INTRODUCTION

In the last class we derived an expression for the magnetic induction, B, on the axis of a circular current loop. Skipping the derivation which you can find in your physics text, we’ll look at how the formula applies to an ideal dipole and to the earth.

B FIELD OF DIPOLE

It’s most convenient to express the field of a dipole using spherical coordinates (radius, longitude and latitude). Given the circular symmetry of the situation, we can express the field in terms of a radial component (vertical, positive up) and a tangential one (horizontal, positive north).

Long ago Arago showed that a bar magnet has the same external magnetic field as a current loop except very close to the magnet. You are all familiar with the pattern made by the "lines of force" around a little bar magnet.

B FIELD OF THE EARTH AS AN AXIAL DIPOLE

Back in the days of Queen Elizabeth I, her physician William Gilbert, noted that the magnetic field directions around a magnetic sphere (magnetite?) were about the same as those around the earth as revealed by early navigators and explorers. Additional data bears him out and at present about 80 % of the field is accounted for by an axial dipole. Averaged over thousands of years, the field directions fit the axial dipole pattern even better. The field strength is about 60, 000 nanoTeslas at the geographic poles and about half that at the equator.

For an axial dipole there is a simple relationship between latitude and inclination of the field. The tangent of the inclination is twice the tangent of the latitude. This equation is used to calculate paleolatitudes based on inclinations recorded in rocks and sediments. We’ll explain how the field is recorded in another lecture.

B FIELD OF THE EARTH

Several maps show that the earth’s field is not exactly an axial dipole at the present time and that it changes rapidly with time. At present the best dipole is inclined about eleven degrees with respect to the geographic axis and accounts for about 80 % of the field.

Be careful about the term "magnetic pole" because it has several different and confusing meanings. In the old cgs system of units, the "poles" were considered the physical source of the field. Geographic poles are where the axis of the earth intersects the surface. "Dip poles" are places where the field is vertical; there are lots of these. The poles of a dipole (best fitting, axial, etc) are where the dipole axis intersects the surface of the earth. Be careful how you use the term.

Because the temperatures within most of the earth are too hot for material to be magnetized and because the magnetic field changes rapidly with time the field cannot be caused by permanent magnetization (like a little bar magnet). Fluctuating electrical currents in the conductive, liquid metal outer core are responsible. Modern computer models of flow in the core are close to providing a rigorous, quantitative explanation of the field and its changes with time.

FORCE ON A CURRENT ELEMENT; TORQUE ON CURRENT LOOP

Applying the Lorenz Law to a little current element we derive a simple expression for the magnetic force on the element. The force is proportional to B and to the current and acts at right angles to the current flow.

Recalling that torque is given by the cross product of a force times a lever arm, we derive an expression for the torque on a current loop. The torque turns out to equal the cross-product of the dipole moment times the B field.

With these expressions we can see how to use a current loop as a compass and as a magnetometer. Next time we’ll apply these ideas to explain the magnetization of rocks and minerals.