If I have an antenna (let's say a short dipole antenna) it radiates an EM field in both longitudinal and transverse directions, I am aware that the longitudinal component will decrease faster (1/r^2 against 1/r) and the transverse component will be the only important one for long distances, but the longitudinal field will never disappear entirely, so this means that in all space there will exist a longitudinal field. I find this contradictory to the idea that in free space E and B fields are always transverse to the direction of propagation What am I missing?
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