High Energy Accelerator Research Organization
Executive summary
Question
There is a prevailing misconception that the magnetic field between the plates of a capacitor carrying alternating current is generated by the displacement current density existing there, and that this directly leads to the generation of electromagnetic waves. This is incorrect. Then, what about the crucial role of displacement current density in Maxwell’s prediction of electromagnetic waves? Apparently a sufficiently coherent and integrated picture of the displacement current density has not yet been available.
Findings
The frequency (ω) dependence of the Coulombic field and induced field components of the displacement current density between the plates of a parallel-plate capacitor carrying alternating current has been investigated. The results (Figure 1) revealed that the amplitude of the Coulombic field component remains constant regardless of frequency. However, the amplitude of the induced field component is extremely small at low frequencies, increases proportionally to ω², and eventually reaches a magnitude comparable to that of the Coulombic field component.
Meaning
The present paper revieled a fundamental nature of the displacement currenr density that has not been examined since Maxwell’s time, providing a coherent and unified view on the displacement current density. It will help avoids unnecessary current and future debates arising from misunderstandings about the phenomena involving dislpacement current density.
Overview
In Maxwell’s equations, the electric field is represented by a single symbol E, but it is actually the sum of Coulomb’s electric field EC existing around electric charge and Faraday’s induced electric field EI generated by varying magnetic field,
E = EC + EI.
Everybody knows this, but not much attention has been paid to the fact that the displacement current density, which is the time derivative of E multiplied by the electric permittivity ε0, also has two components, JC and JI, each related to each electric field,
| ε0 ∂E∂t | = | ε0 ∂EC∂t | + | ε0 ∂EI∂t | = | JC + JI. |
As a consequence, there exists a prevailing misconception that the magnetic field between the plates of a capacitor carrying alternating current is generated by the displacement current density existing there, and that it directly leads to the generation of electromagnetic waves as the frequency increases. This is incorrect because the displacement current density at low frequencies where the magnetic field is usually measured is dominantly JC which cannot in principle generate a magnetic field.*) The magnetic field is actually generated directly by the current flowing in the leads to the plates and in the plates. It was also demonstrated that the displacement current density involved in the generation of electromagnetic waves at high frequencies is not JC but JI.
Then questions may arise: what about JI in the low-frequency region, and what about JC in the high-frequency region? Apparently a sufficiently coherent and integrated picture of the displacement current density is not yet available. The present paper has addressed this issue.
It has been investigated how the amplitude JC0 of JC and the amplitude of JI0 of JI in the space between the plates of a parallel-plate capacitor vary with angular frequency ω when an alternating current, I0 sin ωt flows thouth it. We assume that the plates are circular disks of radius r and the distance between the plates is sufficiently small so that the edge effects of the plates can be neglected.
The results reveal that when the amplitude I0 of the current is constant (or normalized to I0), JC0 remains constant regardless of the frequency ω. On the other hand, the amplitude JI0 is extremely small at low frequencies, increases proportionally to ω2, slows its increase rate around ω ~ c/r, and eventually reaches a magnitude comparable to JC0. Here, c is the speed of light, and c/r is the reciprocal of the time that light takes to travel a distance r. As the plate radius r decreases, the frequency ω ~ c/r increases further, and JI0 becomes even smaller.
Figure 1 shows the frequency dependence of JC0 and JI0 for the case of r = 5 cm (diameter 10 cm) as a representative size for parallel-plate capacitors where magnetic field measurements are usually performed. While JC0 is spatially uniform in the area, JI0 is not. Therefore, the actual comparison is made between their spatial average over a flat surface of radius r of the capacitor plates.
It is seen that even at quite high AC frequencies in everyday term like f ~ 1 MHz (ω = 2πf ~ 6MHz), the amplitude JI0 is still only 10-7 of the amplitude JC0. This confirms the argument is valid that the displacement current density at frequencies where magnetic fields are measured corresponds to the JC, which does not generate a magnetic field. As ω increases, JI —the source of electromagnetic waves—grows larger. This makes the intensity of the electromagnetic waves increase and gradually become detectable. (The frequency at which detection becomes possible is rather a matter of the detector’s sensitivity.) It is also shown that JC, which is not the source of either the magnetic field between the plates or the electromagnetic waves, always exists with its amplitude JC0 remaining constant and is always larger than or comparable to JI0.
The present paper has demonstrated a fundamental property of the displacement current density that has not been sufficiently examined since Maxwell’s time. The coherent picture provided helps avoid current and future unnecessary debates involving displacement current density. In the Editor’s note for the paper published in the American Journal of Physics, it was evaluated as: “This would make an appropriate supplementary topic for advanced-level electromagnetism lectures.” It is also expected to be added to textbooks.
*) T. Hyodo, “Maxwell’s displacement current and the magnetic field between capacitor electrodes,” Eur. J. Phys. 43(6), 065202 (2022).
Please refer to the press release for details.
Contact
High Energy Accelerator Research Organization (KEK)
e-mail: press@kek.jp