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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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(a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33.
(b) Use Gauss's law to find the field inside and outside a long hollow cylindrical
tube, which carries a uniform surface charge .Check that your result is consistent with Eq. 2.33.
(c) Check that the result of Ex. 2.8 is consistent with boundary conditions 2.34 and 2.36.
Find the potential inside and outside a uniformly charged solid sphere whose radius is and whose total charge is .Use infinity as your reference point. Compute the gradient of in each region, and check that it yields the correct field. Sketch.
Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin,for some constant k. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.]
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?
Findthe electric field a distance zfrom the center of a spherical surface of radius R(Fig. 2.11) that carries a uniform charge density .Treat the case z< R(inside) as well as z> R(outside). Express your answers in terms of the total chargeqon the sphere. [Hint:Use the law of cosines to write in terms of Rand .Besure to take the positivesquare root:if ,but it'sif .]
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