### Solution : Field in the vicinity of the axis of a uniformly charged hoop - Corrected Exercises Gauss Theorem

1. electrostatic field at a point on the axis of the ring.
The planes containing the axis Oz are symmetry planes of load distribution  ; at a point M of this axis, the direction of the electrostatic field must belong to each plan so they intersect:

with

So:
where:

As the plane containing the hoop is also a load distribution of the symmetry plane was at a point M' is symmetrical of M from the hoop:

with:

Finally, we get to any point on the axis of the hoop:

2. Field in a spot close to the axis.
We work in cylindrical coordinates.
For any point M of space, the plane containing the point M and the axis Oz is a plane of symmetry of the charge distribution. The field is contained in the plan. Furthermore as there invariance of the charge distribution by rotation about axis Oz we can write the electrostatic field in the form:

To determine the field, using the Gauss theorem and is selected as a closed surface generatrix of the cylinder axis Oz of length dz and radius r.

Inside the Gauss surface no loads where:

Considered as the point M is very close to the axis is made ​​the following approximation:
with
for

Moreover dz as the elementary length of the cylinder , we can consider that:
It develops the double integral:

After simplifications and noticing the presence of the differential function
we get :

For a point M of the axis can now write the electrostatic field in the form: