ELECTROSTATICS OF CONDUCTORS

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Macroscopic electrodynamics is concerned with the study of electromagnetic fields in space that is occupied by matter. Electrodynamics deals with physical quantities averaged over elements of volume that are physically infinitesimal and ignore the microscopic variations of the quantities that result from the molecular structure of matter. The fundamental equations of the electrodynamics of continuous media are obtained by averaging the equations for the electromagnetic field in a vacuum. The form of the equations of macroscopic electrodynamics and the significance of the quantities appearing in them depend on the physical nature of the medium and on the way in which the field varies with time. Charges present in a conductor must be located on its surface. The presence of charges inside a conductor would cause an electric field in it. These charges can be distributed on its surface, however, in such a way that the fields that they produce in its interior are mutually balanced. The mean field in the vacuum is almost the same as the actual field. The two fields differ only in the immediate neighborhood of the body, where the effect of the irregular molecular fields is noticeable, and this difference does not affect the averaged field equations.

§1 The electrostatic field of conductors

Macroscopic electrodynamics is concerned with the study of electromagnetic fields in space that is occupied by matter. Like all macroscopic theories, electrodynamics deals with physical quantities averaged over elements of volume which are “physically infinitesimal”, ignoring the microscopic variations of the quantities which result from the molecular structure of matter. For example, instead of the actual “microscopic” value of the electric field e, we discuss its averaged value, denoted by E:

¯=E. (1.1)

The fundamental equations of the electrodynamics of continuous media are obtained by averaging the equations for the electromagnetic field in a vacuum. This method of obtaining the macroscopic equations from the microscopic was first used by H. A. Lorentz (1902).

The form of the equations of macroscopic electrodynamics and the significance of the quantities appearing in them depend essentially on the physical nature of the medium, and on the way in which the field varies with time. It is therefore reasonable to derive and investigate these equations separately for each type of physical object.

It is well known that all bodies can be divided, as regards their electric properties, into two classes, conductors and dielectrics, differing in that any electric field causes in a conductor, but not in a dielectric, the motion of charges, i.e. an electric current.†

Let us begin by studying the static electric fields produced by charged conductors, that is, the electrostatics of conductors. First of all, it follows from the fundamental property of conductors that, in the electrostatic case, the electric field inside a conductor must be zero. For a field E which was not zero would cause a current; the propagation of a current in a conductor involves a dissipation of energy, and hence cannot occur in a stationary state (with no external sources of energy).

Hence it follows, in turn, that any charges in a conductor must be located on its surface. The presence of charges inside a conductor would necessarily cause an electric field in it;‡ they can be distributed on its surface, however, in such a way that the fields which they produce in its interior are mutually balanced.

Thus the problem of the electrostatics of conductors amounts to determining the electric field in the vacuum outside the conductors and the distribution of charges on their surfaces.

At any point far from the surface of the body, the mean field E in the vacuum is almost the same as the actual field e. The two fields differ only in the immediate neighbourhood of the body, where the effect of the irregular molecular fields is noticeable, and this difference does not affect the averaged field equations. The exact microscopic Maxwell’s equations in the vacuum are

?e=0. (1.2)

?e=−(1/c)∂h/∂t, (1.3)

where h is the microscopic magnetic field. Since the mean magnetic field is assumed to be zero, the derivative ∂h/∂t also vanishes on averaging, and we find that the static electric field in the vacuum satisfies the usual equations

?E=0,curl?E=0, (1.4)

i.e. it is a potential field with a potential ϕ such that

=−gradϕ, (1.5)

and ϕ satisfies Laplace’s equation

ϕ=0. (1.6)

The boundary conditions on the field E at the surface of a conductor follow from the equation curl E = 0, which, like the original equation (1.3), is valid both outside and inside the body. Let us take the z-axis in the direction of the normal n to the surface at some point on the conductor. The component Ez of the field takes very large values in the immediate neighbourhood of the surface (because there is a finite potential difference over a very small distance). This large field pertains to the surface itself and depends on the physical properties of the surface, but is not involved in our electrostatic problem, because it falls off over distances comparable with the distances between atoms. It is important to note, however, that, if the surface is homogeneous, the derivatives ∂Ez/∂x, ∂Ez/∂y along the surface remain finite, even though Ez itself becomes very large. Hence, since (curl E)x = ∂Ez/∂y − ∂Ey/∂z = 0, we find that ∂Ey/∂z is finite. This means that Ey is continuous at the surface, since a discontinuity in Ey would mean an infinity of the derivative ∂Ey/∂z. The same applies to Ex, and since E = 0 inside the conductor, we reach the conclusion that the tangential components of the external field at the surface must be zero:

t=0. (1.7)

Thus the electrostatic field must be normal to the surface of the conductor at every point. Since E = − grad ϕ, this means that the field potential must be constant on the surface of any particular conductor. In other words, the surface of a homogeneous conductor is an equipotential surface of the electrostatic field.

The component of the field normal to the surface is very simply related to the charge density on the surface. The relation is obtained from the general electrostatic equation div e = 4πρ, which on averaging gives

?E=4πρ¯, (1.8)

being the mean charge density. The meaning of the integrated form of this equation is well known: the flux of the electric field through a closed surface is equal to the total charge inside that surface, multiplied by 4π. Applying this theorem to a volume element lying between two infinitesimally close unit areas, one on each side of the surface of the conductor, and using the fact that E = 0 on the inner area, we find that En = 4πσ, where σ is the surface charge density, i.e. the charge per unit area of the surface of the conductor. Thus the distribution of charges over the surface of the conductor is given by the formula

πσ=En=−∂ϕ/∂n, (1.9)

the derivative of the potential being taken along the outward normal to the surface. The total charge on the conductor is

=−14π∮∂ϕ∂ndf, (1.10)

the integral being taken over the whole surface.

The potential distribution in the electrostatic field has the following remarkable property: the function ϕ(x, y, z) can take maximum and minimum values only at boundaries of regions where there is a field. This theorem can also be formulated thus: a test charge e introduced into the field cannot be in stable equilibrium, since there is no point at which its potential energy eϕ would have a minimum.

The proof of the theorem is very simple. Let us suppose, for example, that the potential has a maximum at some point A not on the boundary of a region where there is a field. Then the point A can be surrounded by a small closed surface on which the normal derivative ∂ϕ/∂n < 0 everywhere. Consequently, the integral over this surface (∂ϕ/∂ϕ) df < 0. But by...