1. 3.1 Laplace’s Equation Common situation: Conductors in the system,
which are a at given potential V or
which carry a fixed amount of charge Q.
The surface charge distribution is not known.
We want to know the field in regions, where there
is no charge.
Reformulate the problem.

2. + Boundary conditions. (e.g. over a surface V=const.)
Important in various branches of physics:
gravitation,
magnetism,
heat transportation,
soap bubbles (surface tension) …
fluid dynamics

3. One dimension
Boundary conditions:

4. V has no local minima or maxima.

5. Two Dimensions Partial differential equation.
To determine the solution you must
fix V on the boundary – boundary condition.
V has no local minima or maxima
inside the boundary.
Rubber membrane
Soap film
A ball will roll to the boundary and out.

6. Three Dimensions Partial differential equation.
To determine the solution you must
fix V on the boundary, which is a surface,
– boundary condition.
V has no local minima or maxima inside the boundary.
Earnshaw’s Theorem:
A charged particle cannot be held in a stable equilibrium
by electrostatic forces alone.

7. First Uniqueness Theorem
The solution to Laplace’s equation
in some volume V is uniquely
determined if V is specified on
the boundary surface S.
The potential in a volume V is uniquely determined if
the charge density in the region, and
the values of the potential on all boundaries are specified.

8. Second Uniqueness Theorem
In a volume surrounded by conductors and containing a specified
charge density, the electrical field is uniquely determined if the
charge on each conductor is given.

10. Image Charges What is V above the plane? Boundary conditions:
There is only one solution.

11. Image charge
The region z<0 does not matter. There, V=0.

12. Induced surface charge:
Force exerted by
the image charge
Force on q:
Different
from W of
2 charges!!
Energy:

14. Example 3.2
Find the potential outside the conducting grounded sphere.