The dielectric constant and the refractive index

The refractive index of a material, n, is defined as the ratio of the speed of light in a vacuum to the speed of light in that material.
Equation where c is the speed of light in a vacuum and cm the speed of light in the material.
It is possible to derive another equation for the speed of light, this time in terms of the electric permittivity (ε) and magnetic permeability (μ) of the material. For this, we need Maxwell’s equations.
      Equation        (1)
     Equation      (2)
Taking the curl of both sides of (1) allows us to combine (1) and (2):
Equation
In general for any vector a:
Equation
Now in a vacuum, Equation. In this case the above equation becomes:
Equation
Which is the wave equation in three dimensions. Let us consider the 1D equivalent, as this is easier to solve.
Equation
A possible solution to this equation is a sinusoidal wave of wavelength λ and speed c:
Equation
Differentiating with respect to x and t:
Equation
And substituting back into the 1D wave equation above:
Equation

Which can be simplified and rearranged to give an expression for c, the speed of light in a vacuum:
Equation
It turns out that a similar equation is applicable to the speed of light in any material, cm:
Equation
For a material that is not magnetic the permeability is μ0, so that:
Equation for any non-magnetic material.
Using the expressions for c and cm, the refractive index of the material can be expressed in terms of ε and μ.
Equation
Finally, recall the earlier definition of the dielectric constant in terms of permittivity:
Equation
Therefore κ = n2 .

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