Magnetic Inclination

Magnetic inclination is the angle formed between Earth’s magnetic field and the horizontal plane, varying according to latitude, from 0 degrees at the magnetic equator to nearly 90 degrees at the magnetic poles.

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How to measure it in class

FizziQ gives access to the value of the magnetic field measured along a single axis of the smartphone: you look for the orientation that makes this value maximal, because that is the one where the axis is aligned with the field.

Steps:

Move away from any metallic mass and any magnet, then calibrate the magnetometer by tracing a large figure 8 in the air.
First locate magnetic north in the horizontal plane, then position yourself in the vertical plane containing this direction: the inclination can only be measured in this plane.
Slowly rotate the smartphone in this vertical plane while following the field measurement on a single axis, and hold the device still at the maximum value.
Measure the angle between the smartphone and the horizontal with FizziQ’s inclinometer, a protractor or a profile photo: this is the magnetic inclination.
Repeat the measurement five times and calculate the average, then compare with the expected value, about 64° in metropolitan France.
Deduce the magnetic latitude of the location from the relation tan(i) = 2 tan(L) and compare it with the geographic latitude given by the GPS.

Scientific activities on this topic

The smartphone is an interesting tool for studying magnetic inclination because it has a magnetometer, whose precision nevertheless remains modest. Care must also be taken with its calibration because smartphones constantly try to correct for parasitic measurements and these adjustments can affect the measurement of the surrounding magnetic field.

To experiment with inclination and Earth’s magnetic field, you can measure the inclination angle and deduce the magnetic latitude of the location. This experiment is described at this link, and it can also be found directly in FizziQ in the Activities tab > Add an activity

Learn more

The dipole model for Earth’s magnetic field

Earth’s magnetic field can be approximated by that of a magnetic dipole, that is, a giant magnet placed at the center of the Earth, tilted about 11° relative to Earth’s axis of rotation.

In a dipole field, magnetic field lines emerge from a magnetic south pole and plunge toward a magnetic north pole, forming a structure similar to that of a bar magnet. This model explains why magnetic inclination varies with magnetic latitude:

At the magnetic equator: Field lines are parallel to the ground, so inclination is 0°.
At the magnetic poles: Field lines are almost perpendicular to the ground, so inclination reaches 90°.
At intermediate latitudes: Inclination follows a progressive variation according to magnetic latitude.

The relationship between magnetic inclination i and magnetic latitude L in a dipole model is given by the formula:

tan(i) = 2*tan(L)

This equation is an approximation and the exact values of magnetic inclination at a given point can be found by consulting the International Geomagnetic Reference Field website.

How to Measure Magnetic Inclination?

Magnetic inclination can be measured using a magnetometer, which detects the direction and intensity of the magnetic field. By analyzing the horizontal component H and the vertical component Z of the magnetic field, the inclination can be determined using the relationship:

tan(i) = Z/H, where i is the magnetic inclination, Z the vertical component and H the norm of the horizontal component of the field. Caution: H is not the measurement of a single axis of the smartphone, but the length of the vector projected onto the horizontal plane, which is calculated from the two horizontal axes.

It can also be determined using an inclination compass. This device is a compass that operates in the vertical plane, and which makes it possible to know the orientation of the magnetic field in the vertical plane.

Finally, by using a smartphone’s magnetometer and the measurement of the magnetic field along a single axis, you can determine the direction for which the magnetic field is maximal. The angle formed with the horizontal is the magnetic inclination. This measurement can be done in class or in the field.

Determining the Magnetic North Pole

The study of magnetic inclination helped locate the magnetic north pole. In 1831, explorer James Clark Ross measured inclination at different locations in Canada. When he found a point where inclination reached 90°, he deduced that he was at the magnetic north pole.

Since then, scientists have been tracking the evolution of inclination to observe the movement of the magnetic poles, which change position over time.

Orders of magnitude

Inclination of 0° at the magnetic equator, about 30° in Dakar, about 64° in metropolitan France, about 72° in Iceland, close to 90° at the magnetic north pole. In the southern hemisphere, the inclination is counted as negative: the field points upward. With a total field of about 47 µT in France, the 64° inclination gives a vertical component of about 42 µT, twice as large as the horizontal component, of about 21 µT.

Formula

Magnetic inclination is deduced from the ratio between the vertical component and the horizontal component of the field:

tan(i) = Z / H

where:

i: magnetic inclination, angle between the field vector and the horizontal plane (°)
Z: vertical component of the magnetic field (µT)
H: norm of the horizontal component, H = √(Bx² + By²) (µT)

In the centered magnetic dipole model, the inclination is related to the magnetic latitude:

tan(i) = 2 × tan(L)

where:

L: magnetic latitude of the location (°)

The components can be recovered from the total field B and the inclination:

Z = B × sin(i) and H = B × cos(i)

Application examples

In 1831, James Clark Ross located the magnetic north pole by looking for the place where the inclination is 90°, that is, where the field is perfectly vertical.
A hiking compass bought in Europe tilts and rubs against its casing when used in Australia: manufacturers produce needles weighted differently for five inclination zones of the globe.
Sea turtles and certain birds use the field’s inclination as a latitude indicator during their migrations.
A student in Paris measures an inclination of 64° and deduces a magnetic latitude L such that tan(L) = tan(64°)/2 ≈ 1.03, or L ≈ 46°, a value consistent with the geographic latitude of 49°.
Paleomagnetism measures the inclination frozen into a rock as it cooled, which gives the latitude at which it formed and makes it possible to reconstruct the movement of the continents.

FAQ

Q: Inclination or magnetic declination, how to avoid confusing them? A: Inclination is an angle measured in the VERTICAL plane: it is how much the field dips below the horizontal. Declination is an angle measured in the HORIZONTAL plane: it is the azimuth difference between magnetic north and geographic north. A classic compass ignores inclination and is only affected by declination.

Q: Why doesn’t my compass needle dip toward the ground if the field is inclined at 64°? A: Because it is mounted on a pivot that constrains it to remain in the horizontal plane, and often weighted on one side to compensate. It therefore reacts only to the horizontal component of the field. To see the inclination, you need an inclination compass, whose needle pivots in a vertical plane.

Q: What is the inclination in the southern hemisphere? A: It is negative, which means the field points upward and not downward. In Sydney, the inclination is about −64°, almost the opposite of that of Paris.

Q: Is the inclination stable over time? A: No. It changes by a few arcminutes per year at a given location, because the magnetic poles are moving. The magnetic north pole is currently drifting by several tens of kilometers per year toward Siberia.

Q: Why is my measurement off by 10° or more? A: Almost always because the magnetometer is not in the right vertical plane, or because a nearby metallic mass distorts the field. You must first locate the direction of magnetic north, then tilt the smartphone in that plane, away from any ferrous object.

Earth’s Magnetic Field - Magnetic Declination - Magnetometer - Compass