Solving Physics Problems: Homework Statement and Solutions

[unstoppable]

Homework Statement

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.

Homework Equations

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
E(z)=1/2*Eo(x+jy)*e^-j(ko+K)z+1/2*Eo(x-jy)*e^-j(ko-K)z

Poynting Vector: The Vector E X H is the Poynting Vector. It gives the power per unit area that flows at a point;


Loss Tangent:Theta/(omega*epsilon)

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.

 

[borgwal]

You'd better split your question up.

in any case, for the first two questions, 1a and 1b, you need omega as a function of k...

 

[unstoppable]

You'd better split your question up.

in any case, for the first two questions, 1a and 1b, you need omega as a function of k...

Thank you but how do I set that up? Do you mean omega(k)=omega/k=c*omega/(squarerootof(omega^2-omega^2)) like this?
Note: the second omega^2 is the cuttoff frequency.

Also what do you mean I should split up my question?

Thanks

 

[borgwal]

Thank you but how do I set that up? Do you mean omega(k)=omega/k=c*omega/(squarerootof(omega^2-omega^2)) like this?
Note: the second omega^2 is the cuttoff frequency.

None of the equations you write here make sense to me. What I mean is, you are given a relation between omega and k. Transform that relation into a relation of the form

omega=f(k)

where f is some simple function of k.

Also what do you mean I should split up my question?

Thanks

Since your post contains many questions, maybe you could write several posts containing only a few questions.

 

[unstoppable]

None of the equations you write here make sense to me. What I mean is, you are given a relation between omega and k. Transform that relation into a relation of the form

omega=f(k)

where f is some simple function of k.




Since your post contains many questions, maybe you could write several posts containing only a few questions.

Okay I understand what you mean now, however, the question asks that I should write the phase velocity as a function of omega. How do I write that if I you suggest I write omega=f(k)? I would think my equation has to look something like this...Vp(omega)=... correct?

 

[borgwal]

First, how are the phase and group velocities defined?

Second, what are those velocities as functions of k?

Third, you have a relation between omega and k: you can use it to write omega as a function of k, but also to write k as a function of omega.

 

[unstoppable]

First, how are the phase and group velocities defined?

Second, what are those velocities as functions of k?

Third, you have a relation between omega and k: you can use it to write omega as a function of k, but also to write k as a function of omega.

My book says, The function relation between frequency omega and phase constant k in a medium is the dispersion relation. If omega and k are porportional, as they are in a vacuum, then domega/dk=omega/k ( in the derivative of omega/k is domega/dk) are exactly the same. In an example the book says the vacuum relationship between frequency and phase constant, omega^2=k^2c^2 is found to be modified by a constant shift, say omega^2(cutoff frequency) so that the dispersion relation turns out to be omega^2=omega^2(cutoff frequency)+k^2*c^2.
The phase velocity at frequency omega is Vp=Vp(omega)=omega/k=c*omega/(squareroot(omega^2-omega^2) NoteL: the second omega^2 is the cutoff frequency.

To obtain group velocity, we need the derivative of domega/dk at any frequency. Differentiating the complete dispersion relation with respect to k, recalling that omega, solving the derivative omega^2=omega^2(cutoff frequency)+k^2*c^2 yields
2*omega*(domega/dk)=2*kc^2 so that domega/dk=kc^2 /omega or
Vg=Vg(omega)=c*squareroot(omega^2-omega^2(cutoff frequency)/(omega)


I hope this will help you in understanding what is being asked in order for you to help me. I appreciate your assistance borgwal.

 

[borgwal]

You have to figure out which statements from your book are general definitions of phase and group velocities, and which statements apply to specific cases of specific dispersion relations.

I'll help you with one: the phase velocity is always given by omega/k. What would that be in your case?

 

[unstoppable]

You have to figure out which statements from your book are general definitions of phase and group velocities, and which statements apply to specific cases of specific dispersion relations.

I'll help you with one: the phase velocity is always given by omega/k. What would that be in your case?

The general formula I have in the book for phase velocity is here Vp=Vp(omega)=omega/k=c*omega/squareroot(omega^2-omega^2)
Note: The second omega^2 is the cutoff frequency.

if the phase velocity is always given by omega/k, then I believe in my case the phase velocity is given by omega^3/k^2=L^2/T^3? or is it rather just omega^3/k^2? I want to say it's omega^3/k^2. This is the part that confuses me too.

 

[borgwal]

You seem to have problems at a much more basic level. Go back and learn how to read equations, what they mean, and how to manipulate them in the correct way.

 

[unstoppable]

You seem to have problems at a much more basic level. Go back and learn how to read equations, what they mean, and how to manipulate them in the correct way.

I believe what I was saying is correct. Vp=omega/k. In which my case I would solve for k^2 and then with algebra and substitution it will give me my omega/k. Isn't that what you were getting at?

The little bit of help you gave me, helped me gain some better understanding.
Thanks

Have you gotten the chance to take a look at any of the other questions I posted?

 

[gabbagabbahey]

if the phase velocity is always given by omega/k, then I believe in my case the phase velocity is given by omega^3/k^2=L^2/T^3? or is it rather just omega^3/k^2? I want to say it's omega^3/k^2. This is the part that confuses me too.

ummmm...if the phase velocity is always given by omega/k, then the phase velocity IS omega/k!

That means you have $v_p(\omega,k)=\frac{\omega}{k}$ ; that is to say that your phase velocity is a function of both omega and k.

You are also given the dispersion relationship which relates omega and k. You can use the dispersion relationship to solve for k as a function of omega (and the constants T and L) and then substitute that into the above expression for phase velocity, hence eliminating k from the expression and giving you a phase velocity which is a function of only omega (and the constants T and L).

Do you follow this?

 

[unstoppable]

ummmm...if the phase velocity is always given by omega/k, then the phase velocity IS omega/k!

That means you have $v_p(\omega,k)=\frac{\omega}{k}$ ; that is to say that your phase velocity is a function of both omega and k.

You are also given the dispersion relationship which relates omega and k. You can use the dispersion relationship to solve for k as a function of omega (and the constants T and L) and then substitute that into the above expression for phase velocity, hence eliminating k from the expression and giving you a phase velocity which is a function of only omega (and the constants T and L).

Do you follow this?

Yes I do! Thank you! It makes sense. In the book it says Vp=omega/k and then it says Vp=Vp(omega)=c*omega/(omega^2-omega^2)

The second omega^2 is the cutoff frequency. In the problem is says the dispersion relation is omega^3*T^3=k^2*L^2 where T and L are constants. So I believe in the same manner, I will solve for k and do as you said. Thanks for the guidance! By the way, I have three other questions that I have a problem with, it will be great if I could get some direction on those as well. Here they are below:

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(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.






2. Homework Equations

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
E(z)=1/2*Eo(x+jy)*e^-j(ko+K)z+1/2*Eo(x-jy)*e^-j(ko-K)z

Poynting Vector: The Vector E X H is the Poynting Vector. It gives the power per unit area that flows at a point;


Loss Tangent:Theta/(omega*epsilon)