Understanding the penetration of electromagnetic velocity fields into conductors

The electromagnetic fields due to a point charge can be broken into velocity fields and acceleration fields. The acceleration fields give rise to electromagnetic waves whose penetration into ohmic conductors is described by exponential damping with a characteristic skin depth. The velocity fields allow a steady-current limit where the magnetic field penetrates even a good conductor. Here we note the contrasts between velocity fields and wave fields in their interactions with conductors. We derive a new time-integral invariant of the magnetic field in an ohmic conductor. Finally we note the disparate analyses in the literature and suggest the following summary regarding the penetration of the electromagnetic fields of a point charge moving parallel to a conducting surface. (1) The falloff of the electromagnetic fields is algebraic, not exponential, and cannot be characterized by a skin depth. (2) In the limit of low velocity, the magnetic field penetration is independent of the conductivity of the material. (3) Ln general the penetration of the electromagnetic fields depends upon the velocity of the particle and becomes vanishingly small for a perfect conductor, (4) The time integral of the magnetic field at a fixed spatial point inside (or outside) an ohmic conductor is independent of the conductivity of the material; thus as the conductivity of the material becomes larger and the magnetic field inside becomes smaller, the time of penetration becomes longer. The penetration of time-dependent velocity fields into conductors has become of interest largely in connection with the Aharonov-Bohm effect. It is curious that the classical explanation for the Aharonov-Bohm effect depends upon the time integral of the magnetic field, which is independent of the conductivity of any ohmic shielding material. (C) 1999 American Association of Physics Teachers.