1/2 ε E²,
where ε is vacuum permittivity and E is the electric field strength?
Harvesting energy of an electric field with capacitor plates
Let us devise a way to harvest that energy. We may use the following device:
+ + +
-------------- capacitor plate
^
| V
| E W
-------------- capacitor plate
- - -
R
We can place the right amount of charge into the plates and make the device to cancel an electric field E in a given volume V of space. The electric field pulls on the plates and we can harvest the energy W of the electric field E from the volume V.
But does that mean that the energy W originally resided in that volume V? Or was the energy shipped from some other spatial location?
When the plates are pulled apart at a speed v, there is a magnetic field at the sides of the plates. The Poynting vector will differ from zero there. Let us vary the radius R of the round capacitor plates. The total charge of a plate goes as
~ R²,
the electric field strength at the plate edge per a unit charge goes as
~ 1 / R²,
and the electric and magnetic field strength of the plate at the edge is roughly constant.
The harvested energy goes as
~ R²,
while the length of the plate edge goes only as
~ R.
If R is large enough, then the Poynting vector says that any energy flow was small compared to the harvested energy. If we believe the Poynting vector, then the harvested energy W truly came from the volume V.
Practical experiments to test localization of field energy: circularly arranged pendulums do not work
The system is complex and the Poynting vector might misinterpret the location of energy. Measuring the gravity of the energy W is prohibitively difficult. Measuring the inertia of the process might be easier. If the energy flows from a distant location, then the inertia is larger.
On March 25, 2023 we wrote about circularly arranged pendulum clocks. Let us put electric charges to the pendulums. Can we use the clock cycle to determine the inertia?
+ charge
|
|
+ ------ ------ +
| direction of
| oscillation
+
The pendulums will oscillate synchronously. Energy from their combined electric field will flow to each pendulum and back. The pendulums will approximate a spherical shell of charge expanding and contracting.
If the energy comes locally directly from the field, then the inertia of the oscillation is less than if the energy is shipped from far away. Let us try to calculate an estimate.
A large static electric charge is typically one microcoulomb. The electric field of a charge q is
E = k q / r²,
where k is the Coulomb constant
9 * 10⁹ kg m³/(s⁴ A²).
The field at the distance 1 meter is
~ 10,000 V/m.
Its energy content is
1/2 ε E² ~ 0.5 mJ/m³.
The mass involved is energy per c²:
m = 0.5 * 10⁻²⁰ kg/m³.
The mass of a pendulum is ~ 1 kg, and the accuracy of a pendulum clock is 10⁻⁷. We conclude that we cannot measure the inertia this way.
The hydrogen atom and the 2s and 2p orbitals
The electron in a hydrogen atom moves in low electric potentials of the proton, ranging from -27.2 eV to perhaps -511 keV. We would expect the extra inertia to affect the hydrogen energy levels a lot, but the classic Schrödinger equation hydrogen model ignores the inertia of the electromagnetic field entirely, treating the electron as a point particle under a Coulomb attraction.
We should see the analogy of the precession of the perihelion of Mercury, if the inertia of the electron would be larger close to the proton.
How is it possible that the inertia does not change close to the proton?
The hydrogen atom states are stationary. Maybe there is no energy flow which would affect the inertia of the electron?
In a stationary state the electron does not radiate electromagnetic waves. Its behavior is fundamentally non-classical. Could this explain why the extra inertia is missing?
This is a fundamental question. Let us write a new blog post about this.
Conclusions
Measuring the inertia associated with the electric field seems to be hard in a macroscopic setting, with the pendulums.
When there is a current in an electric wire, then the inertia in the system is prominent: the energy of the magnetic field is substantial. But it is not clear how to interpret that as inertia of the electric field.
In a hydrogen atom, the inertia of the electric field should affect the energy levels a lot, but we do not see any effect. This is a fundamental question: why in a stationary state the electron behaves as if the electromagnetic field would not exist at all? There is no energy loss through radiation, and no inertia of the field is visible. However, in our blog post on March 13, 2021 we showed that the Lamb shift may be due to the loss of inertia in the private electric field of the electron.
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