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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Buckminsterfullerine is a molecule of 60 carbon atoms arranged
like the stitching on a soccer-ball. It may be approximated as a conducting spherical shell of radius . A nearby electron would be attracted, according to Prob. 3.9, so it is not surprising that the ion exists. (Imagine that the electron on average-smears itself out uniformly over the surface.) But how about a second electron? At large distances it would be repelled by the ion, obviously, but at a certain distance r (from the center), the net force is zero, and closer than this it would be attracted. So an electron with enough energy to get in that close should bind.
(a) Find r, in . [You'll have to do it numerically.]
(b) How much energy (in electron volts) would it take to push an electron in (from
infinity) to the point r? [Incidentally, the ion has been observed.]
(a) A long metal pipe of square cross-section (side a) is grounded on three sides, while the fourth (which is insulated from the rest) is maintained at constant potential .Find the net charge per unit length on the side oppositeto Vo. [Hint:Use your answer to Prob. 3.15 or Prob. 3.54.]
(b) A long metal pipe of circular cross-section (radius R) is divided (lengthwise)
into four equal sections, three of them grounded and the fourth maintained at
constant potential Vo.Find the net charge per unit length on the section opposite
to .[Answer to both (a) and (b) : localid="1657624161900" .]
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.]
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.)
Here's an alternative derivation of Eq. 3.10 (the surface charge density
induced on a grounded conducted plane by a point charge qa distance dabove
the plane). This approach (which generalizes to many other problems) does not
rely on the method of images. The total field is due in part to q,and in part to the
induced surface charge. Write down the zcomponents of these fields-in terms of
qand the as-yet-unknown -just below the surface. The sum must be zero,
of course, because this is inside a conductor. Use that to determine .
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