INTRODUCTION

In magnetic survey methods we measure spatial differences or temporal changes of the natural magnetic field of the earth. Crude magnetic methods were first used in Sweden in the 1600’s to search for magnetic iron ores. In the same century Halley, of comet fame, sailed all over the Atlantic Ocean mapping field directions in hopes of establishing a magnetic navigation method. These were the first large-scale deep-sea cruises undertaken for purely earth science purposes (i. e. not for geographical exploration or astronomical observations).

Nowadays magnetic surveys are often the first step in a geophysical study because the costs are low, the speed of data collection is high, and the results often very valuable. At one end of the spectrum are detailed surveys looking for steel drums of toxic chemicals and unexploded military ordnance ("UXO"; magnetometer teams are still working to clean up after World War I, not to mention more recent conflicts). Geologic studies include the search for iron ore, studies of sediment thickness and bedrock structures and litho logy. At the other end of the spectrum are studies of paleolatitudes, continental drift and sea-floor spreading.

Before looking at these and other applications, let’s define the term "magnetic field".

MAGNETIC FIELD "B"

In your physics courses you learned about "B", formally called the "magnetic induction" or "magnetic flux density" measured in Teslas. This is the same quantity commonly called "magnetic field".

B can be defined in terms of the Lorenz force law that relates the force on a electrically charged particle to the charge, the electric field, the velocity and the magnetic field. The magnetic force is given by the charge times the vector cross product of the velocity times the magnetic field. Working out the units we see that B is expressed in Newtons per Ampere per meter; this combination is also called a Tesla.

Devices such as cathode ray tubes and mass spectrometers are based on the Lorenz law and clearly indicate the effects of electrical and magnetic forces on electrons and ions.

BIOT SAVART LAW

This law describes the B-field of a tiny current element, for example a short length of current-carrying wire in an electrical circuit. In magnetics, this law is analogous to Newton’s law of gravitational attraction in gravity. The little current element plays the same role as a point mass in gravity. Just as we calculated the gravitational effect of a body by integrating the effects of many point masses, in magnetics we’ll integrate over tiny current elements.

Compared to Newton’s Law the Biot Savart Law is more complex, involving a vector cross-product. This complexity arises because electrical current or current density is a vector quantity whereas mass and density are scalars.

MAGNETIC INDUCTION (B) OF A LONG LINE OF CURRENT

You will probably find this derivation in your physics book. We’ll assume an infinitely long line of constant current surrounded by a vacuum. Then we’ll apply the Biot Savart Law by integrating over tiny elements of the line from minus to plus infinity.

The result is that B is proportional to the current and inversely proportional to the distance from the current line. The magnetic field lines form circles around the current line. The sense of the field is given by a "right-hand rule" that I’ll demonstrate in class.

Somewhere I read that Oersted discovered this relationship while teaching an introductory physics class about two hundred years ago. We’ll see in a few days that we can apply this result directly to find a formula for the magnetic effect of a semi-infinite magnetic sheet. We also use it to explain "tilt anomalies" mapped by the VLF electromagnetic method (ES934 or ES795/895 next semester).

MAGNETIC INDUCTION B OF A CIRCULAR CURRENT LOOP

We’ll soon see that a circular current loop can be used as an analog of orbiting electrons and spinning electrons. At the level of this course, these electrons are the basic cause of magnetization of materials; that is, they form the tiny current elements of the Biot Savart Law. Common proton precession magnetometers use spinning protons to determine the magnitude of B.

Today’s derivation gives the field on the axis of a loop. This is readily found if we take advantage of the symmetry of the problem and are very careful to resolve all the little dB vectors in the axial direction.

Note that we have written the result in terms of "m", the dipole moment of the current loop. This combination equals the current times the area of the loop. We’ll see dipole moments in many situations ranging from electrons, protons, geologic bodies, and even the whole earth.

In the next class we’ll consider the more complicated problem of B at any point with respect to the loop or dipole, not just at the axial ones.