W. Watson wrote:
I'm guessing that one starts with an open ended transmission line, and
looks at how the E and B fields are distributed along it.

It's a standing-wave antenna. That means there is a forward
wave and a reflected wave. The forward current adds to the
reflected current all up and down the dipole. The magnitude
of the reflected current arriving back at the feedpoint is
approximately 90% of the forward current and are assumed to
be approximately equal by Kraus. The forward current
and reflected current are very close to being in phase at
the feedpoint. The forward and reflected current are equal in
magnitude and 180 degrees out of phase at the tip of the
dipole. The feedpoint impedance of a 1/2WL dipole is approximately
(|Vfor|-|Vref|)/(|Ifor|+|Iref|). The angle of the reflected current
is close to equal in magnitude and opposite in sign to the forward
current at any point along the dipole so the angle of the net
current is very close to zero degrees. The net current pictured
on that web page is a standing current wave with the same basic
shape as in a 1/4WL open-circuit stub.

That current distribution is illustrated in any antenna book,
including Kraus and Balanis. It's also in my 15th edition
ARRL Antenna Book.
--
73, Cecil http://www.qsl.net/w5dxp