Re: Energy

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

At 08:57 AM 9/18/01 -0500, RAUBER, JOEL wrote:

> Does the above discussion mean that I can consider (in the context of
> classical E&M theory) generalized current (the sum of so called "conduction
> current" and "displacement current") as an entity that "flows"?

There's an excellent question there.  Before dealing with it, we need to be
super-careful about the wording.

In this business, alas, the word "flow" is used in two ways.
 -- We sometimes say the charge "flows" across the boundary.
 -- We sometimes say the current "flows" across the boundary.

But that does not mean that charge and current are the same thing.  The
notion of flow in the two cases is different.  Same word, different
notion.  The equations expressing the two notions are quite different.

In general, when there is a local conservation law, there is a conserved
quantity (e.g. electric charge) and the associated current (e.g. electric
current).

The current _per se_ is not a conserved quantity.  Sometimes people use the
words "conserved current" when they mean "the current associated with a
conserved quantity".  It's sloppy, like using the word "microwave" as
shorthand for "microwave oven".  I prefer to speak of a "conserving
current" that is associated with a conserved quantity.

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

Anyway:  There are lots of local conservation laws in the world.  And we
can make more:

 -- Given two local conservation laws, such as conservation-of-energy and
conservation-of-lepton-number, we can add them and get a new conservation
law for the combined quantity.  It's not very useful, but it can be done.

 -- Given a local conservation law, we can multiply both sides by 13 and
have a new, slightly different law.  Again, it's not very useful, but it
can be done.

 -- Given a local conservation law, we can take the derivative of both
sides and have a new law.  Weird, but possibly useful.

 -- Given any vector field, we can compute its curl.  The resulting curly
field will obey a local conservation law.  In this case the "conserved
quantity" on the LHS of the equation will be zero.  This is the law of
conservation of zero.  It's boring, but it's true:  zero is conserved.

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

I'm not entirely sure, but AFAICT Joel's conservation law is
valid.  Displacement currents have to do with field lines moving
around.  Some field lines terminate on charges, so there is a conservation
law here:  Write the good old conservation-of-charge law and differentiate
both sides.  Other field lines don't terminate at all;  they just chase
their tails endlessly.  (Such field lines are created by time-varying
magnetic fields.)  The corresponding displacement currents obey the law of
conservation of zero.  Add these two together and I think you've got Joel's
conservation law.

I don't think the new law tells us anything beyond what the old
conservation-of-charge law tells us.