### Solution : Study of a spherical inhomogeneous distribution - Corrected Exercises Gauss Theorem

1. Expression of the electrostatic field.

All the planes containing the center O of the sphere and the point P are the load distribution of planes of symmetry. The electric field must belong simultaneously to all of these plans, it is therefore carried by their intersection which is the line OM.  we obtain.:
As this distribution has a rotational invariance around the point O, the electric field does not depend on the angular variables.

This result does not depend on the position of this point P.
We choose a Gaussian surface centered in O and radius r. The Gauss theorem is:

Then there are two regions of space:
● For r <R (one can put an equal sign here because the density distribution is what ensures the continuity of the normal component (here radial) of the electrostatic field).

Is obtained by applying the Gauss theorem:
The electric field is expressed in this region:
● For r> R, Inner charge on the surface Gauss is:

Is obtained by applying the Gauss theorem:

The electric field is expressed in this region:
2. Maximum field.
The field has an extreme value that we consider it maximum  when:

Where:

The coefficient k is expressed as:

In which r / R = 1/2 are obtained: