3.1b. Historical evolution (continuation):

        Up to 1800 the quantitative study of electric phenomena was practically restricted to the study of balance states, given the nearly instantaneous establishment of these states. With the invention of the voltaic cell new horizons were disclosed making the laws of stationary state easier to be achieved. In 1820 Hans Christian Oersted noticed that a compass suffered a deflection when placed next to a wire conducting a current [14], and this was possibly the greatest discovery in the history of electromagnetism. In less than a month, Biot and Savart measured the exercised force exerted by an electric current upon the pole of a magnetized needle [15] and from such mensurations Laplace deduced the Biot-Savart’s law for a current element traversed by an electric current i,

dF being the element of force acting on the north pole of a unitary magnet, and j the angle between r and ds vectors. Biot-Savart’s law -- in its differential form (equation 3.1) -- has some aspects which similar to Coulomb’s law, but reflecting the behavior of electrons in permanent escape, that is, elements of electric current with constant intensity i (stationary state). In its integral form it is knows as Ampere-Laplace’s law [16].

        It is not easy to understand what an electric current element is. On the one hand it synthesizes electromagnetism under its simplest expression; on the other hand it does not exist except as a product of a brilliant human idea. It is something hypothetical e purely mathematical in its origins; it is a current that flows from nothing to nowhere and -- when flowing through this real world -- it is just a material point leaving us an equation which -- if integrated into the infinite similar ids -- produces a circuital law related to a real circuit. It is the subtlest and ingenious application of the differential calculation established from Newton’s age on.

        Any similarity between an ids current element and an electron in movement is just coincidental for as time goes by the electron follows its course, while the abstract element remains in its position. Anyway, as we shall see it is possible to characterize the current element in a way which is not so abstract, and conclude:

C-5: Corollary 5

Nothing can prevent us -- regarding the genesis of a magnetic field -- from conceiving the electron as an element from an electric current since both have a vectorial nature.         a

        Notice the contrast between C-1 and C-5.

        Corollary 5 has no imposing character but can open a new path to us. This path, so long as we respect its inherent restrictions, can be quite useful.

        Let us suppose a P point in space, arbitrary but fixed, and ids element -- constant in s and in ds -- in the circuit of an electric current. If we take Biot-Savart’s law (3.1) for granted we might state:

in which Y = sinj / r2 = constant. Therefore, if i is given, dF will be defined. Keeping in mind that

and substituting 3.3 in 3.2, we shall have

or

        Two questions then arise: 1) What is the meaning of v and dq? 2) Is the second equality in 3.4 correct? Is this transformation allowed?

        Quoting Spiegel [17], if dt is given, we can determine dq by means of 3.3, that is, dq is a dependent variable determined through the independent variable dt for a given t. This means that dq can assume any value we want, so long as we choose a convenient dt. So the aforementioned transformation will be allowed, so long as the differentials are tied or v = ds/dt may respect the dependence between dq e dt. Then it is convenient to choose a dqe value for dq -- related to the number of electrons contained in ds and responsible for the dF field. Thus dqe will be the electrolytic charge of the circuit measured in a dt’ time in which this charge crosses ds (it is a value of a theoretical nature and unsettled in itself). In this way it is easy to verify that v acquires the drag velocity characteristic vd (vd = ds/dt') of dqe towards ds (v has the dimension of speed). Under these conditions, the expression 3.5 becomes