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

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?

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

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  • $\begingroup$ 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
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    $\begingroup$ 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