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## Continuous distribution of charge, , per unit volume.

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)

1. 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, 2. 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)

3. 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