I am going to present a simple proof
that the direction of the electric force
between two charged particles lies along
the line connecting them. So to start out with
I'm going to have two source charges
which I'm going to arbitrarily call Q1
and Q2 and I'm going to draw the line
connecting them. These are particles so I
don't have to worry about
center-to-center distance and radii and
so forth, it's just the distance between these two
particles and I'm going to use the line
connecting them as an axis which I'm
going to extend in both directions.
And now we're going to rotate this charge
distribution around that axis. Now I hope
it's obvious to you that rotating this
charge distribution around that axis
doesn't change the way the charge
distribution looks. It looks exactly the
same so here we see an example of a
symmetry which basically means that I
can do something to my system and the
system does not change. So here I've rotated
my system around an axis and it looks
the same before and after the rotation.
So let's assume that the force on Q2
due to Q1 does not lie along that line. In
other words let's assume it has a
nonzero vertical component so that I
have to draw it like this.
If I were to now rotate this charge
distribution around the same axis that I
used before we would now find that
although the charge distribution looks
exactly the same, the force doesn't, so
we've broken the symmetry. So either it
must be true that space is not isotropic
or that the force on 2 due to 1 cannot
have a vertical component. The isotropy
of space is well established by
experiment and it would be sort of silly
to abandon that so we're left with the
realization that the only possible
direction that F21 can have is along the
line joining the two charged particles.
