A non conducting square plate 75 cm on a side carries a uniform surface charge density. The electric field strength 1 cm from the plate, not near an edge, is 45 kN/C. Whats the approximate field strength 15 m from the plate?

 E = 5.03 10⁻³ N / C

Explanation:

For this exercise we can use Gauss's law

         Ф = E. dA = /ε₀

Where as a Gaussian surface we select a cylinder

         Ф= E A = q_{int} /ε₀

Let's use the concept of surface charge density

         σ = q / A

         q = σ A

         E A = σ A /ε₀  

         E = sig /ε₀

Since the plate is non-conductive, the fux does not flow through the plate

With this equation the electric field does not depend on the distance, so the field at 1 cm

              E = 45 10³ N/C

             σ = E ε₀  

             σ = 45 10³  8.85 10⁻¹²

             σ = 3.98 10⁻¹⁰ C / m²

As the point where the field is requested is very far from the plate we can not ignore the finite size of it

At r = 15 m the plate has a very small size, so let's calculate the field as created by a point load at this distance

           E = k q / r²

The plate area is

            A = L L

            A = 0.75²

            A = 0.5625 m²

            q = σ A

            q = 3.98 10⁻¹⁰ 0.5625

            q = 1.26 10⁻¹⁰ C

Let's calculate

            E = 8.99 10⁹  1.26 10⁻¹⁰ / 15²

            E = 5.03 10⁻³ N / C

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