in vector notation (see Fig. 24). An equal and opposite force acts on the first charge, in accordance with Newton's third law of motion. The SI unit of electric charge is the coulomb (C). The magnitude of the charge on an electron is C. The universal constant is called the

(162) |

Coulomb's law has the same mathematical form as Newton's law of gravity.
Suppose that two masses, and , are located at position vectors
and . The gravitational force acting on the second mass
is written

(164) |

(165) |

(166) |

(167) |

In summary,
there are two long-range forces in the Universe, electromagnetism and gravity.
The former is enormously stronger than the latter, but is usually ``hidden'' away
inside neutral atoms. The fine balance of forces due
to negative and positive electric charges starts to break down on atomic scales.
In fact, interatomic and intermolecular forces are all
electrical in nature. So, electrical forces
are basically what prevent us from
falling though the floor. But, this is electromagnetism on the
microscopic or atomic scale--what is usually termed *quantum electromagnetism*. This course is about
*classical electromagnetism*. That is, electromagnetism on length-scales much
larger than the atomic scale. Classical electromagnetism
generally describes phenomena
in which some sort of ``violence'' is done to matter, so that the
close pairing of negative and positive
charges is disrupted. This allows electrical forces to manifest
themselves
on macroscopic length-scales. Of course, very little disruption is necessary
before gigantic forces are generated. It is no coincidence that the vast majority
of useful machines which humankind has devised during the last century or so
are electrical in nature.

Coulomb's law and Newton's law are both examples of what are usually referred to as
*action at a distance* theories.
According to Eqs. (161) and (163), if the first charge
or mass is moved then the force acting on the second charge or mass immediately
responds. In particular, equal and opposite forces act on the two charges or masses
at all times. However, this cannot be correct according to Einstein's theory of
relativity, which implies that the maximum speed with which information can propagate through
the Universe
is the speed of light in vacuum. So, if the first charge or mass is moved then there must
always be time delay (*i.e.*, at least the time needed for a light
signal to propagate between the two charges or masses) before the second charge or
mass responds. Consider a rather extreme example. Suppose the first charge or
mass is suddenly annihilated. The second charge or mass only finds out about
this some time later. During this time interval, the second charge or mass
experiences an electrical or gravitational force which is as if
the first charge or mass were still there.
So, during this period, there is an action but no
reaction, which violates Newton's third law of motion.
It is clear that action at a distance
is not compatible with relativity, and, consequently,
that Newton's third law of motion is
not
strictly true. Of course, Newton's third law is intimately tied up with the
conservation of linear momentum in the Universe. This is a concept which most physicists are loath
to abandon. It turns out that we can ``rescue'' momentum conservation by abandoning
action at a distance theories, and instead adopting so-called *field theories* in which
there is a medium, called a field, which transmits the force from one particle
to another. In electromagnetism there are, in fact, two fields--the electric field,
and the magnetic field. Electromagnetic forces are transmitted via these
fields at the speed of light, which implies that the laws
of relativity are never violated.
Moreover, the fields can soak up energy and momentum. This means that even when
the actions and reactions acting on particles are not quite equal and opposite,
momentum is still conserved.
We can bypass some of the problematic aspects of action at a distance by
only considering *steady-state* situations. For the moment, this is how we shall
proceed.

Consider charges, though , which are located at position vectors
through . Electrical forces obey what is known as
the *principle of superposition*. The electrical force acting on a test charge
at position vector is simply the vector sum of all of the
Coulomb law forces from each of the charges taken in isolation. In other
words, the electrical force exerted by the th charge (say) on the test charge is
the same as if all the other charges were not there. Thus, the force acting
on the test charge is given by

(168) |

and the electric field is given by

At this point, we have no reason to believe that the electric field has any real physical existence. It is just a useful device for calculating the force which acts on test charges placed at various locations.

The electric field from a single charge located at the origin is purely radial,
points outwards if the charge is positive, inwards if it is negative, and has
magnitude

We can represent an electric field by *field-lines*.
The direction of the lines
indicates
the direction of the
local electric field, and the density of the lines perpendicular to this direction
is proportional to the magnitude of the local electric field.
Thus, the field of a point positive charge is represented by a group of equally
spaced straight lines radiating from the charge (see Fig. 25).

The electric field from a collection of charges is
simply the vector sum of the fields
from each of the charges taken in isolation. In other words, electric fields are
completely *superposable*. Suppose that, instead of having discrete charges, we
have a continuous distribution of charge represented by a *charge density*
. Thus, the charge at position vector is
, where is the volume element
at . It follows from a simple extension of Eq. (170) that the electric
field generated by this charge distribution is