Where do electromagnetic waves get their energy from?

[radio171]

We know that EMW are generated by accelerating charges.
If I accelerate a charge it should gain kinetic energy equal to half the mass of the charge times velocity squared.
But at the same time it will radiate electromagnetic waves which carry energy itself.
So where does this energy come from?
It seems that it requires more energy to accelerate the charge and some of this energy is radiated.
If so, how do I know how much of it is turned into EMR and how much is turned into kinetic energy.

Please, correct me, because my logic doesn't make sense.

 

[Dale]

You are correct. It takes more energy to accelerate a charged object than it would take to accelerate an otherwise identical object.

 

[sophiecentaur]

There are many ways in which an EM wave can be generated but consider a simple transmitter feeding a dipole. During each cycle, charges (electrons) are accelerated in either direction and most of the energy is stored in the local fields (as each side of the dipole is charged and discharged). This involves both KE and PE. But, as well as the energy per cycle that is sloshing about in the dipole, there is EM energy radiated in the form of a wave and this actually appears (to the transmitter) as a resistance. The energy that's not radiated builds up in the reactances. When the dipole is fed at it's natural resonant frequency, it is easiest to match to the transmitter / feeder but an antenna of any shape or length can be made to radiate.

it should gain kinetic energy equal to half the mass of the charge times velocity squared

It's worth remembering that the actual KE involved in the average movement of electrons in time with the EM wave frequency is actually very tiny and the energy associated with the currents (movement of charge) is actually the form of an oscillating magnetic field. As an exercise, calculate the maximum velocity and displacement of an electron with a mean drift speed of a couple of mm per second (at DC) when the field is being alternated at 10MHz and a field of, say 10V/m.
The displacement of charge is associated with the oscillating electric field.
It seems to me that your worry about where the KE goes can be resolved by considering the Potential Energy that it's converted to. It's only the in-phase components of these fields that actually radiates as a wave. Locally, there is a complicated jumble of oscillating fields in a variety of phases and directions and these don't radiate.