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 [Formula: see text] for a loop source and [Formula: see text] 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 [Formula: see text] and [Formula: see text] Surprisingly, these limits are not interchangeable. First taking L to be infinite and then doing the late‐time asymptotic expansion yields the [Formula: see text] decay of a line source; in contrast, first doing the late‐time expansion gives a decay of [Formula: see text] for the finite wire, which is formally unchanged as the length goes to infinity.
The Diffraction of Transient Electro-Magnetic Waves by a Wedge
