Magnetic forces: - equivalence of current–carrying coils and magnets; magnetic dipoles; - law of attraction and repulsion, deﬁnition of the ampere. 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 (azimuth) and inclination (dip).
The external ﬁeld: diurnal variations, ionospheric currents, magnetic storms, sunspot activity.
The internal ﬁeld: the dipole and non–dipole ﬁelds, secular variations, the geocentric axial dipole hypothesis, geomagnetic reversals, seabed magnetic anomalies,
The dynamo model
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 GEOPHYSICS (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 spin motions of electrons in atoms. 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 magnetic monopole have failed, and the model of permanent magnets as magnetic dipoles with north and south poles is not particularly accurate. Consequently moving charges are normally regarded as fundamental in magnetism.
Basic observations 1. Permanent magnets A magnet attracts iron and steel, 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 electric current exerts a force 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 solenoid 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; angle 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 is deﬁned in terms of magnetic forces: when ﬂowing in each of two inﬁnite parallel wires one metre apart, it 7 7 2 produces a magnetic force of 2 10− newton 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 magnetic dipole. magnetic field 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 (vacuum). 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 density 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 moment. The magnetic moment 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 torque 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 angles to both B and the current element). 1 2 2 The unit of magnetic B–ﬁeld is the tesla (T): 1T = 1N/(A m) = 1kgA− s− = 1Wb/m The weber (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 Gauss (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 equator 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 east 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 magnetometers, of which there are three important types: ﬂuxgate, proton precession, 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)
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 atmosphere (the ionosphere). 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 ocean 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: maps of the Earth’s magnetic ﬁeld intensity or magnetic induction B show that it varies in a • complicated way with latitude, longitude and time. The variations can be modelled using the International Geomagnetic Reference Field (IGRF) equations. These equations involve a large number of coeﬃcients for spherical harmonics. 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 compass 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 degree 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 geomagnetic pole is a magnetic south pole. • 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 map apparent polar wander of the continents. 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 wavelength 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 magnetite, 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 wind (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 age 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. Magnetostratigraphy 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
geomagnetic pole (1980)
90o W GEOPHYSICS (08/430/0012) THE GEOMAGNETIC DYNAMO
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 • Curie temperature 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 the core (e.g. from the sun 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 dynamos 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. (electromotive force) 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 system 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
disk 1 coil 1 GEOPHYSICS (08/430/0012) ENERGY SOURCE OF THE GEOMAGNETIC DYNAMO
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 tides 2.9 × 1012 W × shows that the Earth’s internal heat is orders of magnitude higher than all other internal sources of energy. It is therefore usually assumed that thermal convection drives the eddies in the core responsible for the dynamo action. It is not established whether the source of heat is (a) radioactive decay in the core, or (b) the sinking of solid iron and nickel 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: magnetic anomaly = 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
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.