Our starting point was the second Green identity for the potential and the function 1/r=modulus(x-x'). We showed how the most general solution of the Poisson Equation can be written as a volume integral over the charge density (rho(x')) times the Green Function G(x,x'), plus a surface integral that accounts for either Dirichlet (potential=0 on surface) or Neumann (normal derivative of potential=0 on surface) boundary conditions.
The fact that we can choose among the two types of boundary conditions is is a consequence of the definition of the further degree of freedom that we define, i.e. the function F(x,x'), solution of the Laplace (homogeneous) equation inside the volume V.
We centered most of our discussion on the meaning of Green's Function (which is used for solving a large number of problems in physics, like e.g. potential scattering and many more), by working out in parallel the problem of potential scattering, identifying the Green's function for the solution of the Helmholtz Equation .
We started out with problems that can be solved with the method of images. This method is a ``calculation technique'' that is useful when dealing with a pointlike charge or simple distributions obtained by superposition of pointlike charges such as lines of charge, in the presence of boundary conditions given by the value of the potential on: i) planes, ii) spheres.
We discussed the problems of a point charge and a grounded conducting plane of a point charge in the presence of a conducting spherical surface. We discussed thoroughly the physical quantities that get into play in the grounded case (potential, surface charge density) (Sec. 2.1, 2.2, 2.6 Jackson) and we passed on to the question of ``idenitfying'' the Green's function, charge density, boundaries etc. for this case (Sec. 2.6).
In particular we noted the difference with the problems of finding the field outside a uniformly charged infinite insulating and conducting planes, respectively. We will keep on discussing Scattering Theory in parallel with our electrostatic problems. Notes will be posted on this website.
1) Two point charges q1 and q2 are located at distance r1 and r2 from a (grounded) conducting plane. The charges lie on a line perpendicular to the plane. (i) What is the force on q2? (ii) What is the surface charge density at the intersection with the line?
2) Consider a rectangular charge distribution (the rectangle has sides 2a and 2b and it is placed at the center of the coordinates system). The surface charge density is sigma=k xy (k is a constant). Find the potential at the point P of coordinates (a,0).
3) Consider a spherical cavity of radius `a' inside a (grounded) conducting object. Find the force acting on a pointlike charge `q' placed inside the sphere at a distance `r' from the center. Is the force attractive or repulsive to the spherical surface? How does this compare with the case in which the charge is outside the sphere?
4) Plot with a computer graphics program of your choice the induced surface charge density produced by the pointlike charge on the plane, vs. both the distance of the charge and the distance from the axis defined by the perpendicular coordinate of the charge on the plane. Notice the differences from a constant distribution of surface charge.
