Gauss' law

since the normal to the surface is always parallel to the local electric field. However, we also know from Gauss' theorem that

where is the volume enclosed by surface . Let us evaluate directly. In Cartesian coordinates, the field is written

(189) |

Here, use has been made of

(191) |

This is a puzzling result! We have from Eqs. (187) and (188) that

and yet we have just proved that . This paradox can be resolved after a close examination of Eq. (192). At the origin () we find that , which means that can take any value at this point. Thus, Eqs. (192) and (193) can be reconciled if is some sort of ``spike'' function:

Let us examine how we might construct a one-dimensional spike function. Consider the ``box-car'' function

(194) |

Now consider the function

(196) |

Thus, has all of the required properties of a spike function. The one-dimensional spike function is called the

(198) |

where use has been made of Eq. (197). The above equation, which is valid for any well-behaved function, , is effectively the definition of a delta-function. A simple change of variables allows us to define , which is a spike function centred on . Equation (199) gives

(200) |

We actually want a three-dimensional spike function:
*i.e.*, a function
which is zero everywhere
apart from arbitrarily close to the origin, and whose volume integral is unity.
If we denote this function by
then it is easily seen that
the three-dimensional delta-function is the product of three one-dimensional
delta-functions:

(201) |

(202) |

which is the desired result. A simple generalization of previous arguments yields

(204) |

(205) |

(206) |

Let us now return to the problem in hand. The electric field generated by a charge
located at the origin has
everywhere apart from the
origin, and also satisfies

(207) |

where use has been made of Eq. (203).

At this stage, vector field theory has yet to show its worth..
After all, we have just spent an inordinately long time proving something using
vector field theory which we
previously proved in one line [see Eq. (187)] using conventional
analysis. It is time to demonstrate the power of vector field theory.
Consider, again, a charge at the origin surrounded by a spherical surface
which is centered on the origin.
Suppose that we now displace the surface , so that it is no longer centered
on the origin. What is the flux of the electric field out of S? This is not
a simple problem for conventional analysis, because the normal to the surface is
no longer parallel to the local electric field. However, using vector field theory
this problem is no more difficult than the previous one. We have

Consider charges located at . A simple generalization of
Eq. (208) gives

(210) |

where is the total charge enclosed by the surface . This result is called

Suppose, finally, that instead of having a set of discrete charges, we have a
continuous charge distribution described by a charge density .
The charge contained in a small rectangular volume of dimensions , , and
located at position
is
. However, if we integrate
over this volume element we obtain

(212) |

This is the first of four field equations, called Maxwell's equations, which together form a complete description of electromagnetism. Of course, our derivation of Eq. (213) is only valid for electric fields generated by stationary charge distributions. In principle, additional terms might be required to describe fields generated by moving charge distributions. However, it turns out that this is not the case, and that Eq. (213) is universally valid.

Equation (213) is a differential equation describing the electric field generated
by a set of charges. We already know the solution to this equation when the
charges are stationary:
it is given by Eq. (172),

where use has been made of Eq. (175). It follows that

(216) |

which is the desired result. The most general form of Gauss' law, Eq. (211), is obtained by integrating Eq. (213) over a volume surrounded by a surface , and making use of Gauss' theorem:

(217) |

One particularly interesting application of Gauss' law is *Earnshaw's
theorem*, which states that it is impossible for a collection of charged particles to
remain in static equilibrium solely under the influence of electrostatic forces.
For instance, consider the motion of the th particle in the
electric field, , generated by all of the other static particles.
The equilibrium position of the th particle corresponds to some
point , where
. By implication,
does not correspond to the equilibrium position of
any other particle.
However, in order
for to be a *stable* equilibrium point, the particle
must experience a *restoring force* when it is moved a small
distance away from in *any* direction. Assuming that the
th particle is positively charged, this means that the electric
field must point radially towards
at all neighbouring points. Hence, if we apply Gauss' law to a small
sphere centred on , then there must be a negative flux of
through the surface of the sphere, implying the presence of a negative
charge at . However, there is no such charge at .
Hence, we conclude that cannot point radially towards
at all neighbouring points. In other
words, there must be some neighbouring points at which is directed *away*
from . Hence, a positively charged particle
placed at can always escape by moving to such points.
One corollary of Earnshaw's theorem is that classical electrostatics cannot
account for the stability of atoms and molecules.