We recall from the preceding section on Gauss's Law that the electric
field vanishes inside a conductor *in electrostatic equilibrium*.
Wires carrying current are *not* in equilibrium and we'll learn
about them shortly. Conductors in equilibrium are known by their boring
nature - nothing moves, as there are no unbalanced forces or torques.

Well, if there is no field, and the field is the *derivative* of the
potential:

(17) |

then the potential inside a conductor must be *constant*. This is
very useful to know. We say that a conductor at ESE is *equipotential*. A charge can be moved around anywhere inside without
doing any net work on it.

This has profound global consequences on the shape of the field outside
the conductor. Remember, the potential at *any* part of the surface
of the conductor is the work per unit charge bringing a charge in from
infinity to *any* point on the surface. No matter which path
followed. There may be places where the field is strong, there may be
places where it is weak, but the *path* integral of the field in
from infinity has the same value for all paths. Wow!

There are two important properties of charged conductors that can be
deduced from this simple result. One is how charge is shared between
two conductors that are electrically connected with a conductor and left
until they arrive at *mutual* electrostatic equilibrium (at the same
potential). The other is the way that surplus charge and field strength
is arranged on various curved conducting surfaces at ESE as a function
of the radius of curvature. *Lots* of useful physics follows from
understanding this.

Suppose we have two conducting spheres of radius and and put
a charge on either one of them. We then separate them by a lot
farther than we'd like to walk^{3} and
connect them with a conducting wire long enough for the charge to be
shared between them. How is the charge distributed?

Well, the two conductors must be equipotential. If we remove the wire after it has done its work and the two spheres are far enough apart that the potential of one can pretty much be neglected on the surface of the other, and if a charge was tranferred when they reached ESE:

(18) |

from which we can see that

(19) |

where is the final charge on sphere 1 and is the
final charge on sphere 2. The charge on a given connected conducting
surface at ESE varies *linearly* with the radius of curvature of
that surface. The larger the radius of curvature (the flatter) the more
of the charge. The charge on a pie-plate mostly resides where the shape
is flat, not on the curved surfaces. How interesting.

Even more interesting, consider the *field* strength near the
surfaces 1 and 2.

(20) |

and

(21) |

But , so

(22) |

Don't be confused. In english, this is just ``The field is stronger on the most sharply curved surface'' (which has the smallest radius of curvature). How interesting indeed.

*Charge* likes to concentrate on the least curved surfaces of a
conductor, but the field strength associated with that charge is highest
at points of the greatest curvature. This is because the charge drops
off linearly in the curvature, but the field strength goes up inverse
quadratically, basically. From this we deduce an important consequence:

*Sharp points of a charged electrical conductor have extremely
strong fields.*

For example, the radius of curvature of a needle tip might be on the
order of microns (millionths of a meter). If the needle has any surplus
charge at all on it, The field strength there might be hundreds of times
stronger than on the more gently rounded sides. The field can even be
strong enough to *rip apart air molecules*.

Whoa! Way cool! How does this work?

I'm glad you asked.