Is the Given Function a Valid Electromagnetic Plane Wave?

[HPRF]

Homework Statement

By considering divergence, show whether the expression

B=kB_ze^i(k_zz-wt)

is a valid function for an electromagnetic plane wave.

Homework Equations

divB=0

The Attempt at a Solution

I have found divB=ik_zB_ze^i(k_zz-wt).

Does this satisfy divB=0 because it is imaginary?

 

[gabbagabbahey]

The field you give is usually denoted as

$$\tilde{\textbf{B}}=\hat{\mathbf{k}}\tilde{B}_ze^{i(k_z z- \omega t)}$$

and is itself complex-valued... The actual magnetic field $\textbf{B}$ is taken to be the real part of [tex]\tilde{\textbf{B}}[/itex] and so

$$\mathbf{\nabla}\cdot\textbf{B}=\mathbf{\nabla}\cdot\text{Re}\left[\tilde{\textbf{B}}\right]=0$$

So, only if the divergence of the real part of the field you gave vanishes, can it be a valid magnetic field (The fact that the magnetic field is polarized in the same direction as the propagation should give you a little bit of hesitation here).