Conductors in electrostatic equilibrium

An electrical conductor is a type of material that allows charges to move freely. These substances have free charges (electrons in general) that can move through the material. Conversely, an electric insulator is a material in which the charges are tied to the molecules and therefore cannot flow freely except in a very limited way.

Electric charges can move freely in a conductor material, therefore an electric field will exert a force on them that is given by:

In a situation of electrostatic equilibrium, if a conductor like the plate of the next figure is placed in an external electric field, the plate electrons will be subject to a force in the opposite direction of the electric field. Therefore, the right side of the plate will have an excess of negative charges and the left side an excess of positive charges.

As positive charges are sources of electric field lines and negatives ones are sinks of field lines, an internal electric field (represented in blue in the figure) will appear inside the conductor.

Recall that we are assuming the conductor to be in electrostatic equilibrium. Therefore there cannot be a flow of charges (if there were, it would not be an electrostatic equilibrium), and the internal field must compensate for the external one in such a way that the field inside the conductor has to be zero. This phenomenon is known as Faraday cage.

The electric charge of a conductor resides therefore entirely on its surface and the electric field inside the conductor is zero:

In an electrostatic equilibrium scenario, the electric field lines outside the conductor have to be perpendicular to its surface: if it were not the case, there would be a tangential component to the electric field that would exert a force on the charges and therefore they would move and the conductor would not be in electrostatic equilibrium.

The magnitude of the electric field outside the conductor is calculated using Gauss’s law:

The net flux of an electric field through any closed surface is equal to the net charge q enclosed in the surface divided by the vacuum permittivity ε₀.

The Gaussian surface (a cylinder or a parallelepiped for instance) through which we are going to calculate the flux of the electric field is represented in red dotted lines in the next figure. The left side is the only side of the surface through which the flux is not zero because, as you can see in the figure, no field lines pass through the other ones.

Moreover, the electric field is parallel to the vector dS and its magnitude is the same for all the points of this side, thus the flux is given by:

The definition of the surface charge density is:

The magnitude of the electric field on the external side of the conductor is therefore:

The magnitude of the field outside the conductor is twice as large as that of an infinite thin flat sheet of charge because in the latter there is a charge (and therefore an electric field) at both sides of the sheet.

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When a charged conductor has an arbitrary shape, the surface charge density σ depends on the curvature of the different areas of the shape, but the previous expression still holds for every surface element small enough to be considered approximately flat as you can see in the next figure.

The surface of a conductor in electrostatic equilibrium is an equipotential surface. If it were not the case, the charges would move because the work done by the electric force to move a charge from one position of the surface to another would not be zero.

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