(a) Write down an expression for the phase velocity vp of an electromagnetic wave in a medium with permittivity e and permeability n. (b) The relative permittivity, er, in an ionized gas is given by er = 1-D^2/w^2 where D is a constant and w is the angular frequency. Find an expression for the refractive index n and thus show that w^2 = D^2 + c^2k^2 where k is the wavenumber and c is the speed of light in vacuum. (c) Hence show that vp vg = c^2. (d) In a particular gas, D has the value 1.2x10^11 /s. Determine the phase and group velocities at 20 GHz. Comment on the result.