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I am looking at a question (though my queries here are more regarding th underlying conepcts, which I don't quite understand) about a radio antenna that is polarising a conducting sphere.

For reference, the question is

A radar set to operate at a frequency $\nu$ has an antenna which emitts a narrow beam of radio waves with Ponting flux $N(\theta, \phi)= ...$ (I have avoided putting the actual functional form here, as I am not sure I can redistribute the question). The total emitted power is P, and the receiver input imepedance is R. The antenna is also used as a detector with a detection limits set by V_{n} as the minimum voltage across the matched load that can be detected. Find the maximum range at which the radar set can detect a perfectly conducting sphere of radius a.

My question here is regarding the poyntiny flux and its role. I would have thought that the electric field strength is $E^2 = 2NZ_0$, with $Z_0$ the impedance of free space, straight from the definition of the Poynting flux as $N=E \times H$. And yet, then I would obtain an electric field strength that does not depend on the distance from the source (which I would think is actually correct for a given electromagnetic wave) but then the maximum dipole moment of the sphere would also not depend on the distance from the radar, if I simply use $p=\alpha E$. I would be grateful if someone could explain where the distance dependence comes in here.

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Your assertion that electric field strength does not depend on distance is incorrect. Poynting flux decreases from the source as $1/r^2$. Thus each of the electric and magnetic field amplitudes vary as $1/r$. For an ideal isotropic radiator, the flux is just the radiated power divided by the total solid angle, 4$\pi$. With a directional antenna, the power is radiated into the solid angle corresponding to the beam width. The ratio $4\pi/\Omega$, where $\Omega$ is the solid angle of the beam, is called the antenna gain. You also have to consider how the target scatters the incident radiation, and determine what fraction of the scattered signal is captured by the beam width of the receiving antenna. In addition, the scattered flux decreases by $1/r^2$ on its return trip to the antenna.

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