Magnetic Inclination and Intensity

The /learn/magnetic-declination-explained pillar focused on declination — the horizontal direction of the field. But the field is a three-dimensional vector with two other observable quantities. This article covers inclination (magnetic dip) and intensity, the geometric relationships, the mapping conventions, and why compasses have to be balanced for the latitude where they'll be used.

Companion to the pillar and to /learn/the-world-magnetic-model.

The full field vector

Earth's magnetic field at a location is a vector with both direction and magnitude. Geomagneticists describe it in several equivalent representations.

Cartesian (X, Y, Z)

The standard navigation frame:

• X = northward component (positive toward true north)
• Y = eastward component (positive toward east)
• Z = downward component (positive into the ground)

These three orthogonal components fully specify the vector. WMM and IGRF coefficient files output X, Y, Z for any latitude / longitude / altitude / date.

Spherical (F, I, D)

The navigation-friendly representation:

• F = total intensity (magnitude), always positive
• I = inclination (dip), positive below horizontal
• D = declination, positive east of true north

Relationships:

F = √(X² + Y² + Z²)
H = √(X² + Y²)        (horizontal magnitude)
I = atan2(Z, H)
D = atan2(Y, X)

So:

X = F · cos(I) · cos(D)
Y = F · cos(I) · sin(D)
Z = F · sin(I)
H = F · cos(I)

These conversions are straightforward but easy to get backward; the standard reference is the WMM technical documentation, which fixes the convention.

Cylindrical (H, D, Z)

A hybrid sometimes used:

• H = horizontal magnitude
• D = declination (the horizontal direction)
• Z = vertical component

The horizontal pair (H, D) describes what a compass measures; Z describes the dip component separately.

Inclination (magnetic dip)

Inclination (I), also called magnetic dip, is the angle of the field below horizontal at a location.

The geometry:

• At the magnetic equator (aclinic line): I = 0°. The field is exactly horizontal; a freely suspended needle stays level.
• In the Northern Hemisphere: I > 0°. The field tilts downward; the north end of a freely suspended needle dips below horizontal.
• At the Magnetic North Pole: I = +90°. The field is exactly vertical (pointing down); a compass needle stands on end.
• In the Southern Hemisphere: I < 0°. The field tilts downward on the south end of a needle.
• At the Magnetic South Pole: I = -90°. Field exactly vertical (pointing up).

Typical inclination values at well-known cities (from WMM2025, approximate):

| City | Inclination | | ------------------------ | ----------- | | Reykjavík, Iceland | +75° | | Anchorage, Alaska | +72° | | London, UK | +66° | | New York, USA | +66° | | Beijing, China | +58° | | Honolulu, USA | +37° | | Cairo, Egypt | +43° | | Quito, Ecuador | +9° | | Singapore | -15° | | Sydney, Australia | -63° | | Cape Town, South Africa | -65° | | McMurdo Station, Antarctica | -82° |

The inclination is roughly the magnetic latitude — for a perfect dipole field, tan(I) = 2 tan(magnetic latitude). Earth's non-dipole structure perturbs this relationship somewhat.

The aclinic line

The aclinic line (or magnetic equator) is the locus of points where I = 0 — the field is exactly horizontal.

The aclinic line does not follow the geographic equator. It's a wavy line that crosses the geographic equator multiple times. Currently:

• Crosses the geographic equator around 30°W (in the Atlantic, off Brazil)
• Dips south to ~5°S in central Africa
• Recrosses around 40°E (in Africa)
• Dips south to ~7°S in India
• Crosses the geographic equator near Sumatra
• Dips south through Indonesia
• Returns north through the Pacific to cross again near the dateline.

The aclinic line moves with the rest of the field. It has shifted noticeably since the mid-20th century.

The aclinic line is important for compass operation: compasses with horizontal needles work best near the aclinic line because there's no dip to fight against. Away from the aclinic line, the needle wants to tilt; a counterweight is needed to keep it level.

Intensity

Intensity (F) is the magnitude of the field vector in nanoteslas (nT). The SI unit is the tesla (T): 1 T = 1 V·s/m² = 1 N·A⁻¹·m⁻¹. Earth's field is so weak that nanoteslas are the practical unit.

Surface intensity ranges:

• Magnetic equator (aclinic line): ~25,000 nT
• Mid-latitudes: ~40,000–55,000 nT
• Magnetic poles: ~60,000–65,000 nT
• South Atlantic Anomaly center: ~22,000 nT (anomalously weak)
• Maximum recorded: ~67,000 nT at high northern latitudes

Intensity follows roughly the dipole formula:

F ≈ F_equator · √(1 + 3 sin²(magnetic latitude))

The factor √(1+3) = 2 explains why the poles are about twice as strong as the equator.

Comparison with familiar magnets

| Source | Approximate intensity | Notes | | ------ | --------------------- | ----- | | Earth's field (mid-latitude) | 50,000 nT | What a compass measures | | Refrigerator magnet | 5,000,000 nT (50 gauss) | Localized; falls off quickly | | Cow magnet (cattle medication) | 50,000,000 nT (500 gauss) | Localized; very strong | | MRI scanner | 1,500,000,000 nT (1.5 T) to 7,000,000,000 nT (7 T) | Lab-grade | | Strongest pulsed magnet | 100,000,000,000 nT (100 T) | National High Magnetic Field Lab | | Neutron-star surface | 10¹⁶ to 10¹⁹ nT (10⁸ to 10¹¹ T) | Among the strongest known fields |

Earth's field is weak by laboratory standards but spans the planet uniformly. Its weakness is why magnetic anomalies (a few nT) can be detected; a strong background field would mask them.

Historical units

The historical CGS unit for magnetic field strength is the gauss (G), named after Carl Friedrich Gauss. 1 gauss = 100,000 nT = 10⁻⁴ tesla. Earth's surface field is 0.25 to 0.65 gauss.

In older geomagnetism literature, the gamma (γ) is sometimes used: 1 γ = 1 nT. Identical to nanotesla; the gamma symbol was dropped in modern SI conventions.

Modern geomagnetism uses nT universally; older papers may use gauss or gamma. Conversions:

1 T = 10⁴ G = 10⁹ nT = 10⁹ γ

The dip needle

A dip needle is a compass needle pivoted on a horizontal axis so it swings in a vertical plane. The angle of the needle below horizontal is the inclination.

The first dip needle was constructed by Robert Norman, an English compass maker, around 1576. Norman noticed that his newly magnetized needles consistently tilted downward, and he investigated by suspending one so it could swing both horizontally and vertically. He measured the dip in London at about 71° (consistent with the value WMM2025 gives for London latitude today — ~66°, the difference reflecting secular drift over 450 years).

Norman published his findings in 1581 in The Newe Attractive. The discovery contributed to William Gilbert's 1600 De Magnete, which used Norman's measurements to argue that Earth itself is a giant magnet.

The dip needle remained an important geomagnetism instrument through the 19th and early 20th centuries. Modern electronic magnetometers (fluxgate, proton precession, optical pumping) measure all three field components directly with arcsecond precision, superseding the dip needle for scientific use. Dip needles persist as demonstration instruments and in some teaching laboratories.

Mapping conventions

Isoclinic lines

Isoclinic lines are lines of constant inclination, mapped on geomagnetic charts. They're roughly parallel to the aclinic line (magnetic equator) but distorted by the non-dipole structure of the field.

Conventional plotting: contours every 5° or 10° of inclination. The aclinic line (I = 0) is the “magnetic equator”; the +75° isoclinic line runs through Iceland, central Russia, and northern Canada; the -75° line runs through Antarctica and southern Indian Ocean.

Isodynamic lines

Isodynamic lines are lines of constant total intensity. Plotted every 5,000 or 10,000 nT.

The maximum-intensity region (~67,000 nT) is over the Canadian Arctic and Siberia; the minimum-intensity region (~22,000 nT) is the South Atlantic Anomaly off Brazil.

Isogonic lines

Isogonic lines are lines of constant declination (covered in /learn/magnetic-declination-explained). The zero isogonic — the agonic line — is where declination = 0.

Field-component maps

WMM2025 publishes maps for all five primary quantities (D, I, F, H, Z) plus the secular variation rate of each. These are the printed declination-and-inclination charts that surveyors and navigators use as quick-reference plots.

Compass balancing

Because inclination varies with latitude, a freely suspended compass needle wants to tilt downward in the Northern Hemisphere (north end down) and the opposite in the Southern Hemisphere.

To keep the needle horizontal so it can rotate freely:

Northern Hemisphere compass: small counterweight on the south end of the needle to balance the dip-down on the north end.

Southern Hemisphere compass: counterweight on the north end to balance the dip-down on the south end.

A Northern Hemisphere compass taken to the Southern Hemisphere works poorly — the counterweight is on the wrong side, doubling the imbalance. The needle either drags on the housing or fails to rotate freely.

Quality compass manufacturers (Suunto, Brunton, Silva, Recta) ship zone-specific compasses, typically calibrated for five global zones:

• Zone 1 (Magnetic Northern): most of North America, Europe, North Asia.
• Zone 2 (North Equatorial): northern parts of Africa and Asia, Caribbean.
• Zone 3 (Equatorial): equatorial Africa and Asia.
• Zone 4 (South Equatorial): southern Africa, South America, Australia.
• Zone 5 (Magnetic Southern): southern South America, southern Africa, New Zealand, Antarctica.

A compass sold for Zone 1 won't work properly in Zone 5 — the counterweight is wrong. For travellers spanning zones, global compasses with no counterweight (relying on a precisely balanced needle and viscous damping) are available but typically less precise.

Aviation magnetic compasses use a different solution: the needle floats on a pivot in a viscous fluid that damps oscillations. Aircraft compasses are calibrated for the typical latitude range of the aircraft's operations.

Crustal vs main field

The numbers above (25,000–65,000 nT) describe the main field generated by the geodynamo. Local crustal field anomalies can add or subtract hundreds to a few thousand nT.

Notable crustal anomalies:

• Kursk magnetic anomaly (Russia): one of the strongest; intensity locally exceeds 200,000 nT due to massive iron-ore deposits.
• Bangui anomaly (Central African Republic): intensity ~+1,000 nT above background due to unknown geological cause.
• Australian Crustal Anomaly: long-wavelength anomalies across the continent from ancient geological structures.

Mineral exploration uses magnetometer surveys (airborne or ground-based) to map crustal anomalies and infer subsurface structure. Modern airborne magnetometers achieve ~0.01 nT precision.

For navigation, crustal anomalies typically don't affect declination by more than 1° (small compared to the main-field secular variation). The WMM doesn't include crustal anomalies; the Enhanced Magnetic Model (EMM) does.

Common misconceptions

“The dip needle is obsolete.” It's historically important — Norman's 1576/1581 work launched the formal study of geomagnetism. As a research instrument, it's been superseded by fluxgate and proton-precession magnetometers, but educational dip needles remain in physics laboratories.

“Intensity and declination are independent.” They're both expressions of the same underlying field vector. A given field configuration produces specific values of all three quantities (D, I, F) simultaneously. WMM and IGRF coefficient fits produce them as a coupled set, not independently.

“A compass works the same everywhere.” Without zone-specific balancing, a compass needle either tilts (in the wrong hemisphere) or sticks against the housing. Zone-specific compasses are an unavoidable consequence of inclination's latitude dependence.

“Magnetic intensity is constant.” It varies geographically (factor of ~3 across Earth), temporally (~0.1% per year secular variation), and on short timescales (geomagnetic storms can shift intensity by hundreds of nT for hours).

“Earth's field is the strongest natural field around us.” A refrigerator magnet at close range is ~100 times stronger; an MRI scanner is ~30,000 times stronger. Earth's field is special not for its strength but for its uniformity and range — it covers the entire planet and extends into space as the magnetosphere.

“Nanotesla and gauss are interchangeable.” 1 gauss = 100,000 nT. Convert carefully — confusion between the units is common. Modern publications use nT; older sources use gauss.

“Inclination is the same as latitude.” For a perfect dipole, tan(I) = 2 · tan(magnetic latitude) — close to but not the same as geographic latitude. The non-dipole field structure adds additional perturbation. London at +51° geographic latitude has inclination +66°, not +51°.

“A vertical compass works at the magnetic pole.” Near the magnetic pole, the horizontal component is nearly zero, so the compass needle has nothing to align with. Specialized vertical compasses (dip needles) work, but standard horizontal compasses become useless within ~1,000 km of the pole. Pole- region navigation uses sun compasses, gyrocompasses, or GPS.

“Intensity is irrelevant to navigation.” Intensity is part of the WMM/IGRF output and matters for mineral exploration, archaeology (paleomagnetic dating), and some scientific applications. For day-to-day navigation, declination is the dominant correction; inclination determines compass design; intensity is mostly background.