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Poincare's Lemma and Electromagnetics

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Poincare's Lemma and Electromagnetics

If B and A are vector fields such that B = curl A, then it can be verified by direct calculation that div B = 0.
The converse, namely div B = 0 implies the existence of (at least) a vector field A such that B = curl A is stated in Electromagnetics books without proof.
I have seen elsewhere some references stating that the converse is based on Poincare's lemma. That lemma is then stated in modern terminology that is pretty obscure,about contractible manifolds.
Can someone provide a description in simple terms, just like Poincare would have done, also with a reference to the original source?
Also, a counterexample in the case of a "non-contractible" manifold would illuminate the concept better.


Thank you very much.
Is there a typo here?
Because the lemma no longer applies, we can have B=0 even though A=0,

It should say that we can have B=0 even though A does not equal 0.

> Can someone provide a description in simple terms,
> just like Poincare would have done, also with a reference
> to the original source?

Here is a simple physical explanation. Take a rope and
make a lasso from it, then pull on the rope to shrink the
loop. If the space is such that, no matter how you thread
your loop, it is pull the lasso shut all the way, the lemma
applies. If instead, there are some loops for which, no
matter how cleverly you try, it is impossible to shrink the
lasso completely because the rope somehow gets stuck around
some hole, then the lemma does not apply to that space.

For original sources, Maxwell talks about this in the first
chapter of his "Treatise on Electricity and Magnetism".
For further reading, he refers to Chapter 1 of Lamb's
Hydrodynamics, which provides a down to earth, easy to
understand account of this topic as it was understood in
the days before modern manifold theory. The ultimate origins
of this lemma date back about a century before Poincare to
at least the times of Green and Gauss.

> Also, a counterexample in the case of a "non-contractible"
> manifold would illuminate the concept better.

Consider the space outside a solid donut. If you pass your
lasso through the hole in the donut, you cannot pull the loop
shut because it gets stuck on the donut.

Moreover, this example is extremely relevant here, because it
is an idealization of a toroidal inductor, such as the ones
which are found all over radios and other electronic gizmos:

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html#c1

http://www.n7un.com/p/my-projects.html

The math here has directly measurable consequences. Because
the lemma no longer applies, we can have B=0 even though A=0,
which is exactly what happens --- in fact, this is a main
reason why people bother with all the fuss of winding wire
around ferrite donuts. If we hold a compass outside a toroidal
inductor, it will barely budge even when there is a large
current flowing through the wire. Suppose that we allow our
current to vary slowly with time (slowly enough that we can
neglect radiative effects) and make our lasso using, not a rope,
but a wire connected to an AC voltmeter. If you make the loop
so as not to pass through the hole in the donut, the meter
will read zero. However, if you wind it through the hole, the
meter needle will jump once we close the switch because the line
integral of A is not be zero.

Passing DC through our inductor, it is possible to demonstrate
that A is not zero even though B is using quantum mechanical
interference; this is called the Bohm-Aharonov effect:

http://galileospendulum.org/2011/03/31/the-aharonov-bohm-effect-or-how-i...

Instead of wires, we use a beam of electrons which is blocked by a
sheet with two holes and falls on a screen some distance away
where we can see the diffraction pattern. If we place our
toroid in front of the sheet so that one hole is in front of
the hole of our donut and the other hole is outside the donut,
we will notice that the diffraction pattern changes when we
connect the toroid to a current source. The reason for this
is that the quantum mechanical phase of the electron beam
depends on A as opposed to B so the A field outside the toroid
will affect the phases of the two beams passing through the
holes so as to change the diffraction pattern.

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