Henri Poincare
There are two fundamental and equally important definitions in physics:
F dot dS = dE (1)
and
F dt = dP. (2)
In these two equalities F stands for “force”, S stands for “displacement”, E stands for “energy”, t stands for “time”, and P stands for “momentum”.
An example where Eq. 1 applies is an elastic membrane, stretched in every direction parallel to its plane. A force must be applied in every direction on each tiny patch of the membrane, but there is no change in the momentum of each patch. What changes is the energy of the patch. That is, the stretched membrane can do work on the stretching agent if the stretching is allowed to return to the unstretched state. This simple example provides the answer to a time-honored conundrum that puzzled some of the best minds of the post-Maxwellian period.1 Following is a description of the conundrum.
Imagine we have a spherical shell of charge, q, with radius R. The electric field energy in this sphere’s electric field is
Uelec = q2/8πεoR. (3)
By considering the momentum in q’s electromagnetic field when the sphere moves with constant velocity of magnitude v<<c, we can also ascribe an electromagnetic mass of
melec mag = q2/6πεoRc2. (4)
This indicates that the total energy of the sphere should be
Utotal = q2/6πεoR, (5)
which is greater than Uelec in Eq. 3! Whence the additional energy?
As Maxwell himself might have pointed out, had he lived, a miniscule patch of electric charge is elastic. Work must be done to decrease the patch’s area.
Now before shrinking our spherical shell of charge, we can describe the shell as a contiguous collection of tiny patches, arrayed in the shell surface. If we shrink the shell, then the tiny patches overlap one another. In order to arrive at a final shell whose surface density is uniform, each tiny patch must (a) be displaced radially inward and (b) each patch must be shrunk. This shrinking process requires an amount of work, dW. And
dW = Utotal – Uelec. (6)
Poincare, standing on Maxwell’s shoulders, was reportedly the first to suggest that stresses were necessarily part of the solution for the conundrum that puzzled Einstein and others. For some reason his suggestion was not accepted by much of the physics community, (including Feynman who, in his fashion, claimed that the whole conundrum is an example of how Maxwellian theory ultimately “falls on its face”).
- See The Feynman Lectures on Physics, V2, Chap 28.