Hi I'm having trouble with these problems. I'll greatly appreciate it if someomne could assist me.
1) A certain wave-propagating system has a dispersion relation that can be expressed by omega^3*T^3=k^2*L^2 where L and T are given constants.
Note: omega is a greek letter and it stands for the frequency.
(a) Give the phase velocity of the wave as a function of frequency omega.
(b) Give the group velocity of the wave as a function of frequency omega.
(c) What is the product of the phase and group velocities, at frequency omega?
(d) What is the ratio of the group velocity to phase velocity at frequency omega?
(e) Basoned Relativity Theory, what is the lowest frequency omega at which this system can possibly support wave propagation?
2) A researcher claims that she can convert a circularly polarized plane wave in air into a linearly polarized one by simply reflecting it from a suitably chosen lossless nonmagnetic dieletric. Assume the dielectric she will use has refractive index n. Is this feat possible for normail incidence, as she claims? Calculate the reflected wave to justify your answer. If she is right, is there a mininum value of n for this to work? If she is wrong; briefly explain why, based on your calculation.
3) A source of unidirection plane waves operates within a medium with moderate conductivity sigma. Suppose we measure the complex electric field amplitudes at the source and at some distance z and find that E(z)/E(0)-0.3-j0.4.
(a) Calculate the loss tangent sigma/(omega*epsilon) of the medium. (Give a numerical value)
(b) What is the ratio of complex magnetic field amplitudes H(z)/H(0) for the same z?
4) A perfect planar mirror in the xy-plane has normally incident and reflected electromagnetic plane waves in the vacuum region z<0 in front of it, at frequency omega. The magnetic field at the mirror surface is circularly polarized:H=H(x + jy)
(x and y are the unit vectors along x and y).
(a) Find the complex electric field amplitude E(z) in the space z<0.
(b) Find the complex Poynting vector(give magnitude and direction) in the space z<0.
equation for phase velocity: Vp=Vp(omega)=omega/k= c*omega/squarerootof(omega^2-omega^2) Note: The second omega^2 is the cuttoff frequency
equation for group velocity: Vg=Vg(omega)=c*squarerootof(omega^2-omega^2)/(omega)
Note: The second omega^2 is the cuttoff frequency.
c=the speed of light 3 X 10^8 m/s
Circular Polarization: E(0)=Eox=1/2Eo(x+jy) +1/2Eo=(x-jy)
Note: j=the imaginary complex number
Poynting Vector: The Vector E X H is the Poynting Vector. It gives the power per unit area that flows at a point;
The Attempt at a Solution
I started the first problem by dividing both sides to get omega^3/k^2=L^2/T^3 and then I got lost into how to apply into the formula for phase velocity. Please help.
The Attempt at a Solution