PHYS 2212
Lab Exercise 02:
Charge Distributions & Integration
PRELIMINARY MATERIAL TO BE READ BEFORE LAB PERIOD
I.
Density Functions:
When we consider a charge distribution consisting of
very many
point charges, it is
convenient to introduce the idea of a
charge density
, which will allow us to treat the distribution as being
continuous
.
A “purist” might object to this, pointing out that ultimately, at the atomic scale, all distributions are
composed of individual point charges (protons and electrons).
However, if we view the distribution at a
macroscopic
scale, even a region that
we
would consider to be pointlike—e.g., a box of length 0.1 mm on a
side—is of vast size in comparison to atomic scales, and would consist of tens of millions of individual point
charges.
In that context, treating a general charge distribution as being continuous is not all that unrealistic.
A continuous
charge density function
, then, is a means of describing how a certain amount of charge is
spread
out
over some particular region.
Depending upon the
type
of region that is being spread over, we have three
different “categories” of density functions:
•
If an amount of charge is spread throughout a three-dimension volume, we have a
volume charge
density
, conventionally denoted by the symbol “
ρ
”.
(This is, perhaps, suggestive of the familiar
mass
density
that describes distributions of matter.)
A volume charge density describes the
amount of
charge found charge per unit volume examined
.
Thus, if you were to state that the volume density at
some point were 15
μ
C/mm
3
, you would essentially be asserting that a
tiny
box around that point
(again, lets say, of dimension 0.1 mm on a side) would contain a total charge of:
(15 x 10
–6
C/mm
3
)
×
(0.1 mm)
3
= 15 nC.
•
If an amount of charge is spread over some two-dimensional
surface
, we have a
surface charge
density
, and will use the symbol
η
to represent such a distribution.
It is important to recognize that we
are treating such a distribution as having “zero thickness”—even though, technically, there is
some
non-zero thickness to a typical “
η
”.
Provided, however, that the thickness “
t”
is very small compared
to the area of the surface, we can neglect
t
and treat the distribution as if it were truly “2D”.
Surface
charge density is then calculated in a manner comparable to volume charge density.
In this case,
however, we measure the amount of charge found
per unit area of the surface examined.
Keep in mind, as well, that there is no
a priori
reason why the surface in question has to be
flat
—it is
completely possible to deal with a situation where charge is spread over the curved surface of a sphere
or cylinder.