11.10 EXERCISES
11.12 Show that the expression below is equal to the solid angle subtended by a
rectangular aperture, of sides 2aand 2b, at a point on the normal through its
centre, and at a distancecfrom the aperture:
Ω=4
∫b0ac
(y^2 +c^2 )(y^2 +c^2 +a^2 )^1 /^2dy.By settingy=(a^2 +c^2 )^1 /^2 tanφ, change this integral into the form
∫φ 104 accosφ
c^2 +a^2 sin^2 φdφ,where tanφ 1 =b/(a^2 +c^2 )^1 /^2 , and hence show thatΩ=4tan−^1[
ab
c(a^2 +b^2 +c^2 )^1 /^2]
.
11.13 A vector fieldais given by−zxr−^3 i−zyr−^3 j+(x^2 +y^2 )r−^3 k,wherer^2 =x^2 +y^2 +z^2.
Establish that the field is conservative (a) by showing that∇×a= 0 ,and(b)by
constructing its potential functionφ.
11.14 A vector fieldais given by (z^2 +2xy)i+(x^2 +2yz)j+(y^2 +2zx)k. Show that
ais conservative and that the line integral
∫
a·dralong any line joining (1, 1 ,1)
and (1, 2 ,2) has the value 11.
11.15 A forceF(r) acts on a particle atr. In which of the following cases canFbe
represented in terms of a potential? Where it can, find the potential.
(a) F=F 0[
i−j−2(x−y)
a^2r]
exp(
−
r^2
a^2)
;
(b)F=F 0
a[
zk+(x^2 +y^2 −a^2 )
a^2r]
exp(
−
r^2
a^2)
;
(c) F=F 0[
k+a(r×k)
r^2]
.
11.16 One of Maxwell’s electromagnetic equations states that all magnetic fieldsB
are solenoidal (i.e.∇·B= 0). Determine whether each of the following vectors
could represent a real magnetic field; where it could, try to find a suitable vector
potentialA, i.e. such thatB=∇×A. (Hint: seek a vector potential that is parallel
to∇×B.):
(a)B 0 b
r^3[(x−y)zi+(x−y)zj+(x^2 −y^2 )k] in Cartesians withr^2 =x^2 +y^2 +z^2 ;(b)B 0 b^3
r^3[cosθcosφˆer−sinθcosφeˆθ+sin2θsinφˆeφ] in spherical polars;(c) B 0 b^2[
zρ
(b^2 +z^2 )^2eˆρ+1
b^2 +z^2eˆz]
in cylindrical polars.11.17 The vector fieldfhas componentsyi−xj+kandγis a curve given parametrically
by
r=(a−c+ccosθ)i+(b+csinθ)j+c^2 θk, 0 ≤θ≤ 2 π.
Describe the shape of the pathγand show that the line integral∫
γf·drvanishes.
Does this result imply thatfis a conservative field?
11.18 A vector fielda=f(r)ris spherically symmetric and everywhere directed away
from the origin. Show thatais irrotational, but that it is also solenoidal only if
f(r)isoftheformAr−^3.