$$\overrightarrow{E} = (-10 \hat x + 4 \hat z) e^{-j(2x+5z)}$$

I recently started to study electromagnetics, but I'm having a hard time following up.

May I ask how to calculate the wave vector $\overrightarrow{k}$ and the wave number, please?

Also, how can I calculate the magnetic field vector in the frequency domain?

I really appreciate your help guys.

ADDED: Thanks for the comments, guys. I noticed these answers. https://physics.stackexchange.com/a/189504/232791 https://physics.stackexchange.com/a/287991/232791

So it seems like the wave vector $\overrightarrow{k}$ is $$\overrightarrow{E} (r) = \overrightarrow{E_0} e^{j(\overrightarrow{k} \cdot \overrightarrow{r})}, \overrightarrow{k} = k_x \hat{x} + k_y \hat{y} + k_z \hat{z}, \overrightarrow{r} = x \hat{x} + y \hat{y} + z \hat{z}, \overrightarrow{k} = -2 \hat{x} + -5 \hat{z}$$ And the wave number, the vector size is $$k = \frac{w}{v_p}, \overrightarrow{k} = k_x \hat{x} + k_y \hat{y} + k_z \hat{z}, |\overrightarrow{k}| = \sqrt{k_x^2 + k_y^2 + k_z^2} = \sqrt{(2\pi/\lambda_x)^2 + (2\pi/\lambda_y)^2 + (2\pi/\lambda_z)^2} $$ equal to $\sqrt{k_x^2 + k_y^2 + k_z^2} = \sqrt{29}$, right?

Since $\overrightarrow{E_0} = E_{0_x} \hat{x} + E_{0_y} \hat{y} + E_{0_z} \hat{z} = (-10 \hat x + 4 \hat z)$ has no $y$ dependence, meaning that this must line in the $zx$ plane,

I thought $\overrightarrow{k}$ should be perpendicular to the $zx$ plane.

  • Have I correctly got both the wave vector and the wave number?
  • "Differentiating and integrating wrt time", do you mean I have to apply Faraday's Law $\overrightarrow{\nabla} \times \overrightarrow{E} + \frac{\partial \overrightarrow{B}}{\partial t} = 0$?

Or $\overrightarrow{\nabla} \cdot \overrightarrow{E} = 0, \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0$ in free space?

Just like your first comment, I don't see time, nor frequency as well. May I please ask for help?


closed as off-topic by ZeroTheHero, G. Smith, Qmechanic May 24 at 9:08

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ZeroTheHero, G. Smith, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I voted to closed because, as states, this is an obvious assignment problem. $\endgroup$ – ZeroTheHero May 23 at 23:12
  • $\begingroup$ I don't see time, nor frequency. What is this, a static field? Then, what wave vector? $\endgroup$ – acarturk May 24 at 1:02
  • $\begingroup$ @acarturk it’s the phasor form; classic notation from E&M. The time component is implicit in a time exponential. $\endgroup$ – ZeroTheHero May 24 at 1:10
  • $\begingroup$ The fact that I cannot see any frequencies around makes it difficult to believe so, but assuming existence of an $\omega$ you can simply say that the exponent in the time dependent $E$ must be $e^{i(\omega t - \mathbf{k}\cdot\mathbf{r})}$ for this to be a wave, or equivalently the exponent in the phasor domain must be $e^{-i\mathbf{k}\cdot\mathbf{r}}$ $\endgroup$ – acarturk May 24 at 1:58
  • 1
    $\begingroup$ Differentiating and integrating wrt time and spatial coordinates is easy in phasor domain, so make use of Maxwell's equations to find $B$ $\endgroup$ – acarturk May 24 at 2:00