Inertia of an expanding and contracting shell of charge in an external electric field

In the previous blog post we studied a collapsing shell of charge. Let us now look at a related problem of a pulsating spherical shell of charge in an external electric field.

                     ___

                  /        \

                  \____/  -                               ● +

  expanding/contracting            external

         shell of charge                     charge

Is there extra inertia in the movement of the shell? The field outside the shell does not change at all. There is no flow of field energy there. But what about the field at the surface of the shell?

                       shell              shell

                       wall               wall

   . . . . . . . . . . . .| -----------------  |  =======  ● +

                          -                        -

                         -->                   <--

   weak field     medium field     strong field

Schematically, the electric field looks like the one in the diagram. The arrows tell that the shell is contracting. The left wall gains energy from the field since it makes the field weaker. The right wall consumes energy because it makes the field stronger.

Our previous blog post claims that the walls gain/lose energy from the adjacent electric field. There is no shipping of energy over large distances. We conclude that there is no extra inertia in the contracting/expanding movement of the spherical shell.

If we would be moving a pointlike test charge, then there would be extra inertia. There would be an energy flow in the combined field of the charges.

Actually, here we have a strong argument for the existence of extra inertia in an electric interaction!

Claim. In an external electric field, the inertia of a test charge is larger when it is a single pointlike charge than when the test charge is a part of a pulsating spherical shell of charges.

Heuristic proof. Let us have a large external charge C and a test charge c.

          <--------  E

                 • -                           ● +

                 c              r             C

             

When a pointlike test charge c moves, there is an energy flow in the combined electromagnetic field of c and C. This adds some inertia H, which depends on the electric field E at c, as well as the distance r. That is, the inertia is proportional to the negative potential of c in the field of C.

Suppose then that c is a part of a pulsating shell of charge. Could there be some mechanism which adds the same inertia H to the test charge c when it is a part of a pulsating shell?

That is implausible. Why would the shell be interested in the distance r and put just the right amount of extra inertia H, which depends on r?

Q.E.D.

Our claim refutes the "geodesic hypothesis" for electromagnetism. The inertia of a test charge depends on the configuration of test charges. In general relativity, the inertia does not depend on the configuration of test masses, but it is determined by the external gravity field and its metric.