Generating Circular Polarization II

[jeff1evesque]

Statement:
Consider two dipole antennas, oriented 90degrees apart [imagine the x-y plane, let "a" be the dipole oriented along the x-axis, and the "b" be the dipole oriented along the y-axis]. If "a" dipole radiates [tex]cos(\omega t)[/tex] and "b" dipole radiates [tex]sin(\omega t)[/tex], the field radiated by the two antennas will be circularly polarized:

[tex]\vec{E}(z, t) = E_{0}[cos(\omega t - \beta z)\hat{x} + sin(\omega t - \beta z)\hat{y}][/tex] (#1)

Relevant Question:
Ok I almost get it now. Now in terms of a specific distance, say in the [tex]\hat{x}[/tex] direction, the cosine function has traveled a distance [tex]\omega t[/tex] (as did the sine function in it's respective axis). But I don't understand why To find electric field at a given location in the [tex]\hat{z}[/tex] direction, we subtract the distance traveled [tex]\omega t[/tex] by the wave number times distance in z, or [tex]\beta z[/tex] - for each component [tex]\hat{x}, \hat{y}[/tex]. The wave number is the wavelength of the sinusoid per unit distance. What happens when we take this wave number [tex]\beta[/tex] and multiply it by [tex]z[/tex]? What does that represent, I cannot see the relation between the two ([tex]\omega t[/tex] and [tex]\beta z[/tex])?

Does one unit length of [tex]z = 1[/tex] for [tex]\beta z \hat{x}[/tex] and [tex]\beta z \hat{y}[/tex] correspond to a length of [tex]\frac{2\pi}{\lambda}[/tex] in the [tex]\hat{z}[/tex] direction?

 

[Redbelly98]

I think we should talk about the simpler case of linear polarization, since your question is with the concept of traveling waves rather than about circular polarization.

So ... let's represent an electromagnetic wave by

E_x(z,t) = E₀ cos(ωt - βz)
​

and the electric field happens to be polarized in the x-direction.

Now consider points in space such that

ωt - βz = constant ≡ φ​

where we are calling this constant "φ".

We can say 2 things about what this represents:

• it represents points moving along the z-direction at constant velocity, which we can see by solving the above equation for z:
  
  z = (ω/β)t - (φ/β)​
• it represents a fixed electric field strength of E₀ cosφ

Combining those two facts, it means that the entire electric field pattern moves along the z-direction at a speed (ω/β). And that's just what a traveling wave is.

 

[Mene]

The wave number, represented by \beta, is defined as the number of cycles per unit distance, or the number of wavelengths per unit distance. Multiplying \beta by z gives us the total number of wavelengths that have passed through the given distance z. This is why we subtract \beta z from \omega t in the equation for the electric field - it accounts for the phase difference between the two dipole antennas at the given location in the z direction.

To answer your question about the relation between \omega t and \beta z, it is important to understand the concept of phase. In this context, phase refers to the position of the wave at a given point in time. As the wave travels, it goes through a full cycle of its sinusoidal motion, and at the end of each cycle, the wave has moved a distance of one wavelength. So, at any given time t, the wave has moved a distance of \omega t in the x direction and \beta z in the z direction.

In terms of your second question, the relationship between one unit length of z and a length of \frac{2\pi}{\lambda} in the z direction is correct. This is because \frac{2\pi}{\lambda} represents one full cycle of the wave, and as mentioned before, one full cycle corresponds to one wavelength. So, for each unit length of z, the wave has completed one full cycle and moved a distance of one wavelength.

Overall, the combination of the wave number \beta and the distance z allows us to accurately describe the phase of the wave at a given location in the z direction, which is crucial in understanding the circular polarization of the electric field in this scenario.