Going back to the analogy in an earlier posting of birds sat on high voltage overhead power cables where they suffer no ill effects, I still have difficulty in understanding why this should affect a swimmer who is swimming freely and touching nothing other than the water surrounding them. For current to go through their body, there surely must be a potential difference between the point of entry and the point of exit, otherwise there would be no flow, and like the birds on the power cable, no ill effect.
Quite right! And there is a potential difference when a live wire dangles off a boat. When I was taught this back in Germany, it was called a
Spannungstrichter (just look at the illustration, it really explains it all). The online dictionary translates this into "potential gradient", which sounds about right.
So the potential on the point where the boat has a wire dangling into the water would be 240V or something similarly dangerous. The potential of the ground (at shore, in the water, etc.) would be 0V. Between the two, a current flows and voltage drops, creating this potential gradient, meaning between the 240V point and the 0V area (a few meters away most likely), there is a gradient of decreasing voltages between 240V and 0V. So for example about 2m away from the cable, the voltage in the water relative to ground might be 90V (just an example, I do not know the actual numbers and it would depend on several factors).
This again, is harmless if you're small, like a fish, because you're only 30cm long and therefore you are only bridging a few volts between the ends of your body. If you're a bit longer, say around 1.8m, you are bridging a longer distance and therefore a higher potential difference between the ends of your body. This is what makes it dangerous, and that's what the illustration above depicts on land - if you keep your feet together, the potential difference between them is much smaller than if you stand with legs spread, or worse, start doing push-ups (or swim directly towards/away from the dangling wire).