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

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Equivalence of Current–Carrying Coils and Magnets; Magnetic Dipoles; - Law of Attraction and Repulsion, Definition of the Ampere GEOPHYSICS (08/430/0012) THE EARTH'S MAGNETIC FIELD OUTLINE Magnetism 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 Reasons against an origin in the crust or mantle 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 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 diácult 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 ² R I ² 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 ¶0=(2¹). The ampere 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 ¶ = 4¹ 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 ¶ I I = 0 1 2 l 2¹r 7 2 where ¶ = 4¹ 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 ² Area ² A I ² - AK B A ? L = m B L = m B¢ sin ² j j j jj j 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 ² where ² 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 (1­ = 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 ¡ ³7 ¡ ¡ ¼ V H Z ¡ ¡d ¡ ³ ¡ ¼ The horizontal component can itself be resolved Z~¡ ³¡ ¡ ¡ ¡ ¡ ¡ ¡³¡ into an easterly BE and a northerly component ¡³ ¡ ¡ ¡³-¡ ¡¡ 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 ¡ ³³ ¡ d is the azimuth of BH , i.e. the angle between A BSB d³³ A S AU B ¡ north and BH ; i is the dip of B, i.e.
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