If you write the electric field as
the gradient of the electrostatic
potential and then substitute that into Gauss' Law you can obtain
Poisson's Equation. Laplace's equation applies in situtations
where the volume space charge density is zero. We'll be using
Laplace's equation in class on Thursday. A handout
with vector differential operators in cartesian, cylindrical, and
spherical coordinate systems was distributed in class.
We spent the rest of the class
looking at a couple of instruments used to measure thunderstorm and
lightning electric fields.
The first is an electric field mill
used to measure static and slowly time varying electric fields.
Referring to the figure below at left (from Uman's 1987 The Lightning
Discharge book). The sensors (referred to as studs in the figure)
are covered by a
rotating grounded plate. The rotating plate is notched or slotted
so that the sensors are periodically exposed to and covered (shielded)
from the ambient electric field. A photograph of the field mill
shown in class is shown below at right.
The two photographs below are closeups of the top of the field mill
The stator plates are exposed to the E field at left and covered
in the photograph at right.
The next figure shows currents flowing into and out of the sensor
plate in response to an incident E field.
The sensor plate is covered at Point 1. At Point 2 the
sensor is uncovered and we assume the ambient field points upward (to
negative charge in the lower part of a thunderstorm perhaps).
Positive charge flows up to the sensor plate. The current flows
from the sensor in Point 3 because the sensor has been covered and
shielded from the E field. Points 4 and 5 are similar except the
polarity of the E field has been changed.
Note the current signals at Points 2 & 5 even though the field
polarities are reversed. You must keep track of when the sensor
is covered and uncovered if you are going to determine the polarity of
the incident E field.
It is a relatively simple matter to relate the amplitude of the
signal current to the intensity of the incident E field.