The thorny way of truth - Free Energy Community

59MARINOV'S COMMENTS TO THE PREVIOUS PAPER BY H. GRASSMANNAt the time of Gr^mann vector algebra and vector calculus were in the cradle. Thusit is very difficult for us to follow Grassmann's reasonings and we can only wonder,seeing that without the appropriate mathematical tools and with a very limited experimentalbasis, Grassmann arrives at the discovery of his formula which is the fundamentalformula for the magnetic interactions (the "cardinal formula" of electromagnetism).Grassmann's formula in vector form is more simple than Ampere's formula and it opensimmediately the door for introducing the magnetic potential A and thus for introducingthe magnetic intensity B. However if vectors will be not used, then Ampere's formulalooks more simple. And if we take Ampere's formula in its vector form (omitting theconstant factor depending on the system of measuring units)f = {3(r.a)(r.b) - 2(a.b)r^}r/r^, (A)where f is the force with which the current element a acts on the current element b, wecome at once to the scalar formula (1) of Grassmann's paper, if putting r.a = racosa,r.b = rbcosB, a.b = abcose, and f/f = - r/r.However it is not at all as easy to come from Grassmann's formula in vector formto the scalar formula (4) in Grassmann's paper.f = bx(axr)/r^ (B)I shall do it, in order to spare the time of the reader.Let us take the current element a at the origin of the reference frame lying in thexy-plane, i.e., a = ax + a^, assuming a = acosa >0, a = acos(7r/2 - a) = asina > 0.X jr X yLet us take the vector r connecting a with b and pointing from a to b along the x-axis,i.e., r = rx, assuming r > 0. The orientation of the vector b must be arbitrary, i.e.,b = b X + b^ + b z, but for simplicity's sake let us assume b > 0, b > 0, b > 0.X y Z i\Ji.Now formula (B) can be writtenf = bx(axr)/r^ = (b^x +b^ +b^z)x{(a^x +ayy)xrx}/r^ = - (b^£ + b^z)xayZ /r^ =- bjasina(£xz)/r ,(C)where b. = b.jt = (b + b') £ is the component of b in the xy-plane and 1 is the unit)6 j6 X yvector along the direction of this component.We see that the unit vector -£xz lies in the xy-plane and as b_ concludes with x anangle < 3' < 7t/2, the vector- fxz concludes with x an angle 7t/2 < 3" < "fr, as3" = tt/2 + 3'. Thus the force f points from the end of the vector r to the end of thevector a (und zwar erfolgt die Bewegung senkrecht gegen b^^ in der durch a und r gelegtenEbene nach derjenigen Seite hin, nach welcher der Schenkel a des Winkels a von dem anderenSchenkel aus (i.e., from the "Schenkel" r) betrachtet liegt).I tried to find also Ampere's original paper where he introduces his famous formula(Ampere, "Memoire sur la theorie mathematique des phenomenes electrodynamique", MEMOIRES