Field Exercise Solution created by a hemisphere surface charge


1. O electrostatic field.
The z axis is the axis of symmetry of load distribution. Field O is carried by this axis.
Consider a surface element of the half sphere centered at a point P. Elemental electrostatic field created by this element to the point O of the axis Oz is expressed as:


Only the projection of this vector onto axis Oz contributes to the field at the point O:

The field component O is:


2. Electrostatic Field in M.
The z axis is the axis of symmetry of load distribution. The field M is carried by this axis.
Consider a surface element of the half sphere centered at a point P.


Elemental electrostatic field created by this element at the point M of the axis Oz is expressed as:

Only the projection of this vector onto axis Oz contributes to the field at point M:


Gold :
and
where:

By integration:

Gold :
Either:
Let:
For :


The limit is calculated


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