Chapter 23, Problem 49CP

Review. A slab of insulating material (infinite in the y and z directions) has a thickness d and a uniform positive charge density ρ. An edge view of the slab is shown in Figure P23.45. (a) Show that the magnitude of the electric field a distance x from its center and inside the slab is E = ρx/ϵ₀. (b) What If? Suppose an electron of charge −e and mass m_e can move freely within the slab. It is released from rest at a distance x from the center. Show that the electron exhibits simple harmonic motion with a frequency

$f=\frac{1}{2\pi }\sqrt{\frac{{\rho }^e}{m_e{ϵ}_0}}$

Figure P23.45