EM wave propagation direction.

[Lavabug]

Homework Statement

Given the following EM wave propagating in vacuum, find the direction along which the electric field oscillates and the direction of propagation of the wave:

[tex] \vec{E} = (-3\hat{i} + 3\sqrt{3}\hat{j}) 10^4 e^{i[\frac{\pi}{3} (\sqrt{5}x + \sqrt{5/3} y10^7 - 8.1246 *10^{15} t]} [/tex]

Btw how can I prevent the thread template from reappearing every time I click preview post? It is quite annoying.

The Attempt at a Solution

I understand the electric field oscillates along the direction [itex]-1\hat{i} \sqrt{3}\hat{j}[/itex], by looking at the amplitide. But how do I determine the direction of propagation from the phase argument? Do I just treat the x and y in the phase as if they were unit vectors of [itex]\vec{k}[/itex]?

 

[vela]

Pretty much. You know a wave propagating in the k direction has a phase given by [itex]i(\vec{k}\cdot\vec{r}-\omega t)[/itex], so just pick out the components of k.

 

[Lavabug]

Thanks, I see it now. One more question, if I am asked to find the associated [itex] \vec{H}[/itex] field by calculating [itex] \frac{1}{\mu_0 c} \hat{u}\times \vec{E} [/itex], where û is the normalized wave vector, how would I split up the components of E for the determinant? Would it be like:[tex] E_x = -3\cdot10^4 e^{i[\frac{\pi}{3} (\sqrt{5}x + \sqrt{5/3} y10^7 - 8.1246 *10^{15} t]}[/tex][tex]E_y = 3\sqrt{3}\cdot10^4 e^{i[\frac{\pi}{3} (\sqrt{5}x + \sqrt{5/3} y10^7 - 8.1246 *10^{15} t]} [/tex]

for example? Or do I also have to split up the phase/exponent for each component of E?

 

[vela]

You can keep the exponential part separate; it's part of the amplitude of the electric field. I would write the electric field as the product of a magnitude and a unit vector:
[tex]\vec{E} = \left(-\frac{1}{2}\hat{i}+\frac{\sqrt{3}}{2}\hat{j}\right)(3\times10^4~\mathrm{V/m})e^{i(\vec{k}\cdot\vec{r}-\omega t)}[/tex]The only part you need for the determinant is the unit vector. The rest just scales the result of the cross product.

 

[paranormal]

As a scientist, it is important to be precise and clear in our language and notation. In this case, the notation used for the electric field is not standard, so it is important to clarify what is meant by the vector components. Assuming that the electric field is given in Cartesian coordinates, the direction of oscillation would be along the vector (-3, 3√3) in the xy-plane. However, it is not clear what is meant by the phase argument in the exponential term. Is it a function of x, y, and t or is it a constant? This needs to be clarified in order to determine the direction of propagation. Additionally, it would be helpful to specify the units for x, y, and t in order to properly interpret the phase argument. Once these details are clarified, we can determine the direction of propagation by considering the direction of the wavevector, which is given by the gradient of the phase.