don’t need to put in terms likexˆy, because they don’t contribute to
the divergence.) Define a new set of coordinates (u,v,w) related to
(x,y,z) by
x=u+p
y=v+q
z=w+r,
wherep,q, andrare constants. Show that the field’s divergence is
the same in these new coordinates. Note thatxˆanduˆare identical,
and similarly for the other coordinates.
54 Using a techniques similar to that of problem 53, show that
the divergence is rotationally invariant, in the special case of ro-
tations about thezaxis. In such a rotation, we rotate to a new
(u,v,z) coordinate system, whose axes are rotated by an angleθ
with respect to those of the (x,y,z) system. The coordinates are
related by
x=ucosθ+vsinθ
y=−usinθ+vcosθ
Find how the uandvcomponents the fieldEdepend onuand
v, and show that its divergence is the same in this new coordinate
system.
55 An electric field is given in cylindrical coordinates (R,φ,z)
byER = ce−u|z|R−^1 cos^2 φ, where the notationER indicates the
component of the field pointing directly away from the axis, and
the components in the other directions are zero. (This isn’t a com-
pletely impossible expression for the field near a radio transmitting
antenna.) (a) Find the total charge enclosed within the infinitely
long cylinder extending from the axis out toR=b. (b) Interpret
theR-dependence of your answer to part a.
56 Use Euler’s theorem to derive the addition theorems that
express sin(a+b) and cos(a+b) in terms of the sines and cosines of
aandb. .Solution, p. 1044
57 Find every complex numberzsuch thatz^3 = 1.
.Solution, p. 1044
58 Factor the expressionx^3 −y^3 into factors of the lowest possible
order, using complex coefficients. (Hint: use the result of problem
57.) Then do the same using real coefficients.
59 A dipole consists of two point charges lying on thexaxis, a
charge−qat the origin, and a +qatx=`. The dipole is immersed
in an externally imposed, nonuniform electric field withEx=bx,
wherebis a constant. Add the forces acting on the dipole. Verify
that the total force depends only on the dipole moment, not onqor
`individually, and that the result is the same as the one found by a
fancier method in example 7 on p. 589. .Solution, p. 1044666 Chapter 10 Fields