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- 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?

\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?