Method of image charges

The mirror image charge or charge is a mental auxiliary support to illustrate the behavior of a charge Q in front of a conducting body at a distance R. The entire influenzierte charge is summarized graphically for a point charge. For reasons of symmetry this point charge is referred to as image-charge. It is therefore a special case of the load center of gravity of a load influenzierten.

The corresponding principle of mirror charge is a non-systematic method for solving electrostatic boundary value problems.

In a broader understanding of the term mirror charge in connection with any non-conductive objects appear. If such a previously uncharged object ionized, then leaving a positive charge for each missing electron. The generated charges are also called mirror charges.

Basics

A conductive body is brought into an external electric field, are usually at the beginning of the field lines are not perpendicular to the surface. This leads to potential differences along the surface, which cause the free electrons to move so that the field lines perpendicular to the surface meet ( external electric fields within conducting bodies take exponentially with time ). Since the external electric field also causes potential differences in the body, the electrons move within the body so that everywhere there is the same potential. Theoretically, therefore, to the surface charge densities. Microscopically closely at keep the electrons to each other at a distance and therefore the influence charges are always very close to the surface, but are not true surface charges.

Since the electrical field lines are perpendicular to the surface of the conductive body changes the electric field in such a way that its surface coincides with an equipotential surface. The mathematical treatment, a second electric field is thus present on the external electric field is introduced with the boundary condition that the field is perpendicular Superimposed on the surface. This is equivalent to the electrical potential at the surface with the constant demand throughout for the sake of simplicity is 0.

The converse of this approach is that you cut off the field at any given electric field along the equipotential surfaces and placed there conducting surfaces (of course, preserving the shape of the equipotential surface ). This procedure is provided with means of function theory a very powerful tool for the solution of certain boundary value problems.

Application Examples

Metal plate

Induction by an opposite charge is produced by the charge Q in the metal plate. The electric field lines and the Coulomb forces between generating charge and plate behave as if a charge -Q is at a distance R behind the metal plate. The charge is generating so - metaphorically speaking - mirrored on the metal surface.

For the arrangement of charges influenzierten surface charges are accepted on the metal surface., The charge density on the surface at the position through which the charge Q is at its maximum, and falls vertically from the radially outward. Assuming an infinitely thin metal plate, this is how the surface charge density indicate a formula as follows:

Other conductive objects

Other conducting objects, such as a metal ball, the mirror surface is not necessarily on the surface. However, it can in most cases a single point charge mental construct in which one can imagine combines the total charge ( load center).

In the case of a conducting sphere of radius R, the mirror can charge via the inversion in a circle ( mentally reflection on the circle surface ) find. Then applies to the distances of the image-charge:

Then for the amount of the mirrored charge: