A uniformly charged conducting plate with area a has a total charge Q which is positive. The figure below shows a cross-sectional view of the plane and the electric field lines due to the charge on the plane. The figure is not drawn to scale. Find the magnitude of the field at point P, which is a distance a from the plate. Assume that a is very small when compared to the dimensions of the plate, such that edge effects can be ignored. Two uniformly charged conducting plates are parallel to each other. They each have area A. Plate # 1 has a positive charge Q while plate #2 has a charge - 3 Q. Using the superposition principle find the magnitude of the electric field at a point P in the gap.

a)  E = σ / 2 ε₀ =   Q / 2A ε₀, b)  E = 2Q/A ε₀

Explanation:

For this exercise we can use Gauss's Law

        Ф = E. dA = / ε₀

Let us define a Gaussian surface as a cylinder with the base parallel to the plane. In this case, the walls of the cylinder and the charged plate have 90 degrees whereby the scalar product is zero, the normal vector at the base of the cylinder and the plate has zero degrees whereby the product is reduced to the algebraic product

              Φ = E dA = q_{int} / ε₀       (1)

As they indicate that the plate has an area A, we can use the concept of surface charge density

             σ = Q / A

             Q = σ A

The flow is to both sides of loaded plate

            Φ = 2 E A

Let's replace in equation 1

             2E A = σA / ε₀  

             E = σ / 2 ε₀ =   Q / 2A ε₀

This is in the field at point P.

b) Now we have two plates each with a load Q and 3Q respectively and they ask for the field between them

The electric field is a vector quantity

         E = E₁ + E₂

In the gap between the plates the two fields point in the same direction whereby they add

         σ₁ = Q / A

         E₁ = σ₁ / 2 ε₀

For the plate 2

         σ₂ = -3Q / A = -3 σ₁

         E₂ = σ₂ / 2 ε₀  

         E₂ = -3 σ₁ / 2 ε₀

The total field is

         E = σ₁ / 2 ε₀  + 3 σ₁ / 2 ε₀  

        E = σ₁ / 2 ε₀  (1+ 3)

        E = 2 σ₁ / ε₀

        E = 2Q/A ε₀