How the charge imbalance in a plasma changes with time

[ergospherical]

Homework Statement

There's a neutral plasma of protons and electrons of density $\rho$. If the number of electrons per unit volume fluctuates around its mean value by a small amount $\delta(\mathbf{x}, t)$, show that $\ddot{\delta} + (\rho e^2/\epsilon_0 m_e m_p) \delta = 0$.

Relevant Equations

N/A

I put the net charge density $\rho_q = e\delta$ so that $\nabla \cdot \mathbf{E} = e\delta / \epsilon_0$, then I tried Maxwell IV:\begin{align*}
\dot{\mathbf{E}} + c^2 \mu_0 \mathbf{J} &= 0 \overset{\mathrm{div}}{\implies} e\dot{\delta} + \nabla \cdot \mathbf{J} = 0
\end{align*}but this just gives you back the continuity eq. How to start?

 

[jasonRF]

This is often derived with the following logic: the charge deviation causes an electric field to be produced, and then the charges respond to that electric field.