Inertia of an electric current relative to the protons

Our previous blog post shows that the Poynting vector cannot explain the inertia which electrons have relative to the protons in a suitably designed coil, which simulates a rotating charged sphere where the electric field is canceled with an opposite charge. This is expected. For a wire, the electric field typically is negligible, while the magnetic field can be very strong. The Poynting vector has a very small value.

The sliding charged tube

Let us analyze, once again, the sliding charged tube around a charged cylinder, which we presented in the previous blog post:

             E              B               E

          ^  ^  ^                          ^  ^  ^

          |   |   |      × × × ×        |   |   |

                   negative charge

                         -------------

          ++++++++++++++++++  positive charge

                         ------------- 

                              --> v

In this case, we can claim that the Poynting vector explains some of the inertia of the the tube.

Let the charged tube move very slowly. Then the energy density of the magnetic field B is very small. Energy in the electric field E on the right side of the tube must flow to the left side, as the tube progresses on its journey. Presumably, this energy flow is described by the Poynting vector, and momentum is associated with it.

In the diagram above if we make the right hand forefinger to point to the direction of E and the middle finger to B, then the thumb points left: that is, energy flows to the left, as expected.

We can very crudely explain the inertia of the tube relative to the cylinder in the following way:

1.   The tube carries the mass-energy of its own electric field (even though that energy is "hidden" by the opposite charge of the cylinder), and

2.   an equivalent amount of inertia arises from the energy flow from the electric field on the right side of the tube to the left side.

The momentum associated with item 1 flows to the right, and the momentum associated with 2 flows to the left. The total momentum, measured in the laboratory frame, is zero. This explains why the momentum held by a pure magnetic field B without an electric field E is zero.

                 rope

                    --> v

              _________

           O_________O  wheel

           |     v <--     |

       ==============

                 frame

Let us have a pulley made of two wheels and a loop of rope, like in the diagram. If we make the pulley to rotate, the system has a zero momentum relative to the laboratory frame. However, the pulley has inertia relative to the frame. The inertia arises from two movements to opposite directions.

The electric field E generated by a decreasing magnetic field B

Faraday's law of induction says

       ∇ × E  =  -dB / dt.

If the magnetic field B in the above diagram decreases, then an induced electric field E' has its curl pointing to the screen. If the right hand has its thumb pointing to the screen, then the fingers tell us the direction of the field E':

             ×   ×   ×    -dB / dt

              <------

                  E'

             ------------    --> v

        ++++++++++++   cylinder

             ------------ 

                tube

The electric field E' "resists" the change in the magnetic field B. The electric field tries to push the negatively charged tube to the right and the positively charged cylinder to the left. Can we somehow explain this process with inertia?

The following model might do the job: in the pulley diagram in the previous section, the lower rope "pushes" positive charges to the left and the upper rope pushes negative charges to the right. In this model, the magnetic field B describes momentum stored in the pulley system.

A current in a closed wire loop: "topology" or a flow of negative energy?

              current

                 <--- I  

                 ______

               /             \

              |                |

                \______/

              wire loop

The sliding charged tube model qualitatively described the momenta associated with a magnetic field. It was crucial that the Poynting vector tells about an energy flow from the right to the left.

But if we have a current in a closed wire loop, then the electric field E is essentially zero everywhere. The Poynting vector does not describe any significant energy flow.

In this case, we might claim that there still is an energy flow, but it is not captured by the Poynting vector.

In the sliding tube model, we may interpret the "hole" in the electric field E around the tube containing negative energy which moves along with the tube. Using this model, switching to a closed wire loop is no dramatic change: we still have negative energy moving along with the electrons.

The Poynting vector does not understand these energy flows: it sums the negative energy flow to the positive energy flow associated with the electric field of the tube, and thinks that the total flow is zero. It is not aware of the separate flows of positive energy and negative energy.

In this model, a pure magnetic field B involves separate flows of positive and negative energy. These flows make up the "pulley" that we described in a preceding section.

Conclusions

On December 17, 2023 we asked where the energy and momentum of a pure magnetic field B is stored. "Pure" in this context means that there is no significant electric field E present, and, consequently, the Poynting vector is essentially zero.

Our pulley model / negative energy model offers an explanation for where the momentum is stored. There is a flux of positive energy to the direction where the charges carrying the current move, and a simultaneous flux of negative energy to the same direction. We obtain zero by summing these energy fluxes  –  that is why the Poynting vector is zero. But the system is dynamic, even though the Poynting vector is not aware of that.