THE PHYSICS OF MAGNETISM

1 CHAPTER 1 THE PHYSICS OF MAGNETISMBACKGROUND: Read chapters on MAGNETISM from your favorite college PHYSICS bookfor is the study of the magnetic properties of rocks. It is one of the mostbroadly applicable disciplines in geophysics, having uses in diverse fields such as geo- MAGNETISM , tectonics, paleoceanography, volcanology, paleontology, and the potential applications are varied, the fundamental techniques are remark-ably uniform. Thus, a grounding in the basic tools of paleomagnetic data analysis canopen doors to many of these applications. One of the underpinnings of paleomagneticendeavors is the relationship between the magnetic properties of rocks and the Earth smagnetic this chapter, we will review the basic physical principles behind MAGNETISM : whatare magnetic fields, how are they produced, and how are they measured?

2 Although manyfind a discussion of scientific units boring, much confusion arose when paleomagnetistsswitched from cgs to the Syst`eme International (SI) units, and mistakes aboundin the literature. Therefore, we will explain both unit systems and look at how toconvert successfully between them. There is a review of essential mathematical tricksin Appendix A, to which the reader is referred for WHAT IS A MAGNETIC FIELD?Magnetic fields, like gravitational fields, cannot be seen or touched. We can feel thepull of the Earth s gravitational field on ourselves and the objects around us, but wedo not experience magnetic fields in such a direct way. We know of the existence ofmagnetic fields by their effect on objects such as magnetized pieces of metal, naturallymagnetic rocks such as lodestone, or temporary magnets such as copper coils that carryan electrical current.

3 If we place a magnetized needle on a cork in a bucket of water,it will slowly align itself with the local magnetic field. Turning on the current in acopper wire can make a nearby compass needle jump. Observations like these led tothe development of the concept of magnetic )b)irwireHFIGURE ) Distribution of iron filings on a flat sheet pierced by a wire carrying a currenti. [From Jiles,1991.] b) Relationship of magnetic field to current for straight wire. [Photo by author.]Electric currents make magnetic fields, so we can define what is meant by a mag-netic field in terms of the electric current that generates it. Figure a is a picture ofwhat happens when we pierce a flat sheet with a wire carrying a are sprinkled on the sheet, the filings line up with the magnetic field producedby the current in the wire.

4 A loop tangential to the field is shown in Figure , whichillustrates theright-hand rule. If your right thumb points in the direction of (positive)current flow (the direction opposite to the flow of the electrons), your fingers will curlin the direction of the magnetic magnetic fieldHpoints at right angles to both the direction of current flow andto the radial vectorrin Figure The magnitude ofH(denotedH) is proportionalto the strength of the currenti. In the simple case illustrated in Figure , themagnitude ofHis given by Amp`ere s law:H=i2 r,whereris the length of the vectorr. So, now we know the units ofH:Am `ere s Law, in its most general form, is one of Maxwell s equations of electro- MAGNETISM : in a steady electrical field, H=Jf,whereJfis the electric currentdensity (see Section in the appendix for review of the operator).

5 In words, thecurl (or circulation) of the magnetic field is equal to the current density. The originof the term curl for the cross product of the gradient operator with a vector field issuggested in Figure , in which the iron filings seem to curl around the MAGNETIC MOMENTAn electrical current in a wire produces a magnetic field that curls around the we bend the wire into a loop with an area r2that carries a currenti(Figure ), Magnetic Momentthe current loop would create the magnetic field shown by the pattern of the iron magnetic field is the same as the field that would be produced by a permanentmagnet. We can quantify the strength of that hypothetical magnet in terms of amag-netic momentm(Figure ).

6 The magnetic moment is created by a currentiand alsodepends on the area of the current loop (the bigger the loop, the bigger the moment).Therefore, the magnitude of the moment can be quantified bym=i r2. The momentcreated by a set of loops (as shown in Figure ) would be the sum of thenindividualloops:m=ni r2.( )So, now we know the units ofm:Am2. In nature, magnetic moments are carried bymagnetic minerals, the most common of which are magnetite and hematite (see Chapter6 for details).b)irma)c)FIGURE ) Iron filings show the magnetic field generated by current flowing in a loop. b) A current loopwith currentiand area r2produces a magnetic momentm. c) The magnetic field of loops arranged as asolenoid is the sum of the contribution of the individual loops.

7 [Iron filings pictures from Jiles, 1991.] MAGNETIC FLUXThe magnetic field is a vector field because, at any point, it has both direction andmagnitude. Consider the field of the bar magnet in Figure The direction of thefield at any point is given by the arrows, while the strength depends on how closethe field lines are to one another. The magnetic field lines representmagnetic of flux lines is one measure of the strength of the magnetic field: the as the motion of electrically charged particles in a wire (a current) creates amagnetic field (Amp`ere s Law), the motion of a magnetic field creates electric currentsin nearby wires. The stronger the magnetic field, the stronger the current in the Magnetic Flux3mBVoltmetervelocitylma)b)FIGURE ) A magnetic momentmmakes a vector fieldB.

8 The lines of flux are represented by the arrows.[Adapted from Tipler, 1999.] b) A magnetic momentmmakes a vector fieldB, made visible by the iron this field moves with velocityv, it generates a voltageVin an electrical conductor of lengthl. [Iron filingspicture from Jiles, 1991.]We can therefore measure the strength of the magnetic induction (the density of mag-netic flux lines) by moving a conductive wire through the magnetic field (Figure ).Magnetic induction can be thought of as something that creates a potential dif-ference with voltageVin a conductor of lengthlwhen the conductor moves relativeto the magnetic inductionBwith velocityv(see Figure ):V= ,we can derive the unit of magnetic induction: the tesla (T).

9 One tesla is the mag-netic induction that generates a potential of 1 volt in a conductor of length 1 meterwhen moving at a rate of 1 meter per second. So now we know the units ofB:V s m 2= way of looking atBis that if magnetic induction is the density of magneticflux lines, it must be the flux per unit area. So an increment of fluxd is the fieldmagnitudeBtimes the increment of areadA. The area here is the length of the wireltimes its displacementdsin timedt. The instantaneous velocity isdv=ds/dt,sod =BdA, and the rate of change of flux isd dt=(dsdt)Bl=vBl=V.( )Equation is known as Faraday s Law and, in its most general form, is the fourthof Maxwell s equations. We see from Equation that the units of magnetic fluxmust be a volt-second, which is a unit in its own right: the weber (Wb).

10 The weber isdefined as the amount of magnetic flux which, when passed through a one-turn coil ofa conductor carrying a current of 1 ampere, produces an electric potential of 1 definition suggests a means to measure the strength of magnetic induction and isthe basis of the fluxgate Magnetic MAGNETIC ENERGYA magnetic momentmin the presence of a magnetic fieldBhas amagnetostatic energy(Em) associated with it. This energy tends to align compass needles with the magneticfield (see Figure ).Emis given by m Bor mBcos ,wheremandBare themagnitudes ofmandB, respectively (see Section in the appendix for a review ofvector multiplication). Magnetic energy has units of joules and is at a minimum whenmis aligned ma)b)batteryFIGURE magnetic momentmof, for example, a compass needle will tend to align itself with a mag-netic fieldB.