I just studied the concept of 'complex permittivity' in time-harmonic field. One of the Maxwell's equations expressed in phasor-domain,
$$\nabla\times\vec H = \vec J + j\omega \epsilon\vec E$$
($\epsilon$ & $\omega$ stand for permittivity & angular freqeuncy),
can be expressed with complex permittivity like following.
$$\nabla\times\vec H = \sigma\vec E + j\omega \epsilon\vec E = (\sigma + j\omega \epsilon)\vec E =j \omega(\epsilon +\sigma/j\omega) \vec E = j\omega\epsilon'\vec E$$
($\sigma$ stands for conductivity)
Then, taking the divergence of both side,
$$\nabla\cdot(\nabla\times \vec H) = 0 = j\omega\epsilon' \nabla\cdot\vec E$$
So, I got the conclusion $\nabla\cdot\vec E = 0$ which means the charge density is zero.
This sounds "Time-Harmonic electromagnetic fields cannot occur in charged matter".
Is this true? or Which point do I miss?