Week 1: Discrete Charge and the Electrostatic Field 37
electronic background.
Franklin, unfortunately, thought that the flavor of mobile chargein ordinary conductors was
positive. In fact, as noted, it isnegative– associated with moving electrons. This is “Franklin’s
mistake” – the bane of physics students for over two hundred years, where thecurrentin a wire
generally points in theopposite directionto the actual motion of the (negative) electrons in the wire.
This will – rarely – matter in particular problems, so keep it in mind.
Note that all of these elementary charges are quite tiny in terms oftheir mass and physical
extent compared to bulk matter. There is therefore alotof charge in nearly any macroscopic piece
of matter. We can easily estimate how much within a factor of two or three by assuming that
anywhere from nearly 100% (in the case of hydrogen) to roughly 40% (in the case of Uranium) of
the mass of matter consists of theprotonsin the nuclei of the atoms that make it up, and note
that for every proton there is generally an electron. The inverse of the mass of a proton is thus a
good (approximate) measure of the number of charges per unit mass – around 5× 1026 charges per
kilogram of matter! Even amicrogram(a billionth of a kilogram) of matter thus has well over ten
million billion charges.
This makes precisely summing up fields produced by all of these charges in chunks of matter
much bigger than atoms all but impossible, even with computers. It isalso unnecessary – with so
many objects, surely anaveragewould do for most purposes! We will therefore have frequent cause
to “coarse grain” our description of matter – toignorethe discrete particulate nature of charge
and average out thetotalcharge ∆Qin afinite but very smallvolume of matter ∆V. By choosing
∆V small enough that we can treat it like a volume differential but large enough that it contains
a lot of charge, we can define a chargedensity. Similarly, we can associate charge densities with
two dimensional sheets of matter (for example, a charged piece ofpaper or metal plate) or one
dimensional lines of matter (for example, a wire or piece of fishing line). We summarize this (and
define the symbols most often used to represent charge) as:
ρ =dq
dV
σ =dq
dA
λ =dq
dxIn all of these forms, it is better indeed to think of charge as being the “fluid” that Franklin imagined
it to be!
The last property associated with charge that we wish to mention early (although we’ll examine
it in more detail later) is that various materials can often be categorized, broadly speaking, into one
of three types with quite distinct properties:
- Insulators. The charge in the atoms and molecules from which an insulating material is built
tends tonot be mobile– electrons tend to stick to their associated molecules tightly enough
that ordinary electric fields cannot remove them. Surplus charge placed on an insulator tends
to remain where you put it. Vacuum is an insulator, as is air, although neither is aperfect
insulator. Insulators still respond measurably to an applied field, however – the charges in
the atoms or molecules distort as the moleculespolarize, and the resulting microscopic dipoles
modify the applied field inside the material. Since we live in air (a material) we do not generally
see thetrueelectric field produced by a charge but one that is very slightly reduced by the
polarization of the air molecules through which the field travels. This iscalleddielectric
responseand we’ll discuss it extensively later. - Conductors. For many materials, notably metals but also ionic solutions, at least one electron
per atom or molecules is onlyweaklybound to its parent and can easily be pushed from one