Answer to Question #114050 in Electricity and Magnetism for Ali

Question

We have an infinite, non conducting, sheet of negligible thicknress carrying a negative uniform surface charge density  and, next to it, an infinite parallel slab of thickness D with positive uniform volume charge density  (see sketch). All charges are fixed. Calculate the direction and the magnitude of the electric field.
(a) Above the negatively charged sheet.
(b) In the slab
(c) Below the slab
(d) Make a plot of E as a function of distance, z, from the sheet.

Answer 1

(a) The direction of the electric field at distance h above the negatively charged sheet: the field from the sheet looks downward, the field from the slab looks upward if the negative sheet is on top of the slab.

$$\vec{E}=\bigg(\frac{-\sigma}{2\epsilon_0}+\frac{\rho D}{2\epsilon_0}\bigg)\hat{y}.$$

(b) The field in the slab consists of the field from the negative sheet (looks down) and from the slab (looks upward at distance d from the sheet).

$$\vec{E}=\bigg(\frac{\sigma}{2\epsilon_0}+\frac{\rho (D-d)}{2\epsilon_0}-\frac{\rho d}{2\epsilon_0}\bigg)\hat{y}.$$

The field below the slab: the field from the sheet look upward, the field from the slab looks down.

$$\vec{E}=\bigg(\frac{\sigma}{2\epsilon_0}-\frac{\rho D}{2\epsilon_0}\bigg)\hat{y}.$$

(d) Without numerical values, the plot can be represented like this: