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I have been reading about how radio dipole antennas radiate "bubbles" of electromagnetic waves, due to the kinking of the field lines from the acceleration of the charges. The bubbles that I have seen are always shown as not extending to the dipole axis.

If an external test charge was placed exactly on the dipole axis, but some distance removed from the antenna, does the external test charge sense the change in the electric field of the dipole? The field of a dipole along the z axis decreases as $p/z^{3}$ and since the dipole moment is changing, my intuition is that this change would be communicated to the test charge.

In Purcell's text, a single accelerating charge generates a sphere of field line radiation that extends through the axis of acceleration. I have seen animations of this as well. I realize that the dipole scenario must be more complicated, but I would think that the field of the dipole would still be detectable in some way along the dipole axis of acceleration.

I'm not asking if a receiving antenna along the dipole axis could reconstruct the signal, just if a test charge would sense any change at all in the field.

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I found a reference from the online MIT course 6.013 about the analytical expression of EM wave radiation from a dipole antenna.

"Hertzian dipoles preferentially radiate laterally, with zero radiation along their axis. The electric field strength Eθ varies as sin θ, which yields a circle in a polar plot, as illustrated in Figure 10.2.2(a)"

David Staelin. 6.013 Electromagnetics and Applications. Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

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