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Chapter 3: Q3.6P (page 124)
A more elegant proof of the second uniqueness theorem uses Green's
identity (Prob. 1.61c), with . Supply the details.
Answer
It is proved as second uniqueness of theorem by using greens identity.
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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).
Charge density
(whereais a constant) is glued over the surface of an infinite cylinder of radiusR
(Fig. 3.25). Find the potential inside and outside the cylinder. [Use your result from Prob. 3.24.]
(a) Show that the quadrupole term in the multipole expansion can be written as
(in the notation of Eq. 1.31) where
localid="1658485520347"
Here
is the Kronecker Deltalocalid="1658485013827" and is the quadrupole moment of the charge distribution. Notice the hierarchy
localid="1658485969560"
The monopole moment localid="1658485018381" is a scalar, the dipole moment localid="1658485022577" is a vector, the quadrupole moment localid="1658485026647" is a second rank tensor, and so on.
(b) Find all nine components of localid="1658485030553" for the configuration given in Fig. 3.30 (assume the square has side and lies in the localid="1658485034755" plane, centered at the origin).
(c) Show that the quadrupole moment is independent of origin if the monopole and
dipole moments both vanish. (This works all the way up the hierarchy-the
lowest nonzero multipole moment is always independent of origin.)
(d) How would you define the octopole moment? Express the octopole term in the multipole expansion in terms of the octopole moment.
A charge is distributed uniformly along the z axis from to. Show that the electric potential at a point r is given by
for .
In Prob. 2.25, you found the potential on the axis of a uniformly charged disk:
(a) Use this, together with the fact that to evaluate the first three terms
in the expansion (Eq. 3.72) for the potential of the disk at points off the axis, assuming .
(b) Find the potential for by the same method, using Eq. 3.66. [Note: You
must break the interior region up into two hemispheres, above and below the
disk. Do not assume the coefficients are the same in both hemispheres.]
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