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Chapter 2: Q45P (page 108)
Let be a normal subgroup of a group and let be a homomorphism of groups such that the restriction of to is an isomorphism . Prove that , where is the kernel of f.
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Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radius and the total charge .
We know that the charge on a conductor goes to the surface, but just
how it distributes itself there is not easy to determine. One famous example in which the surface charge density can be calculated explicitly is the ellipsoid:
In this case15
(2.57) where is the total charge. By choosing appropriate values for a,band c. obtain (from Eq. 2.57):
(a) the net (both sides) surface charge a (r)density on a circular disk of radius R;(b) the net surface charge density a (x) on an infinite conducting "ribbon" in the xyplane, which straddles theyaxis from x=-ato x=a(let A be the total charge per unit length of ribbon);
(c) the net charge per unit length on a conducting "needle," running from x= -ato x= a . In each case, sketch the graph of your result.
A metal sphere of radius R ,carrying charge q ,is surrounded by a
thick concentric metal shell (inner radius a,outer radius b,as in Fig. 2.48). The
shell carries no net charge.
(a) Find the surface charge density at R ,at a ,and at b .
(b) Find the potential at the center, using infinity as the reference point.
(c) Now the outer surface is touched to a grounding wire, which drains off charge
and lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?
Question: Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge . Check your result for the limiting
cases and .
Calculate the divergence of the following vector functions:
Two spheres, each of radius R and carrying uniform volume charge densities +p and -p , respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.]
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