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I read in the book that when we apply high-frequency signal to a conductor, the signal propagates as a sinusoidal current in a conductor. But when the conductor is connected to the antenna, the signal propagates as an electromagnetic wave in free space.

What is the reason behind it?

what it has to do with high frequecy?

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It's because radiation is always made by accelerating charge, and the peak acceleration of a charge moving with amplitude $a$ at angular frequency $\omega$ is $a\,\omega^2$.

J. J. Thomson and Edward Purcell gave us this wonderfully elegant description: we imagine a stationary charge, whose equilibrium electric field line distribution is the radial lines outside the circles in this image. (Thomson thought of the visualization, Purcell used it alone in his highly original derivation of the Larmor radiation formula).

enter image description here

Now the charge begins to move uniformly suddenly, making its steady state field line distribution look like the field inside the circles in the image above. But the changes to the field can only propagate outwards at a maximum speed of $c$ (special relativity), and because the field lines cannot break (since there is no charge in the diagram aside from the accelerated one, and Gauss's law tells us that lines can only terminate on a charge), we must have a configuration like that in the diagram where there is a transition region between the two circles where the E-field is bent nonradially to link the two steady state configurations. This outwardly running kink is the radiation, and it is not hard to see that, not only is acceleration of charge sufficient for radiation, it is also necessary, since, if the charges are all moving uniformly, the field lines will assume a steady state shape and cannot radiate see my answer here for more details.

So now, to make radiation, we need accelerated charges. The more you accelerate them, the more radiation we get. The radiation power varies as the square of the acceleration; indeed you can interpret Purcell's diagram and argument quantitatively as Purcell did and derive the Larmor Formula. See:

Daniel V. Schroeder, Department of Physics, Weber State University, "Purcell Simplified, or Magnetism, Radiation and Relativity", Talk presented at the 1999 Winter Meeting of the American Association of Physics Teachers

The peak acceleration of a charge moving with amplitude $a$ at angular frequency $\omega$ is $a\,\omega^2$. It should now be obvious why high frequency is needed for radiation.

Also, two conductors kept closely parallel bearing high, accelerating current but such that the motions of the charges in the two conductors are opposite does not radiate much. This is known as shielding. What happens is that each conductor radiates as above, but the other acts as a receiving antenna to reabsorb the radiation through the action of the dynamic electric field on its charge back into the circuit.

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