A point charge is a distance from a square surface of side d and is directly above the centre of the square as shown in the figure. We have to find the electric flux through the square.

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We will calculate the flux through the square surface in two ways.

(a)

We will first consider the square surface as one of the six faces of a cube of edge d. Flux through a Gaussian surface of a cube containing charge will be

and therefore the flux through one of the faces, as cube has six faces symmetric with respect to its centre, will be

(b)

Our second answer is a direct calculation of electric field flux at the surface of the square. We will first calculate the electric field vector at point (x, y) on the plane.

Let us consider an infinitesimal element of surface of area at the point (x, y). The magnitude of the electric field at that point due to a point charge will be

,

And the cosine of the angle between the outward normal to the surface and the direction of the electric field at the point (x, y) is

.

Therefore,

.

And,

On performing integration by appropriate substitution, we find

And

On performing integration by appropriate substitution, we find

And

.