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As general, as you raised the question, the answer is yes. Technical development starts by considering specific parameters, e. g. distance between mobile and antenna, antenna size, energy requirement.
It can generate. But you have said that the em energy is generated by the antenna, so take care of the antenna height. To reduce the antenna height use an amplifier. but there also you have a problem, that is be sure about your baterry.
Electromagnetic waves carry energy which reduces with the distance they go. So they can energize a circuit. Already industry is using a technology called RFID. In RFID active-tags with using an inductor the circuit absorbs the electromagnetic energy, rectifies it and uses it for power supply. But circuit must be very low-power.
Yes. but the electromagnetic energy has a great loss when propagate in the space which is proportional to 1/r^2. So, in a far distance only a little will be received which can only drive for example a very small radio system.
Absolutely yes for low power, short duration/duty cycle applications.
Many RFID applications use this principle. For example, here in the US, certain toll highways have RFID transponders that you can attach to your car's windshield so that when you pass a toll booth, the transponder picks up the signal from a transceiver, it powers up the circuit using the energy coupled into the circuit from the transceiver signal, and then transmit its presence back to the transceiver. The transceiver is linked to your toll account and your are charged the toll.
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