From its electromagnetic field, what makes a too short antenna capacitive, and a too long dipole antenna inductive?

Question

I know that at a given frequency and therefore wavelength, a dipole antenna that is too (electrically) short looks capacitive, and one that is too long looks inductive. I can see it on the Smith Chart, and I can counteract it e.g. by "loading" a short antenna with an inductor. There are also some thought experiments out there, e.g. "0 length dipoles", that show why a short antenna must be capacitive to make sense.

But what I don't understand is what actually happens to the electromagnetic field to make the antenna look like an inductor or capacitor at the given wavelength.

I do (roughly, at least) know how current is distributed along a mismatched dipole: On a short dipole, the current is roughly triangularly distributed, because as you get closer to the extreme slope of the sine lobe the slope looks more and more linear. On a long dipole, you get larger and larger (and then more and more) opposite sign current lobes that cancel the electric field.

But how that makes the dipole effectively a capacitor or an inductor, I don't understand. I guess there must be some analogy on how the fields look and/or behave to how they do on either component. Is there some intuitive, maybe even visual explanation?

Answer 1

This is a good question, but we can only see its "field related" answer in case of infinitesimal dipoles only.

In short, using the full field expressions (in the near region) of the infinitesimal dipole, we can derive the complex power propagating radially from the source using:

$$ S = \frac{1}{2} \int \int{(\vec{E} \times \vec{H}^*) . d\vec{A}} $$ This is from the complex Poynting theory is: $$ S = P_{rad} + 2j\omega(W_m-W_e) $$ In case of infinitesimal dipole, the imaginary part is negative, indicating more average electric energy than magnetic, exactly like a capacitive reactance.

Without knowing the field expressions in the near region, it is not possible to calculate the imaginary part of S. So, we can use that "field point of view" in case of infinitesimal dipoles only, where we can derive these field expressions in the near region. Otherwise, we look into the antenna input impedance obtained from measurements, simulations... or whatever.