An appealing feature of time-domain electromagnetics is that the transient response simplifies considerably at late time, usually tending to a power-law or exponential decay. In this note, we point out an interesting discrepancy between the late-time asymptotics of a finite loop source over a half-space and its natural two-dimensional (2-D) approximation, which is two line sources of opposite polarity lying on a half-space. Expressions for the transient responses of both loop (Wait and Ott, 1972) and line sources (Oristaglio, 1982) have been derived before; they show that at late times the voltage induced in a horizontal receiving coil decays as t−2.5 for a loop source and t−2 for a line source.
Here we show that the slower decay for the line source is inherently a 2-D effect. To do this, we derive a closed-form expression for the transient voltage induced by a finite wire of length 2L on a half-space—a new result, for which we can separately examine the limits L → ∞ and t → ∞. Surprisingly, these limits are not interchangeable. First taking L to be infinite and then doing the late-time asymptotic expansion yields the t−2 decay of a line source; in contrast, first doing the late-time expansion gives a decay of t−2.5 for the finite wire, which is formally unchanged as the length goes to infinity.