- Show that the Earth’s magnetic field has three components
- Find the latitude from the equation
In the dipole field experiment, we used a bar magnet and iron filings to model the Earth’s magnetic field. The “lines of force” that connected the two poles formed a three-dimensional pattern which can be found from Maxwell’s equations. The field vector can be broken into three components:
- A radial component, directed away from the Earth
- A latitudinal component, directed towards the Earth’s magnetic North pole
- A longitudinal component, directed along a small circle around the Earth’s magnetic axis
The radial component is given by . It varies with the cube of the distance from the Earth (instead of the square) because the magnetic field is a dipole, rather than a monopole, field. The radial component varies as the cosine of the latitude for the same reason.
Similarly, the latitudinal component varies both as the sine of the colatitude () and as the cube of the distance from the Earth’s center.
Because the field of a dipole is symmetric about the axis , the longitudinal component is zero. In reality, because the Earth’s magnetic field is not a true dipole and because there are local concentrations of magnetic materials, a small longitudinal component can be found to the Earth’s magnetic field.
The angle between the magnetic field and the surface of the Earth is called the inclination, shown on the diagram by I. From the geometry of the situation, it is clear that the tangent of this angle is simply the ratio of the radial to the latitudinal component; in math:
Using a simple dip needle, we can measure the three components of the Earth’s magnetic field and find our latitude. A dip needle is just a magnet suspended in a series of gymbals with perpendicular axes. Because the magnet is free to move in any direction, it aligns itself with the local field. By measuring the dip needle’s inclination, we can find the latitude; by measuring it’s declination, we can find magnetic North.
This method is just an alternative to GPS for finding present locations, but it is the only method we can use to find where rocks were at times in their past. When rocks containing iron minerals cool, the minerals record the magnetic field there and then present; this record stays with the rock, even if it is moved on the surface of the Earth. By carefully studying the magnetism recorded in rocks, scientists have been able to decipher the past positions of continents and understand more about continental drift.
To measure the local inclination, you will need:
- Magnetic Dip Needle
Alternatively, you can make a dip needle with:
- Small bar magnet (~1 cm long)
- 20 cm Thread
GlueBefore the demonstration: To make a dip needle, tie the thread around the center of gravity of the bar magnet. (For a bar magnet, this is the center of the longest axis.) Secure the thread to the bar magnet with a dab of glue, and let it dry.
1. Hold up the dip needle. Point out that it has three perpendicular axes, so that it can rotate freely in space. Ask a student to measure the dip of the needle with the protractor and write the value on the board.
2. Now slowly move toward the wall. Again ask a student to measure and record the dip of the needle with the protractor.
3. Now slowly move toward a desk. once more ask a student to measure and record the dip of the needle with the protractor.
4. Using the equation for latitude, find the latitude of your room from the dip of the needle. Point out the large errors which you would get if you used the values from near the wall or desk.
Why did moving the magnet near the wall and the desk change the orientation? (HINT: How could this be used to prospect for iron ore?)
Fowler, C. M. R., The Solid Earth, An Introduction to Global Geophysics, Cambridge University Press, 472 p., 1990.