Magnetic potentials

Various types of work in addition to pV work are frequently involved in experimental studies. Research on chemical equilibria for example may involve surfaces or phases at different electric or magnetic potentials [11], We will here look briefly at field-induced transitions, a topic of considerable interest in materials science. Examples are stress-induced formation of piezoelectric phases, electric polarization-induced formation of dielectrica and field-induced order-disorder transitions, such as for environmentally friendly magnetic refrigeration. 

As an 11th suggestion, multivalued magnetic potentials arise naturally in magnetics theory [60,61] as well as in other potential theory. This is particularly true during phase transitions, where multivalued potentials seem to be the rule rather than the exception [62]. Theoreticians do all in their power to minimize or eliminate their consideration [61]. However, if deliberately used and optimized, the multivalued magnetic potential... 

The difference in the magnetic potential at two points in a magnetic field is measured by the work necessary to move a unit magnetic pole against the field from one point to the other. This difference is sometimes called a magnetomotive force, in analogy to electromotive force... 

Example 3. Consider now a variation of Example 2 addition of scalar magnetic potential. Consider a generic magnetic potential of the form... 

A magnet is not a store of energy to be tapped as required until exhausted rather, its function is to set up a magnetic potential field in much the same way as the earth sets up its gravitational field. Work is done as magnetic materials are moved around in the field, but the net energy change is zero. 

So Hs is a irrotational field and can therefore be expressed as gradient of a scalar field % called magnetic potential. 

The magnetic potential of a volume V with a given magnetic polarization J can be calculated by solving the Poisson equation... 

Assuming that the magnetic potential is regular at infinity,... 

The total potential of a stored ion is the sum of the electric and magnetic potential q< > and /iB, respectively. Both , the quadrupole potential, and the magnetic field B depend on the square of the axial coordinate. Thus the axial oscillation remains harmonic. The frequency, however, depends on the sign of fi, determined by the direction of the spin. We have designed the nickel ring electrode in such a way that the difference in the axial frequency for both spin directions is 0.7 Hz in a total frequency of 364 kHz [9]. Fig. 7 shows an example of the slight difference... 

U Magnetic potential difference magnetomotive force A = A-turn... 

The permanent electric multipoles (m = 0) are defined by equation (40), the electric multipoles of first order w = 1 by equation (72), and those of the second order w = 2 by equation (79). Similarly, magnetic potential energy of order m — I can be defined, on replacing in equation (83) E by B and p by m. From the general expression [equation (83)] one immediately and quite easily derives all the energies dealt with in the theory of non-linear molecular processes. ... 

The field of electron—atom collisions involves the observation of energy, momentum and spin changes in colliding systems, governed for smaller nuclei by the elementary electrostatic potential 1/r and for larger nuclei by additional relatively-small magnetic potentials understood in terms of Maxwell s equations and special relativity. It involves one electron in the initial state and one, two or (very occasionally) three electrons in the final state. The observation is considerably aided by photon emission and... 

A magnetic potential may not be real. An example is the spin—orbit potential. This is due to the coupling of the magnetic moment of the electron to the solenoid magnetic field caused by the orbital motion of its... 

For heavier atoms the pure fine-structure effect is expected to break down due to relativistic effects. In the very heavy open-shell target atom thallium (Z=81) the ground-state atoms populate only one of the fine-structure levels, and the effect may be important at low energies. In an R-matrix calculation using magnetic potentials derived from the Dirac equation, Goerss, Nordbeck and Bartschat (1991) showed that... 

Strakhov, V. N., 1970a, Some aspects of the plane inverse problem of magnetic potential (in Russian) Izvestia AN SSSR, Fizika Zemli, No. 9, 31-41. 

Let us consider a problem of inversion of the magnetic potential. According to formula (6.37), we have... 

Let us assume that we have observed some magnetic potential Ur C) on the surface of the Earth. The problem is to determine the function /( ) of complex magnetization distribution. We introduce a complex Hilbert space D of data (magnetic potential) with the metric given by formula (7.2), and a complex Hilbert space M of models (functions /(C)) with the metric... 

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