This thread has presented a clear illustration of the danger of quoting
from a book, even an authoritative one, without fully understanding the
context.

Before I continue, let me stipulate that the conductors being talked
about here are all electrically short (much shorter than a wavelength),
and are placed parallel to the E field. The following statements won't
be generally true if both those conditions aren't met.

I've been saying that the voltage at the center of an open circuited 1 m
dipole immersed in a 1 V/m field is 0.5 volts. Richard has quoted some
references which say that the voltage induced in a 1 m wire immersed in
a 1 V/m field is 1 volt, and implying that it follows that the voltage
at the center of an open circuited dipole of that length is 1 volt. It
isn't, and I'll try to explain why.

The voltage drop, or emf, across a very short conductor in a field of E
volts/meter is E*L volts, where L is the length of the very short
conductor in meters. This is undoubtedly the reason for the statements
like the ones Richard has quoted. If we look at the emf generated across
each tiny part of a one meter long dipole by a 1 V/m field and add them
up, we'll find that they total 1 volt. But what's the voltage across the
dipole, from end to end? The answer to that is it's just about anything
we want it to be. Any time we try to measure the voltage between two
points in space when there's a time-varying H field present, the answer
we get depends on the path we take between the two points. Crudely but
not entirely accurately put, it depends on how we arrange the voltmeter
leads. We can, however, measure the voltage at the dipole center, across
a gap that's arbitrarily small.

So what's that voltage? I can't think of any line of reasoning which
would deduce that it's equal to the total emf along the wire (1 volt for
our example). You can casually open various texts and find the answer to
that -- it's either 0.5 volts, as I've said, or 1 volt, as Richard has
said. To understand the reason for the apparent contradiction requires
digging more deeply into the texts.

Most of the texts I have analyze field-conductor interactions with a
special kind of dipole which has a uniform current distribution -- that
is, the current is the same amplitude all along the dipole's length.
There are a couple of ways you can physically make a dipole like this.
One is to begin with a conventional (but short) dipole and add large end
hats, like a two-ended top loaded vertical. A number of authors (e.g.,
Kraus) show a diagram of such a dipole. Another way to get a
distribution like this is to stipulate that the dipole really be only a
tiny segment of a longer wire. The short conductor with uniform current
is often called a "Hertzian dipole", sometimes an "elemental dipole",
sometimes just a short or infinitesimal dipole, or by Terman, a
"doublet". As Balanis (_Antenna Theory - Analysis and Design_, p. 109)
says, "Although a constant current distribution is not realizable, it is
a mathematical quantity that is used to represent actual current
distributions of antennas that have been incremented into many small
lengths." The voltage at the center of a one meter antenna of this type
in a 1 V/m field is 1 volt.

But this fictitious dipole, used for conceptual and computational
convenience, isn't a real dipole. A real dipole -- that is, just a
single, straight wire with a source or load at its center and no end
hats -- doesn't have a uniform current distribution. Instead, the
current is greatest in the center, dropping to zero at the ends. When
transmitting, the current on a short dipole drops nearly linearly from
the center to the ends. When receiving, the current distribution is
nearly sinusoidal. The net result of this non-uniform current is that
the voltage at the center is less than it is for a uniform-current
dipole -- exactly half as much, actually. Conceptually, it's because the
current near the ends contributes less to the voltage at the center. I'm
going to wave my hands over the significance of the different receiving
and transmitting current distributions, except to say that reciprocity
is still satisfied in all ways, including the transmitting and receiving
impedances being the same. To add confusion, the gain, directivity, and
effective apertures of both types of dipole are the same -- 1.5, 1.5,
and 3 * lambda^2 / (8 * pi) respectively. This means that you can
extract the same amount of power from an impinging wave with either type
of antenna, provided that you terminate each in the complex conjugate of
its transmitting feedpoint impedance. The radiation resistance of the
uniform-current dipole is 4 times that of the conventional dipole.
(Remember, these are all electrically short.)

There's a common term for the relationship between the field strength
and the length of a conductor, called the "effective height" or
"effective length". The voltage at the center of a dipole in a field of
E volts/m is simply E * the effective length. The concept is valid for
any length conductor, not just short ones. The effective length of a
uniform-current dipole is equal to the wire length. The effective length
of a short conventional dipole is 0.5 times the wire length. The
effective length for receiving is the same as the effective length for
transmitting -- in transmitting, it relates the strength of the field
produced to the *voltage* -- not power -- applied across the feedpoint.
If you apply 0.5 volts to a standard dipole and 1.0 volts to a
uniform-current dipole, the power applied to each will be the same
because of the 1:4 ratio of radiation resistance, and the generated
fields will be the same. This is consistent with the antenna gains being
the same.

As I mentioned, most text authors use a uniform-current dipole for
analysis. One which directly derives the voltage of a standard short
dipole is King, Mimno, and Wing, _Transmission Lines, Antennas, and Wave
Guides_. Many others, including Kraus, _Antennas_ (2nd Ed. p. 41),
derive the effective length for a short conventional dipole as 0.5 * the
physical length, from which the open circuit voltage due to an impinging
field can easily be determined.

As a last note on a point of contention, electric field strength is
usually defined not by the voltage induced in a conductor but from the
force between charges using the Lorenz force law. The unit of electric
field strength is found to be newtons/coulomb, which is the same as
volts/meter. Among the texts using this definition are Kraus
(_Electromagnetics_), Terman (_Radio Engineering_), Ida, Majid, and Ramo
et al.

Roy Lewallen, W7EL