### 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:

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