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I fail to understand the true difference between EMR and electric and magnetic fields. When current flows, there is an electric field due to the electron flow and a magnetic field, however no EMR (Save blackbody radiation from the wire) is emitted. Then how can radio broadcasts function if magnetic fields only are emitted from wires? Obviously I am missing something major here.

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The field from a current in a wire is purely magnetic for a static current. When the current varies with time, there will be radiation. What you are missing is that a radio antenna doesn't operate with DC.

You can perhaps understand it like this: the magnetic field from a current loop depends on its magnetic moment, which is current times area. When the current changes, the magnetic field has to change too. Then Maxwell's equations tell us that an electric field will be induced, so in the non-static case we can't have a purely magnetic field. If you accept that the field far from the antenna can't respond instantly to the varying current, it is not too hard to accept that there will be radiation. The field at a distance $r$ at time $t$ will respond to the current as it was at time $t_R = t - r/c$ for some speed $c$, so we get a field traveling outward, that is, radiation.

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