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Re: Energy density; the correct one



Situation 1. Suppose I hold two equally positively-charged particles, q1 and
q2, a distance r12 apart. If I keep hold of q1 and release q2, q2 flies off
to a place infinitely far away and gains kinetic energy equal to E2. I then
release q1, which stays put since q2 is now infinitely far away. The total
energy of the system is E2.

And, therefore, it was also E2 *before* the release. (Just to be
clear, however, you are not including the self energies in the fields
of the two isolated charges. That's fine. We can just say that the
field energy of the initial system is greater than the sum of the
field energies of the two isolated charges by the amount E2.)

Situation 2. Suppose I hold two equally positively-charged particles, q1 and
q2, a distance r12 apart. If I release both particles at the same time,
they fly away from each other. When they are infinitely far away from each
other, each has an energy E12.

Only if they also have the same mass. I will assume that they do.

The total energy of the system is 2*E12.

I'm not sure I understand the logic of your notation but that's fine.
Clearly E12 = (1/2)E2.

In situation 1, I did no work on either q1 or q2.

Ditto for situation 2.

I admit that my muscles converted chemical energy to heat energy in
the process of holding onto q1, but I still did no work *on q1*.

Right. Moreover, you might have simply glued, nailed, or riveted q1
to a rigid surface.

q1's electric field did work on q2. The work done by the q1's
electric field on q2 equals the kinetic energy gained by q2, E2.

Hmm. This may lead to some dubious energy accounting. I'd be more
comfortable saying that the "excess" electric field energy of the
initial *system* was the source of q2's ultimate kinetic energy. In
situation 2, it was the source of the kinetic energy gained by both
charges. In both cases the same amount of electric field energy was
lost and in both cases the same amount of kinetic energy was gained.

In situation 2, I did no work on either q1 or q2. Is E1 equal to 2*E12?

What's E1?

--
John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm