An infinitely large layer of charge of uniform thickness t is placed normal to an existing uniform electric field. The charge on the sheet alters the electric field so that it still remains uniform on both the sides of the sheet and assumes values `E_1 and E_2` as shown in figure. The charge distribution in the layer is not uniform and depends only on the distance from its faces. Find the expressions for the force F per unit experienced by the charge layer.

Explanation

Let E be the external electric field and `E_(layer)` by the electric
field due to the layer. On the left side,
`E-E_(layer) = E_1`………(i)
On the right side,
`E + E_(layer) = E_2`………….(ii)
From Eqs. (i) and (ii),
`2E = E_1+E_2`
Using Gauss's theorem,
`-E_1A + E_2A = Q/epsilon_0.`
Q is the charge inside the Gaussian surface
or `E_2-E_1 = Q/(Aepsilon_0)` .........(iv)
So the force experienced by the charge layer per unit area is
`F/A = (Q/A)E`.........(v)
From Eqs. (iii) and (iv), substituting the values of E and `Q//A`
we get
`F/A = epsilon_0 (E_2-E_1) (E_2+E_1)/2 = epsilon_0/2(E_(2)^(2) - E_(1)^(2)` .