Question

Value of phase constant for uniform plane wave in conducting medium

Answer 1

In the limit  , there is a significant difference in the response of a dielectric medium to an electromagnetic wave, depending on whether the lowest resonant frequency is zero or non-zero. For insulators, the lowest resonant frequency is different from zero. In this case, the low frequency refractive index is predominately real, and is also greater than unity. In a conducting medium, on the other hand, some fraction,  , of the electrons are ``free,'' in the sense of having  . In this situation, the low frequency dielectric constant takes the form

(800)


where  is the contribution to the refractive index from all of the other resonances, and . Consider the Ampère-Maxwell equation,

(801)


Here,  is the true current: that is, the current carried by free, as opposed to bound, charges. Let us assume that the medium in question obeys Ohm's law,  , and has a ``normal'' dielectric constant  . Here,  is the conductivity. Assuming an time dependence of all field quantities, the previous equation yields

(802)


Suppose, however, that we do not explicitly use Ohm's law but, instead, attribute all of the properties of the medium to the dielectric constant. In this case, the effective dielectric constant of the medium is equivalent to the term in round brackets on the right-hand side of the previous equation: that is,

(803)


A comparison of this term with Equation (801) yields the following expression for the conductivity,

(804)