3.1 Laplace’s Equation

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.