Re: corrupting the youth

["John S. Denker" <jsd@MONMOUTH.COM>]

Bob Sciamanda wrote:
> I will hazard an answer here:
> With magnetic monopoles,  E and B would each have a both vector part and a
> circulation part.
> (No?)

No.  It's much simpler than that.

Hint:  I wrote this all out, including the monopole issue,
a couple of weeks ago.  See footnote (3) in:
  http://www.monmouth.com/~jsd/physics/maxwell-ga.htm

Here's some amplification:

The electromagnetic field, F, is a bivector.  Always was.
Always will be.  Bivector and nothing but bivector.

 -- In a particular frame, the electromagnetic field has
one bivector component which is _spacelike_.  This component
is traditionally given the quaint old-fashioned epithet,
"magnetic".  A spacelike bivector has both of its edges
in spacelike directions.

 -- In a particular frame, the electromagnetic field has
another bivector component which is timelike.  This component
is traditionally given the quaint old-fashioned epithet,
"electric".  A timelike bivector has one of its edges in
the timelike direction; the other edge is spacelike.

To see an interesting example of such a field F,
see the example at the end of this note.

The equation
        del F = 4pi J                   (1)
contains the entire content of the Maxwell equations,
where J is an ordinary 4-vector representing the
density & current of electric charges.

If you want to include monopoles, it's not rocket
science;  you just replace equation (1) by
        del F = 4pi J + 4pi K           (2)
where K is a trivector representing the density & current
of monopoles.

The LHS is a vector (del) times a bivector (F) so the
product will contain a vector part and a trivector part.
The trivector part produces two of the Maxwell equations;
the vector part produces the other two.  In the absence
of monopoles two of the Maxwell equations have zero on
their RHS.

But don't confuse (del F) with F.  The field F is still
a bivector.  The electric component is a bivector.  The
magnetic component is a bivector.  Always was.  Always
will be.

==============

BTW note that F is not necessarily (or even usually) just
the wedge product of two vectors, but generally the sum
of such products.

==================

Example solution to equation (1):

For an EM wave linearly polarized in the X direction
and propagating in the Z direction, the electric
component is in the TX plane while the magnetic
component is in the ZX plane.  It is an easy homework
exercise to show that
  F = (g_0 g_1 + g_3 g_1) sin(k z - w t + ph)
is a solution to equation (1), where "g" is pronounced
"gamma" ... g_1 is a unit vector in the direction of
polarization, g_3 is a unit vector in the direction of
propagation, and g_0 is a unit vector in the timelike
direction perpendicular to g_1 and g_3.

It's hard to imagine anything much simpler than that:
it has an electric bivector and a magnetic bivector and
an overall sinusoidal running-wave factor.