Electromagnetics from a quasistatic perspective

It has been tempting to regard $∂B∕∂t$ in Faraday’s law as a source of an electric field, thereby sometimes causing confusion and objections; currents and charges should be considered as the only sources of electromagnetic fields. The integrals expressing the fields in terms of these sources involve, in general, the retarded time so that the news from the sources propagates with the finite velocity $c$⁠. In a quasistatic approximation the interactions are instantaneous so this objection is less valid.

The static continuity equation together with Eqs. (1) and (2) constitute a quasistatic model without capacitive or inductive effects.

The integral (6) converges as is evident from the law of Biot-Savart, which implies that the $B$ field goes to zero as $1∕r2$ when $r→∞$⁠.

We consider systems where the charges and currents do not extend to spatial infinity and may use as boundary conditions that the fields approach zero sufficiently fast far away.

Ampère’s law, $∇×H=Jf$⁠, differs from Eq. (15) if there is a nonvanishing polarization current $∂P∕∂t$⁠. Ampère’s law is, strictly, speaking, a static equation, and we prefer to use Eq. (15) as a time-dependent generalization with the same name.

As in Ref. 2 we use the names Ampère’s law for Eq. (15) and Ampère-Maxwell for Eq. (13). Then it seems natural to use the name Ampère-Darwin for Eq. (16).

There seems to be a widespread misunderstanding that quasistatics, defined by the omission of time retardation, includes Ampère’s law (15) as a valid equation. This misunderstanding is stated or implied by many textbooks. See, for example, Ref. 13, p. 314, and Ref. 35, p. 368. Ampère’s law implies stationary current and thus constant charge density. But a slowly varying charge density does not violate the basic assumption underlying quasistatics because it cannot force us to include time retardation.

The term $∂EC∕∂t$ goes to zero as $1∕r3$ when $r→∞$⁠.