Question

A slab of insulating material (infinite in the y and z-directions) has a thickness d and a uniform positive charge density p. An edge view of the slab is shown in the figure below. (Submit a file with a maximum size of 1 MB.)

 (a) show that the magnitude of the electric field a distance x from its center and inside the slab is  

(b) Suppose an electron of charge -e and mass me 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 given by the following expression. 

Answer 1

As slab is of infinite size along y and z directions, is of uniform thickness, d and charged uniformly, electric field due to slab must be parallel to x-axis and symmetric about the center of the slab, that is the electric field at a distance x from the center on both sides is equal and opposite.

We apply gauss law to find field due to slab. Consider a cylindrical Gaussian, with an axis parallel to the x-axis and symmetric about the center of the slab. The electric flux through the curved surface of the cylinder is zero; the field is parallel to the curved surface. The electric flux through circular end surfaces is equal.