MadSci Network: Physics

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

To answer this new question coherently, I have to present the original 
"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?",

and my original answer:
"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."

Now the new answer:
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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