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Re: [Phys-l] Poynting Vector

-----Original Message-----
From: [mailto:phys-l-] On Behalf Of John Denker
Sent: Saturday, June 28, 2008 1:39 PM
Subject: Re: [Phys-l] Poynting Vector

On 06/28/2008 09:48 AM, Jeffrey Schnick wrote:
Consider a charged simple parallel plate capacitor in a static
downward-directed magnetic field. The capacitor is oriented so that
from our point of view, the electric field between the plates of the
capacitor is directed rightward. Poynting tells us that between the
plates, energy is flowing away from us at a rate proportional to EB.
Where is that energy coming from and going to?

The energy is flowing around and around, chasing its tail. It does
accumulate anywhere, as you can easily verify by showing that ∇•S = 0.

Since it is not particularly easy to visualize the fringing fields of
a parallel-plate capacitor, it may help to consider (as a warm-up
exercise at the very least) a concentric coaxial capacitor, i.e.
where the field is in the gap between two tubes. The situation is
nicely symmetric, and the energy just flows around and around in

For the case in which the magnetic field is collinear with the axis of the capacitor, this is a nice warm-up exercise; great suggestion. (Chuck Britton's dipole suggestion is also useful.) It is the difficulty in visualizing the fringing fields that makes visualization of the energy flow challenging. I think that the energy that flows through a small square cross section, such as the one depicted at:
between the plates disperses over a very wide region after it passes through the plates, curves around to one side, to the right in the case depicted such that by the time it passes through the same plane as the original cross section again it passes through a cross section of the same height as the original cross section, but infinitely wide. As it continues its traversal around the loop, the energy becomes more and more concentrated such that the flow rate through the original cross section is constant. Energy flowing through a cross section between the plates that is to the left of a plane that is parallel to the plates and midway between the plates will curve around to the left. From above, the energy circulates in two big loops, clockwise in the loop to the right and counterclockwise in the loop to the left such that the net angular momentum of the system is zero.