Problem: A very large plate in the y-z plane is at a potential of 600 volts and has a surface charge of 30 nanocoulombs/m². A charged solid sphere of radius 5 cm is centered at x = 3 m (3,0,0); it has a volumetric charge density of 70 nanocoulombs/m³. Find the potential at points along the x-axis between x = 0 and x = 2.95 m.

where ds = dx. The plate produces a field which is uniform (independent of x); the field from the sphere varies with x. To find expressions for the fields, we consider the Gaussian surfaces shown and try to find the field at point P, a coordinate x from the origin.

Here, r is the radius of the spherical Gaussian surface (partially shown) which is being used about the sphere of radius a. x is the coordinate to the position P on the x axis where we wish to find the net field. The sphere's center is 3 meters from the plate and therefore the origin, and so r = 3-x. Because the field is radial, it is perpendicular to the surface at P.

As for the plate, we also take a cylindrical Gaussian surface enclosing an area A of the plate. Knowing that the electric field is uniform means that we need to consider only the circular areas at the two ends of the cylinder (the one on the other side of the sheet of charge is not shown - it is hidden).

Considering the plate, and remembering that E_p comes out of both areas A of the Gaussian surface (hence "2A" in the formula) but the sheet itself contains only "A", we have

Substituting into the basic equation above, and recognizing that the potential of the plate is being maintained at 1 kV, for r >a