\circ$ in the north. Indian Institute of Geomagnetism observatories (Alibag, Hyderabad, Nagpur, Shillong, and others) anchor these maps with ground-truth measurements.

Satellites such as ESA's Swarm trio, and India's own Chandrayaan-2 orbiter (which carries a magnetometer and has been mapping the Moon's residual crustal magnetism since 2019), extend the same technique beyond Earth. The numbers are smaller — the Moon's surface field is about 100\ \text{nT}, 500 times weaker than Earth's — but the three-number description (B, \delta, \delta_d appropriately redefined) still works.

Common confusions

If you came here to understand compass behaviour and the three-angle description, you have everything. What follows is the axial-dipole derivation in full, the secular variation story, and the link to the geodynamo.

Deriving the axial-dipole field from scratch

The formula \tan\delta = 2\tan\lambda depends on the dipole field having a particular shape. Here is the derivation from first principles, starting from the magnetic scalar potential of a point dipole.

Step 1. The magnetic scalar potential of a dipole of moment \vec{m} at the origin, evaluated at point \vec{r}, is:

V_m(\vec{r}) = \frac{\mu_0}{4\pi}\,\frac{\vec{m}\cdot\hat{r}}{r^2} = \frac{\mu_0}{4\pi}\,\frac{m\cos\theta}{r^2}

where \theta is the angle between \vec{m} and \vec{r}. Why: this is the magnetic analogue of the electric-dipole potential V = p\cos\theta / (4\pi\epsilon_0 r^2). It is derived by superposing the fields of two opposite poles separated by a small distance; see magnetic-dipole-moment.

Step 2. In source-free regions (outside the dipole itself), \vec{B} = -\mu_0 \nabla V_m / \mu_0 = -\nabla V_m in units where the \mu_0 is folded into V_m. In spherical coordinates:

B_r = -\frac{\partial V_m}{\partial r}, \qquad B_\theta = -\frac{1}{r}\frac{\partial V_m}{\partial\theta}

Step 3. Compute B_r:

B_r = -\frac{\partial}{\partial r}\left[\frac{\mu_0 m \cos\theta}{4\pi r^2}\right] = -\frac{\mu_0 m\cos\theta}{4\pi}\cdot\left(-\frac{2}{r^3}\right) = \frac{\mu_0}{4\pi}\,\frac{2m\cos\theta}{r^3}

Step 4. Compute B_\theta:

B_\theta = -\frac{1}{r}\cdot\frac{\partial}{\partial\theta}\left[\frac{\mu_0 m\cos\theta}{4\pi r^2}\right] = -\frac{1}{r}\cdot\frac{\mu_0 m(-\sin\theta)}{4\pi r^2} = \frac{\mu_0}{4\pi}\,\frac{m\sin\theta}{r^3}

Why: the derivative of \cos\theta is -\sin\theta; the two minus signs cancel. The factor 1/r from the gradient-in-spherical-coordinates formula combines with 1/r^2 from V_m to give 1/r^3.

Step 5. Ratio:

\frac{B_r}{B_\theta} = \frac{2m\cos\theta}{m\sin\theta} = \frac{2\cos\theta}{\sin\theta} = 2\cot\theta

Step 6. Identify B_r as the vertical component at the surface (outward radial = upward; for a dipole oriented with \vec{m} pointing from Earth's north-magnetic-south-physical pole toward its north-magnetic-north-physical end, the surface field in the northern hemisphere points into the ground, so B_V = -B_r in magnitude terms; the sign keeps bookkeeping straight). Identify B_\theta as the horizontal component pointing toward the pole. Then B_V/B_H = 2\cot\theta, and with \theta = 90^\circ - \lambda:

\tan\delta = 2\tan\lambda. \qquad \square

The derivation shows that the factor of 2 is structural — it comes from the radial derivative of 1/r^2 (which gives -2/r^3) versus the angular derivative of \cos\theta (which gives -\sin\theta). It is not a tunable parameter.

Secular variation and the geodynamo

Earth's dipole moment today is about m \approx 8 \times 10^{22}\ \text{A}\cdot\text{m}^2. It has been decreasing at about 0.05\% per year for the past two centuries. Over 2000 years, that adds up to roughly 10%. Paleomagnetic records from cooled lavas and sedimentary rocks show the moment has fluctuated between about 4 and 12 \times 10^{22}\ \text{A}\cdot\text{m}^2 over the past few million years — and has reversed direction roughly every 200,000–500,000 years on average, though with enormous variability (some "superchrons" last tens of millions of years without a flip).

The mechanism is the geodynamo: convection in the liquid outer core (molten iron-nickel, temperature \sim 5000 K, conductivity \sim 10^6\ \text{S/m}) couples with the Coriolis effect from Earth's rotation to sustain a self-generating magnetic field. The equations governing this — magnetohydrodynamics in a rotating shell — have no closed-form solutions; modern understanding comes from supercomputer simulations (notably the Glatzmaier–Roberts dynamo model of the 1990s, which produced the first numerical reversal).

Two consequences for you:

  1. The dipole axis drifts: the north magnetic pole is currently moving across the Arctic Ocean at about 50\ \text{km} per year, faster than at any time on record. This is why declination maps expire.

  2. The field is not purely dipolar. A spherical-harmonic expansion of the real field has a dominant dipole term, a smaller quadrupole, an even smaller octupole, and so on. The non-dipolar parts move on decade timescales (driven by core flow patterns) and are what produce regional anomalies like the South Atlantic Anomaly — a patch of unusually weak field off the coast of Brazil that is a hazard for low-Earth-orbit satellites passing over it.

Measuring the field — the instruments

A modern fluxgate magnetometer (the workhorse at Alibag, and the kind flown on Chandrayaan-2) reads all three components of \vec{B} to a precision of \sim 0.1\ \text{nT} — that is, one part in 500{,}000 of the total field. It works by driving a high-permeability core into saturation with an AC current; Earth's field biases the saturation asymmetry, which shows up as an even-harmonic signal proportional to B.

A proton precession magnetometer gives total intensity |B| to similar precision by measuring the Larmor frequency \omega_L = \gamma B of hydrogen nuclei in a water sample (gyromagnetic ratio \gamma \approx 2.675 \times 10^8\ \text{rad/s/T}). At B = 46\ \mu\text{T}, \omega_L / 2\pi \approx 1960\ \text{Hz} — a clean audio-frequency signal.

Putting these instruments together at a network of observatories — and cross-checking with satellites — is how the (B, \delta, \delta_d) numbers on every aeronautical chart are kept honest.

Where this leads next