Poincare's Lemma and Electromagnetics

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# Poincare's Lemma and Electromagnetics

Submitted by sawarak on Sat, 04/23/2011 - 19:29

Forums:

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.

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## Re: Poincare's Lemma and Electromagnetics

Thank you very much.

Is there a typo here?

Because the lemma no longer applies, we can have B=0 even though A=0,

## Re: Poincare's Lemma and Electromagnetics

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

## Re: Poincare's Lemma and Electromagnetics

> 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.