Re: field transformations

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

On 03/28/2003 05:38 PM, Joe Heafner wrote:
> Is there a way of deriving the relativistic E and B field
> transformations that would be suitable for an introductory e&m course?
> Four vectors and tensors are out.

I can't imagine deriving or even explaining the
transformations without four-vectors.  For that
matter I can't imagine introductory E&M without
four-vectors.

But if I may answer a slightly different question,
here's how I think of the transformations:

1) The fundamental thing of interest is the
electromagnetic field.  It is a bivector.

2) What you call an electric field is an electromagetic
field that happens to have one of its edges in
the purely timelike direction in your reference
frame.

In another frame, the same physics (the same
electromagnetic bivector) will not be purely
electric.

3) What you call a magnetic field is an electromagnetic
field that happens to have both edges in purely
spacelike directions in your reference frame.

In another frame, the same physics (the same
electromagnetic bivector) will not be purely
magnetic.

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

The quintessential field-transformation exercise
is to understand why a current-carrying wire
produces a magnetic field.  The electromagetic
field of the electrons is purely electric in
the frame comoving with the electrons.  The
electromagnetic field of the positive metallic
ions is purely electric in the frame of the
wire (assumed stationary in the lab frame).

So why does the superposition of these two
fields turn out to be purely magnetic in the
lab frame?

Here's the picture:
http://www.monmouth.com/~jsd/physics/img48/wire-bivector.png

For a discussion at the next level of detail,
refer to:
http://lists.nau.edu/cgi-bin/wa?A2=ind0211&L=phys-l&P=R11678