Vector potentials measurement

For the field calculation it more convenient to use a tx(B) curve than the normal ix(H) curve because the calculated vector potential A is derived from the flux density B. This ii(B) curve however can be calculated easily from the measured values. 

Helicity is defined as a qualitative measurement of how a topological configuration is linked, knotted, or twisted. If A is the vector potential of electromagnetism, the quantity then defines magnetic helicity. In order to obtain a nonnull value for K, the condition A (V A) /- 0 must be verified in the volume... 

In search for an explanation, Aharonov and Bohm worked out quantum mechanics equations based on the measurable physical effect of the vector potential, which is nonnull in a region outside the solenoid. Like many other paradoxes in physics, including the twin paradox, the interpretation of this experiment proposed in 1959 was the subject of an intense controversy among researchers. This controversy is well summarized in a review article [55] and in other references of interest [56-67]. 

If the velocity U of an electron within the beam is constant outside the solenoid, the variation of the vector potential A as a function of time in the medium, and thus also in the solenoid, will induce a modification of the phase, as indicated by the equations written above. This will produce a modification of the boundary conditions on the boundary of the solenoid for the quantities a and b. We must also stress that the modification of the vector potential outside the solenoid is generated by either an external or an internal source feeding the solenoid. This can explain the existence of the Aharonov-Bohm effect for toroidal, permanent magnets. The interpretation of the Aharonov-Bohm effect is therefore classic, but the observation of this effect requires the principle of interference of quantum mechanics, which enables a phase effect to be measured. 

As is well known in classical electromagnetics, the fields described by the Maxwell equations can be derived from a vector potential and a scalar potential. However, there are various forms that are possible, all giving the same fields. This is referred to as gauge invariance. In making measurements at some point... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

Note that as discussed in previous sections, under static conditions, these two antennas give no fields. In going between two static conditions, one can have the same fields at intermediate times, but a change in the electric impulse, this being related to a change in the Lorenz vector potential or to a nonzero time integral of the gradient of the Lorenz scalar potential. However, with no fields, the vector potential has zero curl, which in a QED sense is not measurable. 

We consider now the Aharonov-Bohm effect as an example of a phenomenon understandable only from topological considerations. Beginning in 1959 Aharonov and Bohm [30] challenged the view that the classical vector potential produces no observable physical effects by proposing two experiments. The one that is most discussed is shown in Fig. 10. A beam of monoenergetic electrons exists from a source at X and is diffracted into two beams by the slits in a wall at Y1 and Y2. The two beams produce an interference pattern at III that is measured. Behind the wall is a solenoid, the B field of which points out of the paper. The absence of a free local magnetic monopole postulate in conventional... 

Sadowski etal. [49] have described the use of 3D autocorrelation vectors that are based on the electrostatic potential measured on the molecular surface of a molecule. The electrostatic potential was measured over 12 different distances giving 12 autocorrelation coefficients per molecule. The vectors were calculated for the molecules in two different combinatorial libraries a xanthene library and a cubane library. The compounds were then used to train a Kohonen network. The network was successfully able to separate the libraries. 

In principle, a chemical shift calculation represents a perturbation theory, because of the presence of an external field Bz and magnetic moments due to the dipole character of nuclei. Therefore, perturbations to the Hamiltonian and the wave function have to be considered. The next important point is that the origin of the vector potential Az is not fixed due to the relation Bz = rot Az- Any change of the gauge origin Rq should not change any measurable observable. Therefore, a gauge transformation of the wave function 1%) and Hamilton operator h is essential... 

First, we note that the standard photodetection is a local measurement of the field variables (intensities). At the same time, the Aharonov-Bohm effect represents a topological measurement referred to the properties of vector potential along some loop. In the usual form, the Aharonov-Bohm effect deals with static or slowly time-varying magnetic fields [101]. The effect consists in the appearance of a persistent current in a metallic loop over which the magnetic flux passes. This current is a periodic function of magnetic flux with the period of flux quantum hc/e. Besides that, certain resistance oscillations in the loop incorporated into an external circuit with the same period can occur. 

This boundary was assumed to be a ground plane with zero magnetic vector potential. A constant heat flux of 5.5 kW/m was prescribed for the bottom, which is the measured cooling rate for this brick wall. For the momentum transfer, the bottom wall was regarded as a non-slip wall. 

Since there was no current flow in the refractory walls, the magnetic vector potential. A, was set equal to zero. For heat transfer, a constant heat flux boundary condition, equal to the measured heat loss flux, was specified for this wall. The heat loss fluxes at the side wall for the water-cooled copper panels in Ae bullion and slag were 31.3 kW/m and 1.75 kW/m respectively. The conventional non-slip boundary condition was used for momentum transfer. 

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