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I was looking up dipole antennas on Wikipedia, among other sites. I came across this animation:

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and was wondering why there wasn't any field lines in perfectly vertical direction. In the case of a finite antenna would there not be a field there at an instantaneous moment of time?

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Oscillating currents tend to release radiation only perpendicular to the direction of the current itself. A dipole antenna is maybe the simplest antenna to demonstrate this. There isn't really a good reason why, other than this result pops out from the math. The best explanation I can think of is that currents only tend to create magnetic fields perpendicular to the direction of the current. This is also true in electrostatics, when you study the Biot-Savart law. It's also related to how the magnetic field is given by the curl of the magnetic vector potential, which points in the same direction as the current.

There will actually be fields in the vertical direction, but only non-radiating electric fields, that are closely bound to the antenna.

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    $\begingroup$ Lorentz force, magnetic induction and electric induction all have the same inner reason. Electrons have a magnetic dipole moment and related to this moment an intrinsic spin. The spin has the same properties as a gyroscope. More see my explanations in this paper. $\endgroup$
    – HolgerFiedler
    Commented Nov 30, 2015 at 5:14
  • $\begingroup$ There is an obvious relationship between the field direction and the acceleration of the electrons in the dipole. No transverse component can be seen in the same direction as the electron acceleration. $\endgroup$
    – ProfRob
    Commented Jan 16, 2017 at 14:41

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