The Earth’s magnetism

The Earth as a magnet

Narrated video. Transcript:

This is an extra topic. The Earth’s magnetism is not in the current NCERT syllabus, but it was in the older textbook, and it is useful for competitive exams and for understanding how a compass works.

A compass needle on your desk points roughly north, wherever you are. So the Earth has a magnetic field all around it. Near the surface its strength is of the order of 10⁻⁵ tesla.

Draw its field lines, and they look like those of a giant bar magnet placed at the centre of the Earth. The real cause is different: electric currents in the molten iron of the Earth’s outer core, called the dynamo effect.

The axis of this magnetic dipole is not the same as the axis the Earth spins about. It is tilted by about 11.3°. The north magnetic pole lies in northern Canada, and the south magnetic pole in Antarctica.

Now a twist in the names. At the north magnetic pole, field lines go into the Earth. Field lines enter the S pole of a magnet. So the Earth’s imaginary bar magnet has its S pole in the north, and its magnetic moment points towards geographic south.

That is exactly why the N end of a compass points north: it is attracted towards the magnet’s S pole, hidden under the Arctic.

Worked example. The field at the magnetic equator is about 0.4 gauss, that is 4 × 10⁻⁵ T. The equator is on the equatorial line of the dipole, so B = μ₀m/(4πR³), with R = 6.4 × 10⁶ m.

So m = B R³ × 4π/μ₀ = 4 × 10⁻⁵ × (6.4 × 10⁶)³ ÷ 10⁻⁷. Now (6.4 × 10⁶)³ is 2.62 × 10²⁰. The answer is m ≈ 1.05 × 10²³ A m². A truly giant magnet.

At any place, three numbers describe the Earth’s field: the declination, the dip, and the horizontal component. Let us meet them one by one.

Declination, dip and the horizontal component

Narrated video. Transcript:

Stand at a place and imagine a vertical plane through the true north–south line. This is the geographic meridian. The vertical plane along a compass needle is the magnetic meridian.

The two do not quite agree. The angle between true north and the direction a compass shows is the magnetic declination, D. It is small in India: 0° 41′ east at Delhi, and 0° 58′ west at Mumbai.

Now take a needle pivoted about a horizontal axis, so that it can swing up and down in the magnetic meridian. This is a dip needle. In Delhi, its north end dips down by about 42° below the horizontal.

This angle is the dip, or inclination, I: the angle the total field B_E makes with the horizontal. On the magnetic equator the dip is 0°. At the magnetic poles it is 90°: the needle stands straight up and down.

In the northern hemisphere, the north end of the dip needle points down. In the southern hemisphere, the south end points down.

Split B_E into two parts. The horizontal component is B_H = B_E cos I. The vertical component is Z_E = B_E sin I. Dividing them, tan I = Z_E/B_H.

Worked example. At a place, B_H = 0.26 gauss and the dip is 60°. Then B_E = B_H/cos I = 0.26/0.5 = 0.52 gauss. The vertical component is 0.52 × sin 60°, about 0.45 gauss.

An ordinary compass can only turn in a horizontal plane, so it feels only B_H. Near a magnetic pole, B_H is almost zero, and the compass wanders aimlessly. Explorers there had to navigate by other means.

So D tells us where the magnetic meridian points, I tells us how steeply the field dips, and B_H tells us the strength. These three elements describe the Earth’s field at any place.

Try it: dip and declination

Interactive simulation. Things to try:

• Drag the orange dot round the Earth. The field arrow at the surface turns from horizontal at the equator to vertical at the poles.
• Watch the dip needle in the side view: its north end points down in the northern hemisphere and up in the southern.
• Check B_H = B_E cos I and Z_E = B_E sin I against the readouts.
• Move to a magnetic pole: B_H goes to zero.
• Change the declination: the compass seen from above turns away from true north by D.

Total field from B_H

In the magnetic meridian of a certain place, the horizontal component of the Earth’s magnetic field is 0.26 G and the dip angle is 60°. What is the total magnetic field of the Earth there?

1. 0.13 G
2. 0.30 G
3. 0.45 G
4. 0.52 G

Equal components

At a place, the horizontal and vertical components of the Earth’s magnetic field are equal. What is the angle of dip there?

1. 0°
2. 30°
3. 45°
4. 90°

Summary

• Horizontal and vertical components: B_H = B_E cos I, Z_E = B_E sin I
• Dip: tan I = Z_E / B_H
• Total field: B_E = √(B_H² + Z_E²)

• Extra topic, beyond the current NCERT syllabus. The Earth’s field is like that of a dipole at its centre, tilted about 11.3° to the spin axis, with its moment pointing to geographic south.
• Declination D: angle between true north and the magnetic meridian (small in India).
• Dip I: angle of the total field below the horizontal; 0° at the magnetic equator, 90° at the magnetic poles (about 42° in Delhi).
• A compass feels only B_H. D, I and B_H together are the three elements of the Earth’s magnetism.