Answer to Question #113495 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) Distance h above the negatively charged sheet: the field from the sheet looks downward, the field from the slab looks upward.

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

(b) In the slab at distance d below the negative sheet: the field from the negative sheet looks upward, the field from the slab at (D-d) looks upward, the field at d looks downward.

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

(c) The field at distance x below the slab:

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

(d) The plot how the field may depend on distance is below: