In the general theory of quantum electrodynamics, one takes the vector and scalar potentials as the fundamental quantities in a set of equations that replace the Maxwell equations. E and B are slowly disappearing from the modern expression of physical law; they are being replaced by the vector potential, A and scalar potential, Ø. Feynman says the vector potential is not just a mathematical convenience, but is introduced because it does have an important physical significance (Feynman). Lets review a few of examples:
It is easy to agree that a long solenoid carrying an electric current has a B-field inside - but none outside. If we arrange a situation where the electrons are to be found only outside of the solenoid, we know that there will still be an influence on the motion of electrons - as this is the workings of the common electrical transformer. This phenomena has always been of interest to students, because the induction in the wires takes place in a region of space where the resultant magnetic flux is reduced to zero. How could this be? According to classical physics this is impossible, as the force depends only on B, yet we use this transformer principle in common electronic components.
It turns out, that quantum mechanically we can find out that there is a magnetic field inside the solenoid by going around it - even without ever going close to it. We must use the vector potential, A, as shown in figure 4. Alternatively, if we are not too concerned about the zero B-field in the region of the electron, we can also use Faraday's Law of Induction. This law states that the induced electromotive force is equal to the rate at which the magnetic flux through a circuit is changing, as in
Figure 4. The magnetic field and the vector potential of a long solenoid.
In the case of the long solenoid, it turns out that both the classical and quantum calculations give the same result.
Physical effects on charged particles - in a zero B-field - have been studied since the 1950s. The reader is advised to refer to quantum interference of electrons (S. Olariu and I. Iovitzu Popescu), for further study.
Although this is a very important subject, we encourage the reader to investigate this area for himself. Bohm and Aharanow show in their electron interference experiment that a magnetic field can influence the motion of electrons even though the field exists only in regions where there is an arbitrarily small probability of finding the electrons.
Magnetic flux is constructed from two sources, as in figure 5. Both magnets move uniformly in opposite directions with a speed V producing an Em on the electron, inside the conductor. We can find the total Em field by superposition, as follows:
Since B and V are equal in magnitude for both magnets, we find by vector addition the total induced electrical field, as follows:
We notice that the induced electrical field is twice that from a single magnet, while the sum of B is remarkably - zero. This experiment is easy to setup and verify in any electronics laboratory with a pair of magnets, a wire and a voltmeter. In fact, you may wrap the conductor, in an electrostatic or magnetic shielding, and find the same result.
Figure 5. An electron, in a conductor, experiences a force due to the flux from two moving magnetic sources.
The author has tested a setup by pulsing strong currents, opposite and equal, through multiple parallel conductors. The configuration of the conductors in this type of experiment will cancel the B-fields, while still producing an Em field, in accordance with Eq. 4.2. This is similar to an experiment by Hooper (W. J. Hooper), who successfully predicted and measured the motional electric field - all in zero resultant B-field.
Interestingly, all of the above experiments can influence an electron with a zero B-field, in the region of the electron. This has some profound implications - one of which is that the motional electric force field is immune to electrostatic or magnetic shielding.
Experimentally, it can be confirmed that the motional electric field is immune to shielding and follows the boundary conditions of the magnetic (not electric) field. The only way to shield a motional electric field is to use a magnetic shield around the source of the magnetic flux - containing it at the source. These effects are not startling if one remembers that the motional electric field is a magnetic effect and that a magnetic field has a different boundary condition than the electric field.