Answer Happy

Problem 8.1. Surface charge and force induced by a point dipole on a plane A point dipole p is oriented normal to and at a distance d above a horizontal grounded conducting plane that is infinite in extent. a) Using the method of images, calculate the electric field everywhere in the region above the plane. (Of course, this requires a good picture.) b) Find the surface-charge density induced on the conducting plane by the dipole. c) Using Mathematica, make a three-dimensional plot using Plot3D of the surface-charge density induced on the conducting plane by the dipole. In order to have dimensionless variables, express the surface-charge density in multiples and distances in multiples of d. cuss the reason for the minimum 2πd3 that appears in your plot. d) Show that the force on a point dipole placed in an external electric field E. is of p ext F = Ēext (8.1-1) Note that you will have to show explicity, in terms of Cartesian components, what is meant by V Eext, the gradient of a vector! e) Calculate the force exerted on the plane by the dipole by using the result (d) and the expression for the electric field that you found in part (a). Hint: Note that this is a point dipole, and must not to be turned into a finite- sized dipole (somehow constructed of two point charges separated by some nonzero distance).