## Appendix E Analogy to Electrodynamics  
Field Equation (6) is analogous to [PERSON]'s law with the [PERSON]'s correction term [45]. \[\frac{1}{\mu}\
abla\times\mathbf{B}=\epsilon\frac{\partial\mathbf{E}}{\partial t}+\mathbf{J} \tag{13}\]  
Here \(\mu\) is the magnetic permeability, \(\mathbf{B}\) is the magnetic field, \(\mathbf{E}\) is the electric field, \(\epsilon\) is the electric permittivity, \(\mathbf{J}\) is the conduction current density. The conduction current density can be expressed with the conductivity \(\sigma_{e}\) of the medium as follows. \[\mathbf{J}=\sigma_{e}\mathbf{E} \tag{14}\]  
Being proportional to the conductivity this term causes the electric field to decay when an electromagnetic wave travels through a conductive medium (as you experience your mobile phone's signal is weaker when you are inside a conductive object such as a building). The associated loss in the electromagnetic energy is known as the Ohmic loss. The first term on the right-hand side of Equation (13) is known as the displacement current. (It is not an actual electric current but it induces a magnetic field as a conduction current does.) This term does not cause the electric field to decay. It induces a time-varying magnetic field via Equation (13). Through [PERSON]'s law the time-varying magnetic field in turn induces a time-varying electric field. This synergetic interaction between the electric and magnetic field generates an electromagnetic wave. In this mechanism the conduction current term \(\mathbf{J}=\sigma_{e}\mathbf{E}\) plays a role as an energy damper. Figure 13: Strain vector components expressed with principal and non-principal axis coordinate systems. Apply the above mechanism to an Alternating Current (AC) circuit shown in Figure 14 (See [PERSON]'s equations on p. 322 of ref. [45]). An air-gap capacitor is connected to an AC power supply through a conductive wire. An AC current flows through the wire according to the total impedance. In the air-gap either conduction current, displacement current or both currents flow. Here, some current must flow through the air-gap because otherwise the law of charge conservation (law of current continuity) is broken. The carrier of a conduction current is free charges (electrons) and that of a displacement current is bound charges (electrons). The average of the conduction current is proportional to the conductivity \(\sigma_{e}\) of the air-gap and that of the displacement current is the electric permittivity \(\epsilon\) of the air-gap. The more conductive the air in the gap becomes, the portion of the conduction current increases. While both conduction and displacement currents can satisfy the current continuity condition, there is a crucial difference between them in the way they grow (increase in the magnitude). The displacement current grows when the time derivative \(\partial E/\partial t\) increase. However, if for any reason \(\partial E/\partial t\) changes, [PERSON]'s law induces a magnetic field in such a way that it suppresses the change in the electric field. It is a self-stabilizing effect generally referred to as [PERSON]'s law. In other words, if an electric field increases over time for some reason, a negative feedback algorithm kicks in via [PERSON]'s law and the increase is suppressed. On the other hand, if a conduction current increases in a gas medium, the situation is completely different. As mentioned above, the carrier of a conduction current in a gas medium are free charges. The more free charges exist in a unit volume, the gas medium becomes more conductive. This means that if for some reason a conduction current increases in a gas medium, that makes the medium more conductive. Consequently, the conduction current increases further. In other words, unlike the case of displacement current, a positive feedback mechanism kicks in when the conductivity of a gas medium increases. This can trigger a rapid growth of the conductivity, leading to electrical breakdown of the gas medium known as the arc discharge [60].