### Re: Static EM fields and momentum

Date: Thu Dec 16 08:09:50 2004
Posted By: Jerrold Franklin, Professor Emeritus
Area of science: Physics
ID: 1102709229.Ph
Message:
```
To answer this new question coherently, I have to present the original
question:
"If I have a permanent bar magnet, and I place a static charge in the
middle, (Imagine a box -the magnet- and a point charge q embedded inside).
Then the magnet will produce a static dipole magnetic field, and the
static charge will produce a static electric field.
If you calculate the direction of the Poynting vector S (ExB), it will
seem to be circumferential, ie. it looks like it is circulating around the
bar magnet.  Now EM fields have a momentum associated with them, in the
direction of this Poynting vector S.  If you attach a solar sail that is
attached to the bar magnet frictionlessly, placed so that it can rotate
about the magnet in the same direction as S. Won't the sail be pushed?  I
understand that it takes work to place the charge inside the magnet, etc.
But this is a finite amount of work, and since Magnetic field and Electric
field are static, nothing is changing so none of the energy that is stored
in their fields, (~Integral[E^2 +B^2]all space) is lost.  So where does
the energy come from?  And does it seem to be infinite?",

"You have to be careful in using the EM field momentum vector for static
fields.  In order for the field momentum to exert an impulse on any
object, there must be a traveling wave that reflects from (or is absorbed
by) the object.  In the calculation for sunlight moving a solar sail, it
is the reflection of the traveling wave that produces the impulse.  This
can also be calculated using the force on currents in the sail induced by
the oscillating fields of the wave.  If the wave isn't moving or
oscillating in time, there will be no force."

Of course static fields can produce forces on matter.   An electric charge
attracts polarizable matter.  But, the original question was about a force
of rotation caused by the electromagnetic momentum, and this requires
oscillating fields.  The stress tensor is brought up in the new question.
If the stress tensor is used correctly for the configuration mentioned in
the original problem, there is no force of rotation on a sail placed in
the field.  (But it is a difficult calculation, using the stress tensor.)
A misleading impression is made in the source cited.  The book suggests
that the stress tensor actually corresponds to a stress at a point in
space.  The derivation shows that the stress tensor must be integrated
over a completely closed surface, and only then gives the force on matter
inside the surface.

```

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