EM Field: Angle between H and x-axis; frequency

[tquiva]

Homework Statement

Hello all, could someone please assist me with the following problem?

Homework Equations

f = ω/2π
λ = (2π)/β

The Attempt at a Solution

I've already made an attempt to part (b), (c), & (d) of this problem. Below is my work.

• I'm having trouble with part (a). I drew the coordinate system but am not sure how to go about finding the angle between H and the x-axis. Am I supposed to use the H found in part (b) to answer part (a)?
  From part (b), I notice that H has a component on the x-axis. Does this mean that the angle between H and the x-axis is 0?

• For part (d), I know the equation in obtaining frequency is f = ω/2pi. However, there is no value of ω given in the electric field of this problem. It's just "ω" ... I went through my book, and am still a little lost. Is there a way to find ω?

Much help is needed, and I appreciate anyone taking the time to look at this. Many thanks in advance!

 

[rude man]

a) You know the angle between the E and H fields is π/2, so if the angle between the E field and the x-axis is arc tan (-2√3/2) = -60 deg., how can the angle between the H field and the x-axis be π/2?

I would first establish the direction of the wave along the z axis: is it going in the + or - direction? Because the H field direction will be such that E x H gives the + direction.

Look at sin(wt - kz). Fix t = 0 and draw it along the z axis so now you're looking at the wave along the z axis at different spots. Look at z = 0, what is the function? Then look at a small positive spot z = z₀, what is the function there? So is the function going in the +z or -z direction? Remember, t = 0 in both places.

You know the angle between the E and H fields is π/2 so H is 90 deg. away from E. And now that you know the direction of the wave you can determine the direction (remove the ambiguity) of H. As for magnitudes, you know that E/H = η and you're given E.

The rest is just realizing that k = 2π/λ etc.

 

[rude man]

[*]I'm having trouble with part (a). I drew the coordinate system but am not sure how to go about finding the angle between H and the x-axis. Am I supposed to use the H found in part (b) to answer part (a)?
You really need to do part c first. Otherwise, all you know is that H is orthogonal to E.


From part (b), I notice that H has a component on the x-axis. Does this mean that the angle between H and the x-axis is 0?
[/LIST]
No. E and H are orthogonal in a plane wave..

• For part (d), I know the equation in obtaining frequency is f = ω/2pi. However, there is no value of ω given in the electric field of this problem. It's just "ω" ... I went through my book, and am still a little lost. Is there a way to find ω?

But you know the wave number k = 100π and k = 2π/λ so you know λ and you know c = λf. Get the idea?

Much help is needed, and I appreciate anyone taking the time to look at this. Many thanks in advance!

See above.

 

[tquiva]

a) You know the angle between the E and H fields is π/2, so if the angle between the E field and the x-axis is arc tan (-2√3/2) = -60 deg., how can the angle between the H field and the x-axis be π/2?

I would first establish the direction of the wave along the z axis: is it going in the + or - direction? Because the H field direction will be such that E x H gives the + direction.

Look at sin(wt - kz). Fix t = 0 and draw it along the z axis so now you're looking at the wave along the z axis at different spots. Look at z = 0, what is the function? Then look at a small positive spot z = z₀, what is the function there? So is the function going in the +z or -z direction? Remember, t = 0 in both places.

You know the angle between the E and H fields is π/2 so H is 90 deg. away from E. And now that you know the direction of the wave you can determine the direction (remove the ambiguity) of H. As for magnitudes, you know that E/H = η and you're given E.

The rest is just realizing that k = 2π/λ etc.

Thank you so much for this detailed response. I'm sorry but I'm still a bit lost.

Look at sin(wt - kz). Fix t = 0 and draw it along the z axis so now you're looking at the wave along the z axis at different spots. Look at z = 0, what is the function? Then look at a small positive spot z = z₀, what is the function there? So is the function going in the +z or -z direction? Remember, t = 0 in both places.

What is small positive spot z = z₀? I'm still trying to figure out how to determine if the direction is +z or -z? How do I find this polarity?

Also, I notice that you said sin(wt - kz). Does β = 100π or does k = 100π? Or are these the same? I know β is the propagation direction and k is the wave number? Also, does the negative sign follow the value for β such that β = -100π?

If it's not too much trouble, will you please let me know if I got part (b) and (c) correct?

I have a feeling that the angle between H and the x-axis is 90 degrees?

 

[rude man]

[/B]

Thank you so much for this detailed response. I'm sorry but I'm still a bit lost.

What is small positive spot z = z₀? I'm still trying to figure out how to determine if the direction is +z or -z? How do I find this polarity?

OK, let's look at the wave sin(wt - βz). BTW yes, my k is your β.
Fix t = 0, then draw sin(-kz) along the z axis. It starts by going negative, crosses the z axis at z = π/k, goes positive until it crosses the z axis again at kz = 2π, then repeats.

Then on top of that draw sin(kz - kz₀₎ where z₀ is a small distance to the right of z=0. That's the same wave as seen at a distance z₀ ahead of z = 0. Now pick any point, say at z > z₀ on the axis. Which of the two waves looks like it's ahead of the other?

Another way: fix z = 0, then graph sin(wt - kz) = sin(wt) for two different times, say t = 0 and t = t₁ > 0. Same wave, looked at z = 0, at different times. Which of the two waves looks like it's ahead of the other?
So does sin(wt - kz) go in the + or - z direction?

Also, I notice that you said sin(wt - kz). Does β = 100π or does k = 100π? Or are these the same? I know β is the propagation direction and k is the wave number? Also, does the negative sign follow the value for β such that β = -100π?

k = β, I use k, you use β. k = 100π and is always positive.

If it's not too much trouble, will you please let me know if I got part (b) and (c) correct?

You got β right. If your teacher told you the sign of β gives the direction, that would give it to you. It would not give you insight into how the wave propagates. I prefer to always let β > 0 = 2π/λ. Once you get the direction right I'll tell you the sure-fire way of telling direction just by looking at sin(wt - kz).

Remember c = λf? That should be enough for you to derive w and f.

I have a feeling that the angle between H and the x-axis is 90 degrees?

EDIT: Oops, I misread.

No, wrong.
In a plane wave the E and H fields are perpendicular and E x H gives the direction of the wave, so once you know the direction you know if H is 90 deg. clockwise or counterclockwise to E.