Conductors

0. Conductors and Dielectrics

A conductor is a material, usually a metal such as copper, aluminum, or iron, in which some electrons can move through the object. A dielectric, also called an insulator, is a material such as plastic, glass, rubber, or dry wood. In a dielectric, charge cannot flow freely through the whole object. It can only shift a little inside atoms or molecules, which we will study in the next lesson.

A neutral conductor is not empty inside. It is made of atoms. In a metal, the positive nuclei stay in place as part of the solid, while some electrons are able to move from place to place. In an ordinary neutral piece of metal, the positive charge and negative charge balance in each small region. That means the net charge of the interior is zero, and the whole conductor has no net charge unless extra charge is added or induced.

Use the mode buttons to compare the two responses. In dielectric mode, electrons stay close to their own atoms and only shift a little. In conductor mode, mobile electrons can move across the whole object, so an outside field can create charge imbalance near the surfaces. The animation between modes is there to make the difference visible, but the important idea is simple: dielectric charge shifts locally, conductor charge can move through the material. The next concept studies conductor equilibrium more carefully: the surface statement is about net excess charge, not a claim that every electron leaves the inside.

1. Basic Electrostatic Properties

These properties are statements about electrostatics: the charges have already had time to move, and we are looking at the final equilibrium state where there is no continuing current inside the conductor.

That starting point matters. If a conductor is placed in an external electric field, its mobile charges immediately feel \(\vec{F}=q\vec{E}\). Electrons drift one way. Since the positive parts of the atoms cannot flow through the metal, the other side is left with less negative charge, so it acts positive. This separated, induced charge creates its own field in the opposite direction to the original field. The motion continues only until the cancellation is complete.

1. \(\vec{E} = 0\) inside a conductor. At equilibrium, any nonzero internal field would make the free charges move again. Therefore the total field inside the conductor must be zero.
2. \(\rho = 0\) inside a conductor. From Gauss's Law, \(\nabla \cdot \vec{E} = \rho/\epsilon_0\). Since \(\vec{E}=0\), \(\rho\) must be zero. Here \(\rho\) means the net charge density: positive charge and negative charge added together per volume.
3. Any net charge resides on the surface. Since there is no net charge density in the interior, any excess charge has only one place left to reside: the surface.
4. A conductor is an equipotential. Since \(\vec{E}=0\) inside, \(V(b)-V(a)=-\int_a^b \vec{E}\cdot d\vec{l}=0\) for any two points in or on the conductor.
5. \(\vec{E}\) is perpendicular to the surface, just outside. If \(\vec{E}\) had a parallel component, charges would flow along the surface. In static equilibrium, this cannot happen.

The key is not that conductors are magic; it is that free charges are allowed to rearrange until \(\vec{E}=0\) in the metal. Then the rest follows. Gauss's law gives \(\rho=0\) in the interior, so any excess charge must be on the surface. The zero field makes the whole conductor an equipotential. And just outside the surface, any tangential part of \(\vec{E}\) would start surface charge flowing, so the equilibrium field can only be perpendicular to the surface.

2. Induced Charges

If you hold a charge \(+q\) near a neutral conductor, the field of \(q\) pulls free electrons in the conductor toward it, creating a patch of negative charge on the near side. This leaves a patch of positive charge on the far side.

This process is called induction. The charge distribution on the surface adjusts itself perfectly so that the total field inside the conductor remains zero.

3. Cavities and the Faraday Cage

Suppose we have a conductor with a cavity (a hole) inside it, and there are no charges within the cavity. What is the field inside the hole?

Regardless of the intense electric fields raging outside the conductor, the field inside the empty cavity is identically zero. The conductor shields the cavity completely. This is the principle of the Faraday Cage.

4. Surface Charge and the Force on a Conductor

We know that the field inside a conductor is zero, but the field immediately outside is non-zero and perpendicular to the surface. This discontinuity implies that the surface charges themselves experience a force.

Because the charges on the surface repel each other, they try to expand outward. This creates an electrostatic pressure on the surface of the conductor, pulling it into the vacuum. The force per unit area (\(f\)) is given by:

This force is always directed outward, regardless of whether the surface charge is positive or negative (since \(E^2\) is always positive).

5. Capacitors

A Capacitor is formed by any two conductors separated by an insulator (or vacuum). If we place a charge \(+Q\) on one conductor and \(-Q\) on the other, a potential difference \(V\) develops between them.

The ratio of charge to potential difference is a constant called Capacitance (\(C\)). It depends purely on the geometry of the conductors, not on the amount of charge.

To "charge" a capacitor, work must be done against the electric field to move electrons from the positive plate to the negative plate. This work is stored as potential energy \(W\) in the electric field: