An interesting theory, I've heard of this before, or rather a similar idea. Some years ago, I read that a market garden, not sure where it was, was using power from a local tv or radio transmitter, which was located close by. An apparatus similar to a crystal set was designed, the audio/video content filtered out, the resulting dc voltage being used to power other appliances.
In practice, I think the Crystal Radio trickle charge could work, it'd only charge the battery with micro-amps though. Long time to charge up.
