Magnetometer - the History

The magnetometer is an Instrument for measuring the strength and sometimes the direction of magnetic fields, including those on or near the Earth and in space. Magnetometers are also used to calibrate electromagnets and permanent magnets and to determine the magnetization of materials. Magnetometers specifically used to measure the Earth's field are of two types: absolute and relative (classed by their methods of calibration). Absolute magnetometers are calibrated with reference to their own known internal constants. Relative magnetometers must be calibrated by reference to a known, accurately measured magnetic field.

The simplest absolute magnetometer, devised by C.F. Gauss in 1832, consists of a permanent bar magnet suspended horizontally by a gold fiber. And perhaps we should dwell on the name Gauss first, because of the unit called after him and the for man himself, the outstanding German scientist Carl Friedrich Gauss. The Gauss is a unit of magnetic induction in the centimetre-gram-second system of physical units. One gauss corresponds to the magnetic flux density that will induce an electromotive force of one abvolt (10 volt) in each linear centimeter of a wire moving laterally at one centimeter per second at right angles to a magnetic flux. One gauss corresponds to 10 tesla (T), the International System Unit. The gauss is equal to 1 maxwell per square centimeter, or 10 weber per square metro. Magnets are rated in gauss. Who was this extraordinary man Gauss? Carl Friedrich Gauss, who, with Archimedes and Newton, ranks as one of the greatest mathematicians of all time, at an early age overturned the theories and methods of 18th-century mathematics and, following his own revolutionary theory of numbers, opened the way to a mid-19th- century rigorization of analysis. Although he contributed significantly to pure mathematics, he also made practical applications of importance for 20th-century astronomy, geodesy, and electromagnetism. His own dictum,"Mathematics, the queen of the sciences, and arithmetic, the queen of mathematics," aptly conveys his perception of the pivotal role of mathematics in science. Born on April 30, 1777, in Brunswick, now in Germany, Gauss was the only son of poor parents. Impressed by his ability in mathematics and languages, his teachers and his devoted mother recommended him to the Duke of Brunswick, who granted him financial assistance to continue his education in secondary school and from 1795 to 1798 to study mathematics at the University of Gottingen. In 1799 he obtained his doctorate in absentia from the university at Helmstedt. The subject of his dissertation was a proof of the fundamental theorem of algebra, which was proven only partially before Gauss, which states that every algebraic equation with complex coefficients has complex solutions; moreover, Gauss skillfully formulated and proved this theorem without the use of complex numbers.

Gauss was deeply religious, aristocratic in bearing, and conservative. He remained aloof from the progressive political currents of his time. In Gauss, apparent contrasts were combined in an effective harmony. A brilliant arithmetician with a phenomenal memory for numbers, he was at once a profound theoretician and an outstanding practical mathematician. Teaching was his only aversion, and, thus, he had only a few students. Instead, he effected the development of mathematics through his publications, about 155 titles, to which he devoted the greatest care.

Three principles guided his work: "Pauca, sed matura" ("Few, but ripe"), his favourite saying; the motto "Ut nihil amplius desiderandum relictum sit" ("That nothing further remains to be done"); and his requirement of utmost rigour. It is evident from his posthumous works that there are extensive and important papers that he never published because, in his opinion, they did not satisfy one of these principles. He pursued a research topic in mathematics only when he might anticipate meaningful relationships of ideas and results that were commendable because of their elegance or generality. The golden anniversary of the granting of the doctorate to Gauss was celebrated in 1849. For this event, he prepared a new edition of his earlier proofs of the fundamental theorem of algebra, which, because of his declining health, was his last publication. The honour that gave him the greatest joy, however, was the bestowal of honorary citizenship on him by the city of Gottingen. On the basis of his outstanding research in mathematics, astronomy, geodesy, and physics, he was elected as a fellow in many academies and learned societies. He declined numerous invitations of other universities to become a professor and remained on the faculty of the University of Gottingen until his death on February 23, 1855. Soon after his death, coins were struck in his honour. The title of mathematicorum princeps is a fitting tribute.

Gauss's theorem, also called GAUSS'S LAW, either of two statements describing electric and magnetic fluxes. Gauss's law for electricity states that the electric flux across any closed surface is proportional to the net electric charge enclosed by the surface. The law implies that isolated electric charges exist and that like charges repel one another while unlike charges attract.

Gauss's law for magnetism states that the magnetic flux across any closed surface is zero; this law is consistent with the observation that isolated magnetic poles (monopoles) do not exist. Mathematical formulations for these two laws--along with Ampere's law (concerning the magnetic effect of a changing electric field or current) and Faraday's law of induction (concerning the electric effect of a changing magnetic field)--are collected in a set called Maxwell's equations (q.v.), which provide the basis of unified electromagnetic theory.

Although we strayed off the subject and nearly lost ourselves in history, there is another name mentioned on which I would like to expand, Tesla. Also the unit of magnetic induction or magnetic flux density in the metre-kilogram-second system (SI) of physical units. One tesla equals one weber per square metre, corresponding to 10 gauss. It is named for Nikola Tesla and it is used in all work involving strong magnetic fields, while the gauss is more useful with small magnets. Like Gauss, Nikola Tesla was one of the outstanding scientists of our times. Nikola Tesla was born on July 9th in the year 1856 in Smiljan, Croatia. He died the next century and witnessed the second world war. He died on the seventh of January 1943 in New York City. He was a Serbian-American inventor and researcher who discovered the rotating magnetic field, the basis of most alternating current machinery. He emigrated to the United States in 1884 and sold the patent rights of his system of alternating current dynamos, transformers, and motors to George Westinghouse the following year. In 1891 he invented the Tesla coil, an induction coil widely used in radio technology. Tesla was from a family of Serbian origin. His father was an Orthodox priest; his mother was unschooled but highly intelligent. A dreamer with a poetic touch, as he matured Tesla added to these earlier qualities those of self-discipline and a desire for precision. Training for an engineering career, he attended the Technical University at Graz, Austria, and the University of Prague. At Graz he first saw the Gramme dynamo, which operated as a generator and, when reversed, became an electric motor, and he conceived a way to use alternating current to advantage. Later, at Budapest, he visualized the principle of the rotating magnetic field and developed plans for an induction motor that would become his first step toward the successful utilization of alternating current. In 1882 Tesla went to work in Paris for the Continental Edison Company, and, while on assignment to Strassburg in 1883, he constructed, in after work hours, his first induction motor. Tesla sailed for America in 1884, arriving in New York, with four cents in his pocket, a few of his own poems, and calculations for a flying machine. He first found employment with Thomas Edison, but the two inventors were far apart in background and methods, and their separation was inevitable.

In May 1885, George Westinghouse, head of the Westinghouse Electric Company in Pittsburgh, bought the patent rights to Tesla's polyphase system of alternating current dynamos, transformers, and motors. The transaction precipitated a titanic power struggle between Edison's direct-current systems and the Tesla-Westinghouse alternating-current approach, which eventually won out. Tesla soon established his own laboratory, where his inventive mind could be given free rein. He experimented with shadow graphs similar to those that later were to be used by Wilhelm Rontgen when he discovered X-rays in 1895.

Gauss - the Unit

Before 1932 the name was applied to the unit of magnetic field strength now called the oersted, and it is sometimes still used in this sense (e.g., the Earth may be said to have a magnetic field strength of about one gauss). Measuring the period of oscillation of the magnet in the Earth's magnetic field gives a measure of the field's strength. A widely used modern absolute instrument is the proton-precession magnetometer. It measures a voltage induced in a coil by the reorientation (precession) of magnetically polarized protons in ordinary water. The Schmidt vertical-field balance, a relative magnetometer used in geophysical exploration, uses a horizontally balanced bar magnet equipped with mirror and knife edges.

Magnetic methods.

Measurements can be made of the Earth's total magnetic field or of components of the field in various directions. The oldest magnetic prospecting instrument is the magnetic compass, which measures the field direction. Other instruments include magnetic balances and fluxgate magnetometers. Most magnetic surveys are made with proton-precession or optical-pumping magnetometers, which are appreciably more accurate. The proton magnetometer measures a radio-frequency voltage induced in a coil by the reorientation (precession) of magnetically polarized protons in a container of ordinary water. The optical- pumping magnetometer makes use of the principles of nuclear resonance and caesium or rubidium vapour. It can detect minute magnetic fluctuations by measuring the effects of light-induced (optically pumped) transitions between atomic energy levels that are dependent on magnetic field strength.

Magnetic Surveys
Magnetic surveys are usually made with magnetometers borne by aircraft flying in parallel lines spaced two to four kilometers apart at an elevation of about 500 metros (one metro = 3.28 feet) when exploring for petroleum deposits and in lines 0.5 to one kilometer apart roughly 200 metros above the ground when searching for mineral concentrations. Ground surveys are conducted to follow up magnetic anomaly discoveries made from the air. Such surveys may involve stations spaced only 50 metros apart.

Magnetometers also are towed by research vessels. In some cases, two or more magnetometers displaced a few metros from each other are used in a gradiometer arrangement; differences between their readings indicate the magnetic field gradient. A ground monitor is usually used to measure the natural fluctuations of the Earth's field over time so that corrections can be made. Surveying is generally suspended during periods of large magnetic fluctuation (magnetic storms).

Magnetic effects result primarily from the magnetization induced in susceptible rocks by the Earth's magnetic field. Most sedimentary rocks have very low susceptibility and thus are nearly transparent to magnetism. Accordingly, in petroleum exploration magnetics are used negatively: magnetic anomalies indicate the absence of explorable sedimentary rocks. Magnetics are used for mapping features in igneous and metamorphic rocks, possibly faults, dikes, or other features that are associated with mineral concentrations. Data are usually displayed in the form of a contour map of the magnetic field, but interpretation is often made on profiles.

Rocks cannot retain magnetism when the temperature is above the Curie point (about 500 degrees Celsius for most magnetic materials), and this restricts magnetic rocks to the upper 40 kilometers of the Earth's interior. The source of the geomagnetic field must be deeper than this, and it is now believed that convection currents of conducting material in the outer core generate the field. These currents couple to the Earth's spin, so that the magnetic field--when averaged over time--is oriented along the planet's axis. The currents gradually change with time in a somewhat erratic manner and their aggregate effect sometimes reverses, which explains the time changes in the Earth's field. This is the crux of the magnetohydrodynamic theory of the geomagnetic field.

Measurement of the field.

Magnetic fields can be measured in various ways. The simplest measurement technique still employed today involves the use of the compass, a device consisting of a permanently magnetized needle that is balanced to pivot in the horizontal plane. In the presence of a magnetic field and in the absence of gravity, a magnetized needle aligns itself exactly along the magnetic field vector. When balanced on a pivot in the presence of gravity, it becomes aligned with a component of the field. In the conventional compass, this is the horizontal component. A magnetized needle may also be pivoted and balanced about a horizontal axis. If this device, called a dip meter, is first aligned in the direction of the magnetic meridian as defined by a compass, the needle lines up with the total field vector and measures the inclination angle. Finally, it is possible to measure the magnitude of the horizontal field by the oscillations of the compass needle. It can be shown that the period of such an oscillation depends on properties of the needle and the strength of the field.

Magnetic observatories continuously measure and record the Earth's magnetic field at a number of locations. In an observatory of this sort, magnetized needles with reflecting mirrors are suspended by quartz fibers. Light beams reflected from the mirrors are imaged on a photographic negative mounted on a rotating drum. Variations in the field cause corresponding deflections on the negative. Typical scale factors for such observatories correspond to 2- 10 nanoteslas per millimetre vertically and 20 millimetres per hour horizontally. A print of the developed negative is called a magnetogram. Magnetic observatories have recorded data in this manner for well over 100 years. Their magnetograms are photographed on microfilm and submitted to world data centers, where they are available for scientific or practical use. Such applications include the creation of world magnetic maps for navigation and surveying; correction of data obtained in air, land, and sea surveys for mineral and oil deposits; and scientific studies of the interaction of the Sun with the Earth.

In recent years, other methods of measuring magnetic fields have proved more convenient, and older instruments are being gradually replaced. One such method involves the proton-precession magnetometer, which makes use of the magnetic and gyroscopic properties of protons in a fluid such as gasoline. In this method, the magnetic moments of protons are first aligned by a strong magnetic field produced by an external coil. The magnetic field is then turned off abruptly, and the protons try to align themselves with the Earth's field. However, since the protons are spinning as well as magnetized, they precess around the Earth's field with a frequency dependent on the magnitude of the latter. The external coil senses a weak voltage induced by this gyration. The period of gyration is determined electronically with sufficient accuracy to yield a sensitivity between 0.1 and 1.0 nanotesla.

An instrument that complements the proton- precession magnetometer is the flux-gate magnetometer. In contrast to the proton-precession magnetometer, the flux-gate device measures the three components of the field vector rather than its magnitude. It employs three sensors, each aligned with one of the three components of the field vector. Each sensor is constructed from a transformer wound around a core of high-permeability material (e.g., mu-metal). The primary winding of the transformer is excited with a high-frequency similar to 5 kilohertz) sine wave. In the absence of any field along the transformer axis, the output signal in the secondary winding consists of only odd harmonics (component frequencies) of the drive frequency. If, however, a field is present, it biases the hysteresis loop for the core in one direction. This causes the core to become saturated sooner in one half of a drive cycle than in the other. This in turn causes the secondary voltage to include all even harmonics as well as odd. The amplitude and phase of the even harmonics are linearly proportional to the component of the field along the axis of the transformer.

Most modern magnetic observatories have both a proton-precession magnetometer and a flux-gate magnetometer mounted on granite pillars in nonmagnetic, temperature controlled rooms. The outputs from the instruments are electrical signals, and they are digitized and recorded on magnetic media. Many observatories also transmit their data soon after acquisition to central facilities where they are stored with data from other locations in a large computer database.

Magnetic measurements are often made at locations remote from fixed observatories. Such measurements are commonly part of a survey designed to better define the Earth's main field or to detect anomalies in it. Surveys of this type are routinely carried out by foot, ship, aircraft, and spacecraft. For surveys near the Earth's surface the proton-precession magnetometer is almost always used because it does not need to be precisely aligned. Above the Earth's surface the main field decreases rapidly, and the need for precise alignment is less severe. Thus, flux-gate magnetometers are generally employed on spacecraft. Calculation of components of the vector field in a co-ordinate system fixed with respect to the Earth requires knowledge of the location and orientation of the spacecraft. geomagnetics, a branch of geophysics concerned with all aspects of the Earth's magnetic field, including its origin, variation through time, and manifestations in the form of magnetic poles, the remnant magnetization of rocks, and local or regional magnetic anomalies. The latter reflect the difference between theoretical and observed magnetic intensities at points of measurement with a magnetometer, and, when plotted on a magnetic map (called an aeromagnetic map if the magnetometer was flown across the area), the anomalies provide the basis for inferences about probable subsurface structure and composition. prospecting, search for economically exploitable mineral deposits. Up to the 20th century, prospecting was done by men roaming likely areas on foot and recognizing gold, iron, lead, or other valuable ores by sight. Certain types of mineral deposits are associated with certain types of rock and land forms; copper, lead, and zinc, for example, generally appear in igneous rock formed by cooling of masses of molten minerals at or near the Earth's surface. Geologists can sometimes infer the extent of deposits by mapping outcroppings; drilling is then used to confirm the estimates.

In the 20th century, more sophisticated techniques developed as the result of the maturing of the physical sciences and the need to seek minerals beneath the surface. Some valuable minerals, such as iron and copper, are magnetic: first the compass and later the more sensitive magnetometer have been used to detect them. The gravimeter, an instrument that can detect minute changes in the Earth's gravitational field, can be used to detect certain minerals that have densities different from those of the surrounding formations. Some sulphide mineral deposits have undergone partial oxidation, and the resulting nonuniformity in chemical composition creates an electric potential that causes currents to flow in the surrounding ground; voltmeters can be used to detect them. Another electrical method involves implanting electrodes in the ground and tracing the current between them by means of a galvanometer; the current will seek out conductors in the ground.

Seismic methods utilize information gained from the transmission of natural (earthquake) and artificial shock waves by different underground bodies. In systematic seismic exploration, a hole is drilled and an explosive charge set off in it; seismic waves, traveling to the boundaries between different rock layers and reflected from these layers, can be timed, and the types of rock deduced.

Ores of uranium and thorium give off radiation that can be detected by suitable instruments such as the Geiger counter. Geochemical methods of prospecting involve chemical analysis for traces of metals in soils, vegetation, and stream water or silt. Methods of prospecting for oil and natural gas are similar to those used for minerals.

Nuclear magnetic resonance.

In the absence of atomic motion in rigid lattices (crystals), NMR makes it possible to determine molecular structures not observable by other means. In many solids, even at low temperatures, there occur atomic diffusion and rotation of groups of atoms. These movements affect the shape of the NMR absorption peak. A study of these effects as a function of temperature can supplement other physical measurements. In metals, the nuclei are influenced by an interaction between the spins of the conduction electrons (electrons not bound to atoms that move freely through the metal) and the applied field. This condition results in a shift of the resonant frequency from the value observed for the same nucleus when it is present in an insulator. These so-called metallic shifts provide important information on the magnetic susceptibility, the quantum mechanical wave functions that describe energy states, and the density of states of conduction electrons in the metal. In superconductors, the shape of the NMR spectral peaks provide detailed information on the penetration and internal distribution of the magnetic field. In ferromagnets or antiferromagnets (crystals in which not all electrons are paired), the NMR is influenced by the internal magnetic fields produced by the array of ordered electronic spins. In ferromagnets the shift is a measure of the lattice magnetization; in an antiferromagnet there are at least two shifts that give the magnetization of each antiferromagnetic sub lattice separately, a result unattainable by conventional magnetic measurements. For certain nuclei, the NMR spectrum reveals the existence of nuclear electric quadrupole moments (an electric quadrupole consists of a charge distribution equivalent to a special arrangement of two electric dipoles) that interact with the electric fields that exist at the nuclear sites. These interactions provide information on the microscopic distribution of electric charge around the nucleus. The most important consequence of the extraordinary sharpness of nuclear magnetic resonance (NMR) lines in liquids is the possibility of measuring the chemical shifts--that is, the separations between NMR lines from nuclear spins of the same species but in different molecular environments. The physical origin of chemical shifts is the following: an external magnetic field polarizes the closed electron shells of the atoms and produces a small magnetic field, proportional to the external field, which shifts the NMR line with respect to its position for the bare nucleus--e.g., one that is devoid of electrons. The bare nucleus itself is never observed, but the atomic diamagnetic shifts that correspond to atoms located in different molecular sites are slightly different, and it is their differences that produce the chemical shifts. As an example, the proton NMR spectrum of ethyl alcohol exhibits three peaks, with relative weights or intensities of 3:2:1. In more complicated molecules such spectra contain much chemical information and can help in the determination of unknown molecular structures. The multiplicity of lines is further increased by the interaction between nuclear spins. As already mentioned in connection with motion narrowing in liquids, the usual magnetic dipolar interactions are averaged out by molecular motion and do not split the NMR spectra. There exists, however, an indirect interaction between nuclear spins, caused by the electrons, that splits the resonance line of a specific nuclear spin into many components. High resolution nuclear magnetic resonance has become one of the most prized tools in the fields of organic chemistry and biochemistry. On the experimental side, the requirements to be met by the equipment are severe. In order to match natural line widths of a fraction of a cycle, the applied magnetic fields must have a relative stability and homogeneity throughout the sample better than one part in 10. Special magnets that give uniform fields and are stabilized, devices that twirl samples in order to smooth out the magnetic inhomogeneity, and sophisticated radio- frequency detection equipment are commercially available. The trend toward higher fields (over 100 kilogauss), resulting from super conducting solenoids, improves the resolution by increasing the chemical shift splittings and the signal-to-noise ratio.

The measurement of the precession frequency of proton spins in a magnetic field can give the value of the field with high accuracy and is widely used for that purpose. In low fields, such as the Earth's magnetic field, the NMR signal is expected to be weak because the nuclear magnetization is small, but special devices can enhance the signal 100 or 1000 times. Incorporated in existing portable magnetometers, these devices make them capable of measuring fields to an absolute accuracy of about one part in 1,000,000 and detecting field variations of about 10 gauss. Apart from the direct measurement of the magnetic field on Earth or in space, these magnetometers prove to be useful whenever a phenomenon is linked with variations of magnetic field in space or in time, such as anomalies arising from submarines, skiers buried under snow, archaeological remains, and mineral deposits.

Electron-spin resonance.

In contrast to nuclear magnetic resonance, electron- spin resonance (ESR) is observed only in a restricted class of substances. These substances include transition elements--that is, elements with unfilled inner electronic shells--free radicals (molecular fragments), metals, and various paramagnetic defects and impurity centers. Another difference from NMR is a far greater sensitivity to environment; whereas the resonance frequencies in NMR in general are shifted from those of bare nuclei by very small amounts because of the influence of conduction electrons, chemical shifts, spin-spin couplings, and so on, the ESR frequencies in bulk matter may differ greatly from those of free spins or free atoms because the unfilled subshells of the atom are easily distorted by the interactions occurring in bulk matter. A model that has been highly successful for the description of magnetism in bulk matter is based on the effect of the crystal lattice on the magnetic center under study. The effect of the crystal field, particularly if it has little symmetry, is to reduce the magnetism caused by orbital motion. To some extent the orbital magnetism is preserved against ligand fields of low symmetry by the coupling of the spin and orbital momenta. The total energy of the magnetic center consists of two parts: (1) the energy of coupling between magnetic moments due to the electrons and the external magnetic field, and (2) the electrostatic energy between the electronic shells and the ligand field, which is independent of the applied magnetic field. The energy levels give rise to a spectrum with many different resonance frequencies, the fine structure Another important feature of electron-spin resonance results from the interaction of the electronic magnetization with the nuclear moment, causing each component of the fine-structure resonance spectrum to be split further into many so-called hyperfine components. If the electronic magnetization is spread over more than one atom, it can interact with more than one nucleus; and, in the expression for hyperfine levels, the hyperfine coupling of the electrons with a single nucleus must be replaced by the sum of the coupling with all the nuclei. Each hyperfine line is then split further by the additional couplings into what is known as superhyperfine structure. The key problem in electron-spin resonance is, on one hand, to construct a mathematical description of the total energy of the interaction in the ligand field plus the applied magnetic field and, on the other hand, to deduce the parameters of the theoretical expression from an analysis of the observed spectra. The comparison of the two sets of values permits a detailed quantitative test of the microscopic description of the structure of matter in the compounds studied by ESR. The transition elements include the iron group, the lanthanide (or rare-earth) group, the palladium group, the platinum group, and the actinide group. The resonance behaviour of compounds of these elements is conditioned by the relative strength of the ligand field and the spin-orbit coupling. In the lanthanides, for instance, the ligand field is weak and unable to uncouple the spin and orbital momentum, leaving the latter largely unreduced. On the other hand, in the iron group, the components of the ligand field are, as a rule, stronger than the spin-orbit coupling, and the orbital momentum is strongly reduced. The advent of ESR has marked a new understanding of these substances. Thus, it was formerly thought that in the iron group and the lanthanide group ions of the crystal were bound together solely by their electrostatic attraction, the magnetic electrons being completely localized on the transition ion. The discovery of superhyperfine structure demonstrated conclusively that some covalent bonding to neighbouring ions exists. With few exceptions, the magnetic moments of imperfections such as vacancies at lattice sites and impurity centers in crystals that give rise to an observable ESR have the characteristics of a free electronic spin. In the study of these centers, hyperfine and superhyperfine structure provide a mapping of the electronic magnetization and make it possible to test the correctness of the model chosen to describe the defect. The most widely studied by resonance are those of phosphorus, arsenic, and antimony, substituted in the semiconductors silicon and germanium. Studies of hyperfine and superhyperfine structure give detailed information on the status of these impurities.

Free radicals are ideally suited for study by electron- spin resonance. They can be studied in a concentrated form or in very dilute solutions. The sensitivity of ESR is particularly important for the study of very short-lived species. The ESR of free radicals in solutions gives an extreme wealth of hyperfine lines because the magnetic electron is not localized on one nucleus but interacts with several nuclei of the radical.

Maxwell's equations Four equations that, together, form a complete description of the production and interrelation of electric and magnetic fields. The physicist James Clerk Maxwell in the 19th century based his description of electromagnetic fields on these four equations, which express experimental laws. The statements of these four equations are, respectively: (1) electric field diverges from electric charge, an expression of the Coulomb force, (2) there are no isolated magnetic poles, but the Coulomb force acts between the poles of a magnet, (3) electric fields are produced by changing magnetic fields, an expression of Faraday's law of induction, and (4) circulating magnetic fields are produced by changing electric fields and by electric currents, Maxwell's extension of AmpSre's law (q.v.) to include the interaction of changing fields. The most compact way of writing these equations in the metre-kilogram- second (mks) system is in terms of the vector operators div (divergence) and curl. In these expressions the Greek letter rho, [{rho}], is charge density, J is current density, E is the electric field, and B is the magnetic field; here, D and H are field quantities that are proportional to E and B, respectively. The four Maxwell equations, corresponding to the four statements above, are: (1) div D = rho} (2) div B = 0, (3) curl E = -dB/dt, and (4) curl H = dD/dt + J.

Conclusion

Well, as one might have observed I strayed of the subject. Really, all I wanted to stress is the importance of Gauss and Tesla in our modern day exploration of the surface and interior of our small planet called Earth, Our own design of hardware and software of this application of a magnetometer has only been made possible through these outstanding scientists and mathematicians that paved the industrial road we travel more than a century ago.

BIBLIOGRAPHY.

Gauss, the life of (Encyclopaedia Brittanica, 1996). Inez Hunt and Wanetta W. Draper, Lightning in His Hand: The Life Story of Nikola Tesla (1964), is a complete, authoritative, nontechnical biography. Nikola Tesla Museum, Nikola Tesla 1856-1943: Lectures, Patents, Articles (1956), contains authentic reprints, diagrams, lectures, and considerable detailed information. Nikola Tesla, Experiments with Alternate Currents of High Potential and High Frequency (1904), furnishes Tesla's own story of his Colorado experiments. C.M. van der Vaart; Gauss- a thesis(1981). Various sources of my own days of study.