OK, let's begin:
Firstly, let's denote the electric field at a point P due to a point charge Q located at a distance z from P. Using Coulomb's Law, the formulation for the electric field E at a distance z from point charge Q is E = k*Q/(z^2), where k is Coulomb's constant.
Instead of directly comparing it with the field at some other location, it may be easier to first obtain the average electric field over a spherical surface of radius R surrounding P. Gauss's law states that the flux out of any closed surface surrounding a point charge is equal to the charge enclosed divided by epsilon_0 (where epsilon_0 is the permittivity of free space). The average electric field is this flux divided by the surface area (4*pi*R^2), resulting in E_s = Q / (4 * pi * R^2).
Now, let's analyze both the requested parts:
Part 1: z > R
The point charge Q resides outside the sphere. When you substitute z for R in the equation for the surface average electric field, you have what is virtually the electric field at z. Making this substitution, we get E_part1 = Q / (4 * pi * z^2). As you may observe, this is identical to our original equation for E_p, demonstrating that when z > R, the electric field at P is indeed the same as the average surface field.
Part 2: z < R
This time the point charge sits within the spherical surface. Whilst the total charge enclosed by the surface remains Q, the average electric field over the sphere remains constant, but the distance from the charge (z) is less than the radius of the sphere (R). In this case, we clearly see that E_p, the electric field at point P, will be greater than E_s, the average field over the surface. Here, if z != R, E_s does not equal E_p, indicating that the electric field at P is not the same as the average electric field over the spherical surface.
Thus, in conclusion, the electric field at point P is only the same as the average field over a spherical surface when that field is located outside the sphere (i.e., when z > R).