The phenomenon of breakdown sets an upper limit on the charge which
can be stored on a conductor. There is, however, another important factor which
affects the onset of breakdown. This is best illustrated in the following
simple example. Suppose that we have two charged conducting spheres of radii and
, respectively, which are connected by a long conducting wire. The wire allows charge to
move back and forth between the spheres until they reach the same potential (recall
that the electric potential is uniform in a conductor). Let be the
charge on the first sphere, and the charge on the second sphere. Of course,
the total charge
carried by the two spheres is a conserved quantity.
The electric field generated by each sphere is the same as if the charge on that
sphere were concentrated at its centre. Assuming that the wire is sufficiently
long that the two spheres do not
affect one another very much, the absolute potential of the first sphere
is
, whereas that of the second
sphere is
[see Eq. (97)]. Since , we find that

(101) | |||

(102) |

Note that if the second sphere is much smaller than the first (

(103) |

Clearly, if then the field just above the smaller sphere is far stronger than that above the larger one. Suppose that the total charge on the two spheres is gradually increased until breakdown occurs. Since , it follows that breakdown always occurs above the smaller sphere.

Equation (104) is a special case of a far more general rule:
*i.e.*, the electric
field-strength above some point on the surface of a conductor is inversely proportional
to the local radius of curvature of the surface. It is clear that
if we wish to store significant amounts of charge on a conductor
then the surface of the conductor must be made as smooth as
possible. Any sharp spikes on the surface possess
relatively small radii of curvature. Intense local electric
fields are generated above these spikes whenever the conductor is charged.
These fields can easily exceed the critical field for the breakdown
of air, leading to sparking, and the eventual loss of the charge on
the conductor. Sparking tends to be very destructive because of
its highly localized nature, which leads inevitably to very large electric currents,
and, hence, to intense heating.

Clouds can acquire very large negative charges during thunderstorms. An equal
and opposite positive charge is induced on the surface of the Earth.
The electric field generated between the clouds and the Earth can become
sufficiently large to cause breakdown in the atmosphere, giving rise to
the phenomenon which we call *lightning*. Let us consider the various factors which determine where lightning strikes. Breakdown starts
at cloud level, as a so-called ``dark leader'' of ionized air traces out a path
towards the ground. When it comes within about 10 meters of ground level, a second
dark leader comes up from the ground to meet it. Once the two leaders meet, and
a conducting path is established, the lightning strike proper occurs.
Note that, contrary to popular opinion, the lightning strike
travels *upwards* from
the Earth to the clouds. It is clear that lightning ``strikes'' a particular
object on the ground because the object emits a dark leader: *i.e.*,
because breakdown takes place just above the object.
In a thunderstorm, the ground, and the
objects upon it, acts essentially like a charged conductor with a
convoluted surface.
Thus, any ``spikes'' on the ground (*e.g.*, a person standing in a field,
a radio mast, a lightning rod)
are comparatively more likely to be hit by lightning, because the electric field-strength
above these points is relatively large, which facilitates breakdown.