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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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In Section 3.1.4, I proved that the electrostatic potential at any point
in a charge-free region is equal to its average value over any spherical surface
(radius R )centered at .Here's an alternative argument that does not rely on Coulomb's law, only on Laplace's equation. We might as well set the origin at P .Let be the average; first show that
(note that the in da cancels the out front, so the only dependence on R
is in itself). Now use the divergence theorem, and conclude that if Vsatisfies
Laplace's equation, then,.
Two long, straight copper pipes, each of radius R, are held a distance
2d apart. One is at potential , the other at (Fig. 3.16). Find the potential
everywhere. [Hint: Exploit the result of Prob. 2.52.]
Show that the average field inside a sphere of radius R, due to all the charge within the sphere, is
Where is the total dipole moment. There are several ways to prove this delightfully simple result. Here's one method:
(a) Show that the average field due to a single chargeqat point r inside thesphere is the same as the field at r due to a uniformly charged sphere with
, namely
Where r is the vector from r to
(b) The latter can be found from Gauss's law (see Prob. 2.12). Express the answerin terms of the dipole moment of q.
(c) Use the superposition principle to generalize to an arbitrary charge distribution.
(d) While you're at it, show that the average field over the volume of a sphere, dueto all the charges outside, is the same as the field they produce at the center.
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).
The potential at the surface of a sphere (radius ) is given by
,
Where is a constant. Find the potential inside and outside the sphere, as well as the surface charge density on the sphere. (Assume there's no charge inside or outside the sphere.)
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