Finding Magnetic Field from Electric Field: A Classical Physics Problem

[OhNoYaDidn't]

Hey guys, i just came across this on my classical physics course.
So, I'm given that: $$E(z, t) = {E_{0}}sin(wt)sin(kz)\widehat{x}$$, and I'm supposed to find an expression for the associated magnetic field B.

Usually, i just find the propagation direction, and do it's cross product with the direction of E, and then write it as $$\overrightarrow{B}(r,t)=\frac{1}{c}\widehat{k}\times \overrightarrow{E}$$but in this case, it doesn't seem to be as straightforward.

I thought of using $$\triangledown \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t}$$, finding the curl of E, and then integrate with respect to time, this would give me:

$$B(z, t) = \frac{k}{w}{E_{0}}cos(kz)cos(wt)\widehat{y}$$

Is this right?
Thank you.

 

[Delta2]

Yes seems right to me. It is just that the equation that is given for E, represents a standing wave, so the propagation vector cannot be defined (standing waves do not propagate).