### Solution : Studying a cylindrical charge distribution - Gauss theorem

1. Field Management.
The planes containing the point M, perpendicular and orthogonal to the axis of the load distribution are symmetry of this distribution plans: the electrostatic field must therefore have its direction carried by the intersection of these two planes. The field is carried by the radial vector of the cylindro-polar base.

2. Invariances.
The distribution is cylindrically symmetric, it is therefore invariance by translation parallel to the cylinder axis and rotating around this axis.

3. Vector electrostatic field.
Symmetries and invariances used to write the electrostatic field vector in the form:

It shall, in both cases, for Gauss surface a closed cylinder whose axis mingled with  the distribution ,of height h, and radius r.
As the field vector and the vector ds t so the radial component of the field is constant along the side surface of the cylinder can be written:

Outside distribution:
Inside the distribution:
4. Potential.
As the vector electrostatic field is opposite the gradient of the potential V and it depends only on r (the distribution being cylindrically symmetric) is obtained:
Is obtained by integrating the potential V. The integration constants are obtained by the zero potential condition at r = 0 and by the continuity of the potential at r = R.
Inside the distribution:
Outside distribution:
5. Field inside.
The symmetry and invariance properties are not changed. Considering a Gaussian surface identical to that described in 3 and taking into account the radial dependence of the distribution in the calculation of the inner charge to the Gauss surface are:

Where: