Introductory Physics Volume Two

8 Electric Field 1.2
similar to the gravitational force. Recall that the gravitational force
between two massive objects is inversely proportional to the square of
the distance between the objects and proportional to the weight of each
object. The electric force between two charged objects is also inversely
proportional to the square of the distance between the objects and is
proportional to the charge on each object. This observational fact is
referred to as Coulomb’s Law.
Fact: Coulomb’s Law
The magnitude of the force between charges q a and q b that are a
distance r apart is given by
F = q aq b 1
4πɛ o r 2
1
where
4πɛ o
= 8.987552 × 10 9 N · m 2 /C 2 ≈ 9.0 × 10 9 N · m 2 /C 2 .
It has already been noted that the direction of the
electric force is determined by the sign of the charges:
opposites charges attract, like charges repel.
Fact: Coulomb’s Law in Vector Form
The magnitude and direction of the electric force can be combined
into one vector expression of Coulomb’s Law as follows:
⃗F ab = q aq b ⃗r a − ⃗r b
4πɛ 0 |⃗r a − ⃗r b | 3
where F ⃗ ab is the force on charge a due to charge b and ⃗r a and ⃗r b
are the position vectors of the two charges. Note that if we let
⃗r = ⃗r a − ⃗r b that we can write
⃗F ab = q aq b 1
4πɛ 0 r 2 ˆr
The vectors in the above definition are pictured below.
a
r a
r ab
r b
b
Both vectors are from the origin of the coordinate system, which can
be chosen for computational convenience.