]> 29.2 Faraday's Law and its Consequences

## 29.2 Faraday's Law and its Consequences

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)

$d d t ∬ S B ⟶ · n ^ d S = − c ∮ δ S E ⟶ · d l ⟶$

Stated in words: the derivative of the flux of magnetic field through a surface is proportional to the electromotive force around its boundary (which is the circulation integral of the electric field around it).

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 19 th 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 $S$

$∬ S ( ∂ B ⟶ ∂ t + c ∇ ⟶ × E ⟶ ) · d S ⟶ = 0$

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

$1 c ∂ B ⟶ ∂ t + ∇ ⟶ × E ⟶ = 0 ⟶$

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.

If we take the divergence of the left hand side of the last equation above, we find that the divergence of the time derivative of $B ⟶$ must be 0. This means that if there is a divergence of $B ⟶$ , it must be exactly the same at all times; unless there is a magnetic current density that appears in this equation. No magnetic sources have ever been found, so that as far as we now can tell we have $∇ ⟶ · B ⟶ = 0$ .

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.

For magnetic fields, the direction of the magnetic field at a point represents the direction that a compass needle takes if placed at that point, or the direction that iron filings line up in if placed at that point.

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".

In other words, he found that sources of these or any other vector fields represent places where the divergence of the field is positive, and sinks are places where its divergence is negative.

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.