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If there is no time dependence, so that
,
then we have from (3.2) that
. The
general solution to this equation states that the electric field can be
expressed as the gradient of a scalar function, the electrostatic
potential, as
Hence, (3.4)
becomes
 |
(3.9) |
where
is the charge density.
(3.9) is Poisson's equation. However, when there are no charges
so that
, this reduces to Laplace's equation
 |
(3.10) |
- We will simply state the general solution to (3.9).
Firstly, remember that the potential for a point charge is
If there are more charges we simply add the potentials together.
Thus, it is not too surprising that when there is a continuous
distribution instead of adding we integrate over the volume
containing the charge. Thus,
 |
(3.11) |
where
and
. Thus,
- The flux of
across a closed surface containing charges is
by the divergence theorem, and using (3.4) we get
Hence, we have that the flux of the electric field across the surface is
 |
(3.12) |
- If
and
, (3.9) becomes in
spherical coordinates,
Integrating once we obtain
where
is a constant. Integrating a second time gives the
electrostatic potential as
and the radial electric field component is
 |
(3.13) |
Thus, there is a possible singularity at
and (3.10) fails
at this point. If at
there is a point charge of strength
,
what is the value of the contant
? We can use (3.12) and
draw a small sphere of radius
about the point charge. Thus,
or substituting for
from (3.13) we get
Thus,
in agreement with (3.6).
Next: Electric Dipoles
Up: Electrostatics
Previous: Point Charges
Prof. Alan Hood
2000-11-06