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Equivalence of Current–Carrying Coils and Magnets; Magnetic Dipoles; - Law of Attraction and Repulsion, Definition of the Ampere

Equivalence of Current–Carrying Coils and Magnets; Magnetic Dipoles; - Law of Attraction and Repulsion, Definition of the Ampere



Magnetic : - equivalence of current–carrying coils and ; magnetic ; - law of attraction and repulsion, definition of the . Magnetic fields: - magnetic fields from electrical currents and magnets; magnetic induction B and lines of magnetic induction.

The geomagnetic field

The magnetic elements: (N, E, V) vector components; declination () and inclination (dip).

The external field: diurnal variations, ionospheric currents, magnetic storms, sunspot activity.

The internal field: the and non–dipole fields, secular variations, the geocentric axial dipole hypothesis, geomagnetic reversals, seabed magnetic anomalies,

The model

Reasons against an origin in the or and reasons suggesting an origin in the fluid outer core.

Magnetohydrodynamic dynamo models: motion and eddy currents in the fluid core, mechanical analogues.

Background reading: Fowler §3.1 & 7.9.2, Lowrie §5.2 & 5.4 (08/430/0012) MAGNETIC FORCES

Magnetic forces are forces associated with the motion of electric charges, either as electric currents in conductors or, in the case of magnetic materials, as the orbital and motions of in . Although the concept of a magnetic pole is sometimes useful, it is difficult to relate precisely to observation; for example, all attempts to find a have failed, and the model of permanent magnets as magnetic dipoles with north and poles is not particularly accurate. Consequently moving charges are normally regarded as fundamental in .

Basic observations 1. Permanent magnets A attracts and , the attraction being most marked close to its ends. The centres of these zones where the attraction• is strongest are called the magnetic poles. When freely suspended about a vertical axis, a magnet comes to rest with its two poles aligned along an approximately north–south• direction. The pole pointing north is the north (N) pole and the other is the south (S) pole. magnetic poles occur only in N–S pairs, never as isolated poles. • 2. Current–carrying conductors A conductor carrying an exerts a on iron, steel, magnets and other current–carrying conductors. • A small planar coil carrying a current behaves like a small magnet whose poles are on the axis of the coil and close to its• centre. When freely suspended about a vertical axis, a current carrying coil comes to rest with its axis aligned along an approximately north–south direction; the face of the coil around which the current flows anticlockwise is its “north poleÔ.

a magnet aligns a current–carrying coil The north and south poles itself north–south aligns its axis north–south of a current–carrying coil north N I I  R I  6 ? • N S S © I GEOPHYSICS (08/430/0012) EQUIVALENCE OF CURRENT CARRYING COILS AND MAGNETS

(i) A current carrying coil and a small magnet produce a very similar magnetic forces on other coils and magnets. They produce virtually identical patterns in iron filings sprinkled on a sheet of paper held close to the axis of the coil or magnet. Both produce a dipole magnetic field as illustrated on the left below; the figure on the right shows a close-up of the field in the vicinity of the coil.

(ii) A current carrying behaves like a bar magnet. The magnetic field from a solenoid is portrayed below. The magnetic field from a bar magnet is very similar to this. GEOPHYSICS (08/430/0012) LAWS OF MAGNETIC ATTRACTION AND REPULSION

Like poles attract; unlike poles repel. • A pair of parallel current–carrying wires attract each other if the currents flow in the same direction in both wires: • they repel if the currents flow in opposite directions.

Magnetic forces vary with • distance, decreasing as distance increases; measured from the axis of the coil or magnet; the strength of the magnet or current.

The Biot–Savart law (1820) summarises the observations of forces between parallel current–carrying wires: the force per • unit length between two infinite parallel wires separated by a distance a and carrying currents I1 and I2 is proportional to I1I2/a. The constant of proportionality is written as µ0/(2π).

The ampere

The ampere is defined in terms of magnetic forces: when flowing in each of two infinite parallel wires one apart, it 7 7 2 produces a magnetic force of 2 10− per metre length on each wire. This definition makes µ = 4π 10− N/A . × 0 × GEOPHYSICS (08/430/0012) MAGNETIC FIELDS Magnetic forces are treated by introducing a vector field called the magnetic field. Magnetic fields can be represented by lines of magnetic induction around a current-carrying conductor or magnet. Lines of induction can be tracked by means of a small magnet or a small current-carrying coil. For example, they are circular around a long straight wire.

The magnetic field from a circular coil is like that from a small magnet at the centre of the coil aligned along its axis. It is called a dipole field since it is equivalent to the field from a closely spaced north (N) and south (S) pole. The figure below shows the lines of magnetic induction around a . of a dipole GEOPHYSICS (08/430/0012) MAGNETISM: BASIC CONCEPTS

In magnetism current elements are normally regarded as fundamental; magnetic poles are a useful concept but they have no physically well-defined location, strength or permanence. Ampere’s law for the force between current carrying elements is the basic law describing magnetic forces. The force per unit length on each of two infinite straight parallel wires, distance r apart, carrying currents I1 and I2, is: F µ I I = 0 1 2 l 2πr

7 2 where µ = 4π 10− NA− = the permeability of free space (). The ampere (A) is defined from this law. 0 × Magnetic fields are a conceptual tool for handling magnetic forces and can be represented by lines of magnetic induction (“lines of forceÔ). The tangent to a line of induction gives the direction of the magnetic field and the of the lines of induction indicates the magnitude of the field. The strength of a magnetic field is defined quantitatively by the magnetic induction B. B is sometimes called the magnetic field intensity. B represents the number of lines of magnetic induction passing through unit area perpendicular to the field. Magnetic Dipoles Although the force between current elements can be measured with an apparatus called the current balance, most magnetic measurements use current–carrying coils or magnets. Current–carrying coils and magnets both behave like magnetic dipoles; there are no magnetic monopoles. The strength of a magnetic dipole is measured by its magnetic . The of a current–carrying loop of cross–sectional area A is: m = IA where I is the total current in the loop, i.e. NI1 when there are N turns in the coil, each carrying a current I1. The magnetic moment of a magnet is defined from that of its equivalent coil, i.e. the coil that experiences the same when placed in the same location and orientation in the same magnetic field. GEOPHYSICS (08/430/0012) MAGNETISM: BASIC CONCEPTS (CTD)

The magnetic moment determines both (a) the magnetic field that the current–carrying coil or magnet produces, and

(b) the torque, or turning moment, that acts on the coil or magnet when placed in a magnetic field. Magnetic moment is a vector whose direction is given by the right–hand corkscrew rule. m m = IA  Area 

A I θ - AK B A ? L = m B L = m B× sin θ | | | || | Magnetic Induction or B–field The magnetic B–field (or magnetic induction or magnetic flux density) is a vector field defined from the torque L (turning moment) acting on a magnetic dipole: L = mB sin θ where θ is the angle between the vectors m and B. The torque tends to turn m towards B and has a vector direction given by the right–hand corkscrew rule. The magnetic B-field could equally well be defined from the force on a unit current element. (This force acts at right to both B and the current element). 1 2 2 The unit of magnetic B–field is the (T): 1T = 1N/(A m) = 1kgA− s− = 1Wb/m The (Wb) is the unit of magnetic flux: 1 Wb = 1 T m2 = 1 V s. 9 4 Older units are the gamma (1γ = 1nT = 1nanotesla = 10− T) and the (1 G = 10− T). 4 Because the Earth’s magnetic field is a small fraction of a tesla ( 0.31 0.62 10− T from to pole), the nanotesla is the unit commonly used in magnetic surveying. ∼ − × GEOPHYSICS (08/430/0012) THE EARTH’S MAGNETIC FIELD (1) The magnetic elements These define the Earth’s magnetic field B at any point. The Earth’s magnetic field B is measured with respect to a Cartesian (x, y, z) frame of coordinates, namely geographic •north, geographic and the vertical.

Since B is a vector, it can be resolved into a north N • BN BH vertical B and a horizontal component B . d ¡ 7 ¡ ¡  V H Z ¡ ¡d ¡  ¡  The horizontal component can itself be resolved Z~¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡ into an easterly BE and a northerly component ¡ ¡ ¡ ¡-¡ ¡¡ BN . BE WE A ¡ @I ¡ The direction of B is defined by A@ ¡ • A @ ¡ ¡ (a) its declination d A ¡ @ S N and (b) its inclination i. A @ A i ¡  ¡ d is the azimuth of BH , i.e. the angle between A BSB d A S AU B ¡ north and BH ; i is the dip of B, i.e. the angle S ¡  ¡ ¡ B XXXB S i between B and B. WEB XBXX H ¡ ¡ S ¡ SAB The RH figures show how d and i can ¡ ¡ B •  ¡ SA ¡ ¡  B be measured by magnets mounted on BV ? S SB horizontal and vertical axes.

The magnitude of B is measured by , of which there are three important types: fluxgate, , and• alkali vapour. The magnetic elements are related by: B = B sin i, B = B cos i, B = B cos d, and B = B sin d • V | | H | | N H E H GEOPHYSICS (08/430/0012) THE EARTH’S MAGNETIC FIELD (2)

General features

The Earth’s magnetic field can be mapped by means of isomagnetic charts. These are explained on the next page. • Secular variations are slow changes in the Earth’s magnetic field with time. For example magnetic north drifts gradually • over the years.

Polarity reversals: There is very strong evidence from palaeomagnetism that the Earth’s magnetic field has reversed • its polarity many times through geological time.

The field can be decomposed into an external and an internal field. The external field originates from effects above • the Earth’s surface, mostly from charged particles in the upper (the ). It comprises just a few percent of the total field. The internal field originates within the Earth and it is further subdivided into (i) an inclined dipole field, and (ii) an irregular non–dipole field. The non–dipole field on average accounts for less than 10% of the total field.

On a more local scale the geomagnetic field exhibits various anomalies. On land these anomalies are of interest in • magnetic prospecting for minerals, in mapping faults and structural trends, and in estimating depths to crystalline magnetic rocks (“basementÔ) beneath sediments. The floor shows a characteristic pattern of linear magnetic anomalies that run approximately parallel to the mid–ocen ridges. These anomalies correspond to stripes of permanently magnetised basalts having magnetisation that is either normally or reversely polarised, according to whether the Earth’s field was normally or reversely polarised when the basalt cooled near the spreading centre. GEOPHYSICS (08/430/0012) MAGNETIC CHARTS

Isomagnetic charts contour plots of the magnetic elements.

Magnetic intensity: of the Earth’s magnetic field intensity or magnetic induction B show that it varies in a • complicated way with , and time. The variations can be modelled using the International Geomagnetic Reference Field (IGRF) equations. These equations involve a large number of coefficients for . Although the equations are based on large amount of observational data, the secular variations are notfully predictable and the equations are updated every 10 years.

Isogonic charts show isogonics. Isogonics, or isogonic lines are lines of constant declination. The figure overleaf shows • the isogonic charts for 1955 and 1992.

Isoclinic charts show lines of constant inclination (isoclinics). •

Magnetic meridians are lines whose tangents give the declination. Because they trace the direction of BH, or the • direction in which a needle would point, they help visualise the geomagnetic field, more so than isoclinic charts. GEOPHYSICS (08/430/0012) EXAMPLES OF ISOGONIC CHARTS

Isogonics, or isogonic lines are lines of constant declination. The magnetic dip poles are not the geomagnetic poles: why not? The chart on the left is for 1955, that on the right for 1992. The differences indicate the to which the geomagnetic field is changing with time. GEOPHYSICS (08/430/0012) THE EARTH’S MAGNETIC FIELD (3) The geomagnetic dipole field

The Earth’s magnetic field is mainly dipolar: its shape approximates the field of a dipole, the geomagnetic dipole, whose axis passes through the centre of the Earth. Currently the geomagnetic dipole’s axis is inclined at about 11◦ to the Earth’s axis of rotation.

The geomagnetic dipole’s axis intersects the Earth’s surface at the geomagnetic poles. • The plane through the centre of the Earth perpendicular to the dipole axis intersects the Earth’s surface at the • geomagnetic equator.

The north is a magnetic . • 4 4 The magnitude of the dipolar field varies from 0.62 10− T at the geomagnetic pole to 0.31 10− T at the geomagnetic • equator. Usually the geomagnetic field is given in× nanoteslas: the corresponding values are× 62000 nT and 31000 nT.

The dipole field is obtained by making a best fit to worldwide accurate measurements of the Earth’s magnetic field. • Suitably accurate and widespread measurements date back to circa 1830.

The direction of the dipole’s axis drifts slowly. • When averaged over periods of the order of 10000 years, the mean geomagnetic field appears to be that of an axial • dipole. GEOPHYSICS (08/430/0012) THE EARTH’S MAGNETIC FIELD (4)

The geomagnetic dipole field (ctd)

The inclination of a dipole magnetic field is uniquely related to magnetic latitude. Over periods of several thousand years the variations of the non-dipole field are effectively random and the average geomagnetic field is that of a dipole at the centre of the Earth pointing north. The relation between magnetic inclination and magnetic latitude is used to infer palaeolatitudes and hence apparent wander of the . GEOPHYSICS (08/430/0012) THE EARTH’S MAGNETIC FIELD (5)

The non-dipole field

The non–dipole field superimposes irregular components onto the dipole field which can reach 15000 nT in places • but are mostly just a few percent of the dipole field. ∼ ±

The non–dipole field does not vary systematically with latitude, longitude or time. • The scale of variations in the non-dipole field is very significant. It comprises • (i) gross variations on the scale of 5000 km or more, and (ii) shorter variations with dimensions of 150 km or less. The localised anomalies (ii) originate in the outer 25 km of the Earth: short wavelength variations cannot physically originate at depth. Moreover temperatures beyond∼ this kind of depth exceed the Curie point of , the temper- ature above which magnetite is no longer magnetic. The regional anomalies (i) most likely originate from sources at great depth, thought to be in the outer core. The lack of variation in the 200 to 2000 km range suggests that there are no sources of magnetism in the mantle (although the conductivity of the mantle would damp out any rapidly varying features). GEOPHYSICS (08/430/0012) MAGNETISM: SECULAR VARIATIONS

The external magnetic field undergoes diurnal (daily) variations of magnitude 20–80 nT and it is sometimes disturbed by magnetic storms giving variations up to 1000 nT. These variations originate from∼ ionospheric currents and the solar (from sunspots). The internal field undergoes slow temporal changes called secular variations which have been (a) observed by direct mea- surement since circa 1830 and (b) inferred from paleomagnetic evidence. Direct Observation

Westward drift: the longitude of the geomagnetic pole has changed from about 65◦W in 1830 to near 70◦W for much of this century; its latitude has remained around 78 to 79◦N.

Gradual decrease of the dipole field: at the geomagnetic equator the Earth’s B–field has decreased by about 5% per century.

The non–dipole field: has changed more rapidly; its contours expand and contract with periods of the order of 10 to 1000 years; on average it drifts westwards at about 0.2◦ per year. Palaeomagnetic Evidence Rocks and archaeological artefacts, such as bricks from kilns, containing magnetic minerals record and preserve the direction and magnitude of the geomagnetic field in their magnetic moments. Archaeological artefacts can be used to investigate the geomagnetic field over the past few thousand years. They indicate variations, with a period around 10000 years, that are consistent with observations since 1830. Geomagnetic reversals: a major discovery of palaeomagnetism is that some rocks are reversely magnetised. There are no physical or chemical differences between normally and reversely magnetised rocks. Moreover all rocks in a given band have the same polarity. Thus the explanation lies in reversals of the polarity of the geomagnetic field. In geological terms reversals are almost instantaneous, taking of the order of 10000 years. uses geomagnetic reversals as the basis of a geological time scale. GEOPHYSICS (08/430/0012) GEOCENTRIC AXIAL DIPOLE HYPOTHESIS

The apparent geomagnetic poles (VGPs = virtual geomagnetic poles) estimated from geologically young rocks cluster around the geographic pole. The figure below illustrates this with a selection of VGPs estimated from Recent European and North American rocks.

The geocentric axial dipole hypothesis postulates that over time spans of several thousand years the average geomagnetic field corresponds to that of a dipole at the centre of the Earth with its axis parallel to the axis of rotation, except for those periods when the field is undergoing a polarity reversal.

90o E N.American VGPs European VGPs

180o 0o

geomagnetic pole (1980)


Origin of the geomagnetic field

The geomagnetic field is not the field of a permanent magnet. At depths below 30 km the temperature exceeds the • of iron minerals while the magnetism of rocks in the outer 30 km is too weak to produce the Earth’s field.

Secular variations rule out an origin in the solid part of the Earth. • All modern explanations rely on electric currents in the fluid and conductive outer core to produce the field. The theory • is that of a magnetohydrodnamic dynamo. Its main assumptions are: (a) the fluid outer core is in circulatory motion, and (b) initially there was a small non-uniform magnetic field in (e.g. from the or from electrochemical currents).

The mechanism is that • (1) motion of eddies induces currents in the fluid conductor by dynamo action (a moving conductor in a magnetic field); (2) the induced currents generate a further magnetic field. This self-exciting dynamo has to sustain its own magnetic field.

A mechanical analogue of the geomagnetic dynamo is the disk dynamo. A simple disk dynamo has two possible states • corresponding to two polarities of its magnetic field. Two interconnected disk can simulate spontaneous reversals of the magnetic field. GEOPHYSICS (08/430/0012) DISK DYNAMOS

The upper figure shows a disk dynamo: a disk rotating in a magnetic field parallel to its axle induces an e.m.f. () between the axle and the rim. If contacts are placed on the axle and rim, the e.m.f. generates a current. This current can be passed through a coil to produce in turn its own magnetic field which can form part of two coupled disk dynamos as in the lower figure. In the coupled disk dynamo, the current from disk 1 drives the coil for disk 2 and the current from disk 2 drives the coil for disk 1. The is self exciting provided there is any stray magnetic field in its vicinity. The magnetic field generated by the coupled disk dynamo oscillates and may even reverse its polarity.

Simple disk dynamo


I 1 B 1

Two coupled disk dynamos

coil 2 disk 2


Generating the geomagnetic field would require large currents. These would dissipate large amunts of energy. What keeps the dynamo going? A comparison of possible energy sources, e.g. the geothermal flux 3.2 1013 W the 2.9 × 1012 W × shows that the Earth’s internal is orders of magnitude higher than all other internal sources of energy. It is therefore usually assumed that thermal drives the eddies in the core responsible for the dynamo action. It is not established whether the source of heat is (a) in the core, or (b) the sinking of solid iron and through the outer core. Stacey (1960) suggested that the Earth’s precessional motion may provide a dynamo mechanism. GEOPHYSICS (08/430/0012) SEAFLOOR MAGNETIC ANOMALIES Magnetic anomalies Magnetisation in rocks perturbs the Earth’s magnetic field, producing anomalies where the field strength or direction or both depart from the general trend of the area: = measured magnetic component regional component. 9 − Magnetic anomalies are usually measured in nanteslas (nT): 1 nanotesla = 1 nT = 10− tesla = 1 γ = gamma (old unit) In continental regions magnetic anomalies can range up to thousands of nanoteslas. In oceanic regions magnetic anomalies may exceed 1000 nT near ridge crests but they rarely reach 1000 nT elsewhere. Oceanic anomalies are distinctly linear, usuually parallel or subparallel to mid-ocean ridges and frequently symmetric with respect to the ridges. observed field = anomalies + regional

observed regional

seabed magnetic anomalies +

− polarity of anomalies

reverse−normal (white−black) display GEOPHYSICS (08/430/0012) VINE-MATTHEWS HYPOTHESIS F.J.Vine and D.H.Matthews (1963) proposed that oceanic anomalies were caused by strips of alternately normally and reversely magnetised crust generated at the mid-ocean ridges as newly created rock cools through its Curie point. The same pattern of reversals in seen in seabed sediments and in thick lava flows. This unique pattern is the basis of the magnetostratigraphic time scale. The figure on the left shows how seafloor magnetic anomalies are the product of seafloor spreading and reversals of the Earth’s magnetic field. The figure on the right shows a well-known magnetic anomaly pattern in the north-east Pacific at a complicated conjnction of three lithoshperic plates - the Pacific, North American and Farallon.