The wikipedia article states that this approximate impedance is calculated assuming a sinusoidal current distribution on a centre-fed dipole.
There are no fields in the conductors as this analysis assumes perfect conductors.
This formula is an approximation, and the term
\$Ci(\frac{2ka^2}{L}) \$
which goes as \$log(\frac{2ka^2}{L}) \$ as \$a \rightarrow 0\$
and will dominate the reactance as \$a \rightarrow 0\$ for all element lengths not \$ \lambda/2\$, and the fact that it doesn't have any affect at precisely \$ \lambda/2\$ I think should be seen as a limitation of the approximation.