1.7: Campo Eléctrico D

Hemos estado asumiendo que todos los “experimentos” descritos se han llevado a cabo en vacío o (que es casi lo mismo) en el aire. Pero, ¿y si la carga puntual, la vara infinita y la hoja cargada infinita de la Sección 1.6 están todas inmersas en algún medio cuya permitividad no es\(\epsilon_0\), but is instead \(\epsilon\)? In that case, the formulas for the field become

\[\nonumber E=\frac{Q}{4\pi\epsilon r^2},\quad \frac{\lambda}{2\pi\epsilon r},\quad \frac{\sigma}{2\epsilon}\]

Hay un\(\epsilon\) in the denominator of each of these expressions. When dealing with media with a permittivity other than \(\epsilon_0\) it is often convenient to describe the electric field by another vector, \(\textbf{D}\), defined simply by

\[\textbf{D}=\epsilon \textbf{E}\label{1.7.1}\]

En ese caso las fórmulas anteriores para el campo se convierten en apenas

\[\nonumber D=\frac{Q}{4\pi r^2},\quad \frac{\lambda}{2\pi r},\quad \frac{\sigma}{2}\]

Las dimensiones de\(D\) are Q L^-2, and the SI units are C m^-2.

Esto puede parecer bastante trivial, pero sí resulta ser más importante de lo que puede parecer en este momento.

Ecuación\ ref {1.7.1} parecería implicar que los vectores de campo eléctrico\(\textbf{E}\) and \(\textbf{D}\) are just vectors in the same direction, differing in magnitude only by the scalar quantity \(\epsilon\). This is indeed the case in vacuo or in any isotropic medium – but it is more complicated in an anisotropic medium such as, for example, an orthorhombic crystal. This is a crystal shaped like a rectangular parallelepiped. If such a crystal is placed in an electric field, the magnitude of the permittivity depends on whether the field is applied in the \(x\)- , the \(y\)- or the \(z\)-direction. For a given magnitude of \(E\), the resulting magnitude of \(D\) will be different in these three situations. And, if the field \(\textbf{E}\) is not applied parallel to one of the crystallographic axes, the resulting vector \(\textbf{D}\) will not be parallel to \(\textbf{E}\). The permittivity in Equation \ref{1.7.1} is a tensor with nine components, and, when applied to \(\textbf{E}\) it changes its direction as well as its magnitude.

Sin embargo, todavía no nos detendremos en eso y, a menos que se especifique lo contrario, siempre asumiremos que estamos ante un vacío (en cuyo caso\(\textbf{D}\) = \(\epsilon_0\)\(\textbf{E}\)) or an isotropic medium (in which case \(\textbf{D}\) = \(\epsilon\)\(\textbf{E}\)). In either case the permittivity is a scalar quantity and \(\textbf{D}\) and \(\textbf{E}\) are in the same direction.