Solution : Potential field created by a hemisphere charged on surface - electrostatic potential - Corrected exercises electromagnetism

1. Research of the potential.
  Consider a surface element of the half sphere centered at a point P. The electrostatic potential created at a point M of the axis Oz is expressed as:
By integration over the azimuthal angle and expressing r:
  and :
Can then be expressed:
The potential M is then:
2.Electrostatic field at M
  We have :

3. Potential and field on O.

The potential is written in the form:
Calculating the boundary of V where z tends to 0:
For the electric field, the procedure is the same way: