# Conductors

When a conductor is placed in an electric field, charge carriers from throughout the volume will flow to the surface, polarizing the material and creating a counterfield. While real conductors have a finite number of charge carriers, in practice we can assume that conductors contain an inexhaustible source of charge to create the counterfield.

So when do these charges stop flowing? Well, the reason the charges are moving is because an electric field exists inside the conductor. As the counterfield gets stronger, the net electric field inside the conductor gets weaker. Eventually the counterfield will be exactly as strong as the external field, and the net electric field inside the conductor is zero. No net electric field means that the charges no longer feel a force, and the charges stop moving (besides the random motions due to thermal fluctuations). We call this electrostatic equilibrium, when the charge carriers stop moving, and we note:
Inside a conductor in electrostatic equilibrium, the electric field is zero.

Note that this is different from the case of the charged spherical shell, which created no electric field inside itself, but which did not prevent other electric fields from existing there. The charge carriers in a conductor actively arrange themselves to ensure that the electric field inside is zero, and do so relatively quickly. The only times they are thwarted are when the electric field is changing rapidly over time, or if charge is continuously being added to the conductor.

When a conductor is in electrostatic equilibrium, it exhibits several other properties as well.
• All excess charge lies on the surface; the interior is neutral.

Suppose I place some positive charge on one side of the metal block. Because charge can flow freely through the metal, the positive charges will repel one another and spread as far apart as possible. One might imagine that this means the charges would be evenly distributed throughout the metal, but actually they will all end up on the surface of the metal. We can prove this using Gauss' Law: for suppose there was some free charge somewhere in the interior of the metal. Then we could draw a sphere surrounding that free charge, and because the net flux through that sphere is zero (there's no field inside the conductor, and thus no field lines), the net charge inside the sphere must be zero. We can use that trick everywhere in the metal except on the surface. At the surface, it's impossible to surround a charge without our sphere leaving the conductor (and of course, outside the conductor the field is not necessarily zero).

• The electric field directly outside a conductor is perpendicular to its surface.

While the electric field does not have to be zero outside the metal, the field lines do have to be perpendicular to the surface. The charges sitting on the surface of the metal are able to feel the external electric field, and so feel a force due to it. The charge carriers are unable to leave the conductor, so the perpendicular component of the electric field will have no effect on them. However, if the electric field has a component parallel to the surface, it will cause the charges to flow along the surface, which means the conductor is not in electrostatic equilibrium.

This figure shows a metal sphere placed into an initially uniform electric field. The sphere polarizes, and the counterfield added to the original field causes it to bend so that the field lines enter and leave the sphere in a radial direction. Notice how the field lines terminate at the negative charges on the sphere, and start up again from the positive charges, just as one would expect.

• The electric field inside any empty cavities inside the metal is also zero.

We know that the electric field inside a conductor is zero, but what if there is an air-filled cavity inside the metal, completely surrounded by the conductor? (Think of it as a metal box that is currently closed.) Consider a block of metal placed in an electric field which points to the right. We know that it will polarize, and charge carriers on the surface will create a counterfield making the electric field zero inside the metal. Now suppose we were able to disintegrate a portion of the metal inside, forming an enclosed bubble without doing any damage to the rest of the block. Because the polarized surface charge is untouched by the disintegration, the counterfield is the same in the bubble as it was when it was metal: perfectly equal and opposite to the external field. Thus the electric field inside the bubble is also zero.

This property is called the Faraday cage effect, and has a number of practical applications:

• Experiments and equipment which are sensitive to small electric fields are usually surrounded by Faraday cages, to protect them from stray electric fields in the environment.
• The reason a car is relatively safe in a lightning storm is because its metal frame forms a partial Faraday cage. (The rubber tires don't matter much; if the lightning can travel through miles of air it won't have much trouble with a couple inches of rubber.)
• Faraday cages can block radio signals, because radio waves are a combination of electric and magnetic fields. Sometimes this is desirable (top secret locations are often enclosed in Faraday cages so that information cannot be transmitted out) and sometimes it is not (like when your cellphone doesn't work so well in an elevator).
It should be noted that Faraday cages only block fields which come from outside the cavity. The cavity will contain an electric field if a charge is placed inside.
A $$+3\u{\mu C}$$ charge is placed inside a cavity inside a neutral metal sphere as shown.
What does the electric field of this situation look like?
What is the total charge on the outer surface of the sphere, and on the inner surface of the cavity?
What if I place a $$-1\u{\mu C}$$ charge on the metal part of the sphere? What are the total charges on the outer and inner surfaces then?
• The electric potential is the same everywhere inside the metal.

The electric field points from higher to lower potential. Since the electric field is zero inside a conductor in equilibrium, there must not be any higher or lower potentials: the potential at every point inside the conductor is the same. Note that it is not necessarily zero: a metal ball could be at a potential of 5V, or -4V, or what have you. But the metal ball, if it is in equilibrium, does have a single, well-defined potential. (We will discuss this a bit more in the next section.)