\(R\propto \cfrac{1}{A}\)
We let,
\(R=\cfrac{k}{\pi r^2}\) where \(k\) is a constant of proportionality.
If \(r=\sqrt{\cfrac{d}{x}}\) then
\(R=\cfrac{k}{\pi d}.x\)
then
\(V=IR=I\cfrac{k}{\pi d}.x\)
In which case,
\(E=\cfrac{V}{d}=I\cfrac{k}{\pi d^2}.x\)
Given a constant \(I\),
\(E=D\cfrac{V}{d^2}.x\)
where \(D=\cfrac{k}{R\pi }\) is a constant and \(R\) is the total resistance of the wire.
A plot of \(r=\sqrt{\cfrac{d}{x}}\) is given below, the actual function plotted is (10/x)^(1/2), with d =10,
This graph is rotated about the x-axis to generate a solid that is the antenna. Since the driving signal \(V\) is sinusoidal, it can be applied to the boarder end all the same.
