Material Boundaries

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Metallic interfaces (perfect electric conductors)

By definition, we assume that all fields vanish inside a metallic material. This assumption is made by virtue of the mobile charges that are free to redistribute themselves on the surface when the metal is immersed in an external electric field. This means that the tangential electric field must be zero, i.e.

where is the unit vector directed away from the surface of the conductor.

Since charges redistribute themselves on the surface of the conductor to make the field vanish within the metal, just outside the metal, the field lines are perpendicular to the metal. These field lines terminate on the induced surface charge given by

,

given that ε is the dielectric permittivity of the material enclosing the metal surface.

Dielectric interfaces

Where different non-conducting materials meet, a dielectric interface is formed. To form the boundary condition on the tangential field, let us compute the potential around a closed loop that cuts across the interface (as seen in the accompanying

figure). We know that the potential difference around any closed loop

in an electrostatic field must be zero. As the height of the loop becomes very small, such that the one arm of the loop lies just outside the dielectric and the other arm just inside, the line integral can be approximated as

The condition on the closed-loop potential difference implies that the tangential electric field must be continuous as one crosses the dielectric boundary. This means that

.

are the tangential components of the field (i.e. parallel to the surface of the interface).

This argument is valid also for time-varying problems, because one can always apply this idea on a scale where the electric field appears to be "static" (i.e. on a scale much smaller than a wavelength).

To find the condition on the field perpendicular to the dielectric interface, enclose a small portion of the surface in a cylinder of radius a. The height h of the cylinder is such that one end-face lies just outside the dielectric and the other just inside. Hence, we can approximate Gauss's Law with

πa²(D_n1 - D_n2) = ρ_sπa²,

where is a term representing a surface charge that may exist on the surface. Hence, the normal electric flux density must satisfy

(D_n1 - D_n2) = ρ_s.

If is zero, we see that the normal electric flux density must be continuous across the dielectric boundary.

Method of Images

For problems that exhibit a high level of symmetry, it may be possible to represent the charges induced on metal surfaces by equivalent image charges within the metal. Note that this is a mathematical trick by which we build an equivalent field problem that satisfies the boundary conditions for the region of interest. The solution is only valid in the original domain (x > 0).

In the figure, the simplest case is illustrated: a point charge Q at a distance a in front

of a metallic plane.

If we take away the metallic plane and put another point charge of opposite polarity at a distance a to the left of the origin, the fields for x > 0 behave exactly as in the original problem. In effect, we have simplified the problem by using the notion of an image.