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Chapter 3: Q3.36P (page 160)
Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form
Answer
The given relation is proved.
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Two point charges, 3q and -q, are separated by a distance a. For each of the arrangements in Fig. 3.35, find (i) the monopole moment, (ii) the dipole moment, and (iii) the approximate potential (in spherical coordinates) at large r (include both the monopole and dipole contributions).
Solve Laplace's equation by separation of variables in cylindrical coordinates, assuming there is no dependence on z (cylindrical symmetry). [Make
sure you find all solutions to the radial equation; in particular, your result must accommodate the case of an infinite line charge, for which (of course) we already know the answer.]
A rectangular pipe, running parallel to the z-axis (from to ), has three grounded metal sides, at and The fourth side, at , is maintained at a specified potential .
(a) Develop a general formula for the potential inside the pipe.
(b) Find the potential explicitly, for the case (a constant).
a point charge located inside (same as above, in other words, only with ).(In this case, of course, Laplace's equation does not hold within the sphere.) Show that, in general,
role="math" localid="1657706668993"
where is the potential at the center due to all the external charges, and is the total enclosed charge.
Find the potential in the infinite slot of Ex. 3.3 if the boundary at x = 0 consists of two metal strips: one, from y = 0 to y = a/2, is held at a constant Potential , and the other, from y = a/2 to y = a , is at potential .
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