W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 39

same sign. A perfect rendition of the verbal statement, but now we cancomputethe force in a
specific set ofcoordinates.

The constant:

ke=

1

4 πǫ 0

= 9× 109

N−m^2

C^2

(2)

effectively defines the “size” of the unit of charge in terms of the already known SI units of force
and length, and obviously will vary if we change to a different set of units^29.

Coulomb’s Law may be simple, but it is very, very powerful – it describes the pervasive and
ubiquitous force that holds the atoms and molecules of our experience (and henceus) together.
However, it is also not in a terribly convenient form. We note that Coulomb’s law describesaction
at a distance. We’d like there to be acausefor the observed force that is presentwherethe force
is exerted, and lacking anything better to do we’llinventthe cause and call it theelectrostatic field
just as we similarly defined thegravitationalfield last semester.

Using fields is, as we will see, highly advantageous compared to alwayscomputing forces between
twocharges.

1.3: Electrostatic Field

The electrostatic field is the supposed cause of the electrostatic force between two charged objects.
Each charged object produces afieldthat emanates from the charge and is thecauseof the force the
other charge experiences at any given point in space. This field is supposed to be present everywhere
in space whether or not we measure it.

The fundamental definition of electrostatic field produced by a chargeqat position~ris that it
is the electrostatic force per unit charge on a small test chargeq 0 placed at each point in space~r 0
in the limit that the test charge vanishes:

E~= lim

q 0 → 0

F

q 0

(3)

or

E~(~r 0 ) =kq(~r^0 −~r)

|~r 0 −~r|^3

(4)

If we locate the chargeq at the origin and relabel~r 0 →~r, we obtain the following simple
expression for the electrostatic field of a point charge:

E~(~r) =kq
r^2 rˆ (5)
In general, we’ll work the other way around. First we’ll be given a distribution of charges, from
which we must determine the field. With the field known, we can then evaluate the force these
charges will exert onanother(e.g. test) charge placed placed on the field by means of the following
rule:
F~=qE~ (6)

A common question that students often ask is: “Why all of the hassle with letting test charges
go to zero if you’re just going to divide it out anyway?” The reason is that – as we will see later

• the presence of the test charge exerts a force in turn on thesourcedistribution of charge. If that

(^29) Actually the size of the Coulomb was originally defined in terms of theAmpere– the unit of electrical current –
andmagneticforces. We’ll learn about this in a few weeks when we study magnetism.