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Chapter 3: Q3.23P (page 150)
A spherical shell of radius carries a uniform surface charge on the "northern" hemisphere and a uniform surface charge on the "southern "hemisphere. Find the potential inside and outside the sphere, calculating the coefficients explicitly up to and .
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For the dipole in Ex. 3.10, expand to order ,and use this
to determine the quadrupole and octo-pole terms in the potential.
In one sentence, justify Earnshaw's Theorem: A charged particle cannot be held in a stable equilibrium by electrostatic forces alone. As an example, consider the cubical arrangement of fixed charges in Fig. 3.4. It looks, off hand, as though a positive charge at the center would be suspended in midair, since it is repelled away from each comer. Where is the leak in this "electrostatic bottle"? [To harness nuclear fusion as a practical energy source it is necessary to heat a plasma (soup of charged particles) to fantastic temperatures-so hot that contact would vaporize any ordinary pot. Earnshaw's theorem says that electrostatic containment is also out of the question. Fortunately, it is possible to confine a hot plasma magnetically.]
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).
In Ex. 3.8 we determined the electric field outside a spherical conductor
(radiusR)placed in a uniform external field . Solve the problem now using
the method of images, and check that your answer agrees with Eq. 3.76. [Hint:Use
Ex. 3.2, but put another charge, -q,diametrically opposite q.Let, withheld constant.]
(a) Using the law of cosines, show that Eq. 3.17 can be written as follows:
Whereand are the usual spherical polar coordinates, with the axis along the
line through . In this form, it is obvious thaton the sphere, localid="1657372270600" .
(a) Find the induced surface charge on the sphere, as a function of . Integrate this to get the total induced charge . (What should it be?)
(b) Calculate the energy of this configuration.
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