|
|||||
Faraday discovered that just as current flux caused magnetic circulation, there is current circulation caused by magnetic flux, but the effect is proportional not to the magnetic flux but to its derivative with respect to time. Thus he found the following integral relationship (which is called Faraday's law of magnetic induction): A consequence of this effect is the fact that you can rotate a magnet in such a way as to cause current to flow in a wire, or even more, in a coil of wire. This is the means that generation of electric power is achieved. It should be noted that the notations and concepts of vector calculus were not really existent at the time of these discoveries, and what seem like obvious consequences to us, given Stokes' theorem and the concept of the curl of a vector, were very obscure through most of the 19th Century. If we apply Stokes' theorem to the electric field E here, we can replace the right hand side here by -c multiplied by the flux of the curl of E. If we fix a surface S, the derivative can be brought inside the integral and we obtain, for any fixed surface S:
If we make the physical assumption that a quantity that is 0 when its component in any direction is integrated over any surface is in fact always 0, we get the differential equation:
and this can be considered as a mere restatement of Faraday's Law. This law does not directly imply that there are no magnetic charges, that
serve as "sources" for the magnetic field the way positive electric
charges are "sources" and negative charges are "sinks" for
the electric field. The electric charge density is proportional to the divergence
of the electric field. Faraday also introduced the notion of "lines of force". The electric
field represents the force on a small "test" charge. He suggested linking
the infinitesimal field lines together into paths to which he gave this name.
He found that magnetic field lines have no sources or sinks, and instead form
closed loops. Electric field lines on the other hand originate at the locations
of positive charge, which are its "sources" and end where there is negative
charge, called "sinks". This gives us perhaps the best way to develop an intuitive notion of what the divergence of a vector field is, if you have some idea of the behavior of electrostatic fields. Imagine that your field represents an electric field; its divergence then corresponds to the charge density that would produce such a field. |