Let us look at the problem which we recognized in the previous blog post.
----------------------------- - outer cylinder wall
| | | E
----------------------------- + cylinder wall
----------------------------- + cylinder wall
| | | E
------------------------------ -
---> v
----> x
Let us consider a coaxial configuration where the electric field is nicely isolated between the cylinder walls.
The coaxial system moves to the right at the speed v. There is a magnetic field
B = v / c² * E.
The Poynting vector has the value
ε₀ c² E × B
= ε₀ E² * v.
A half of the value comes from shipping the energy of the field. The other half comes from shipping the x pressure of the field.
The Poincaré stresses which keep the cylinders from exploding cancel the pressure part of the momentum. The Poincaré stresses correspond to a negative pressure which cancels the positive x pressure in the field E.
^ E'
|
-------------- + rod
|
v
Let us then add a static positively charged rod at the center of the cylinder. The field
E + E'
is larger at the location of the rod. The Poynting vector to the right increases in value by
ε₀ c² E' × B
= ε₀ E × E'.
But there is no extra shipping of field energy or pressure in the x direction. How is it possible that the Poynting vector increases?
The answer: as elementary charges in the positively charged cylinder come close to the positively charged rod, they lose kinetic energy to the potential of the field of the rod. The charges the kinetic energy back when they recede from the rod. This is the origin of the energy flow in the field.
Conclusions
There is no paradox in this. We simply forgot to take into account the kinetic energy of the charges which create the magnetic field.
Now that we understand the energy flow in this case, let us return back to the Biot-Savart law.
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