EXERCISES
133
iii.
S_:
1”
cixldx2
[F
*
FI
pm(x1,
x2).
iv.
1:
1:
dxld~2
IF
I;]
pm(x1,xd.
23.
Prove that the monopole and quadrapole moments
of
any &pole (“physical” or
24.
A
quadrapole charge distribution consists
of
four
point charges in the xlx2-plane
“ideal”) are zero.
as shown below.
(a)
Using
Dirac &functions express the charge density,
pc(xl,
x2,
x3),
of this
(b)
The quadrapole moment
of
this
charge distribution is a second rank tensor
distribution
.
given by
- -
Q
=
Qtj
el@,.
The elements
of
the quadrapole tensor are given by the general expression
Q,
=
Im
dxl
/:
dx2
1;
dx3
p&l,
x2,
x3)
[~xJ,
-
(-Q%)~J]
--m
where
a,,
is
the Kronecker delta. In particular,
Q22
=
/=
dxl
Irn
dx2
Sy
dx3
pc(xl,
x2,
x3)
[kxz
-
(x?
+
~2’
+
x?)]
.
Evaluate all the elements
of
the
quadrapole tensor for the charge distribution
shown in the figure above.
-x
-‘x
--m
(c)
Does this charge distribution have
a
dipole moment?
(d)
Find the coordinate system in which
this
quadrapole tensor is diagonal.
25.
An
ideal quadrapole has a charge density
p,(x,
y, y)
that
is
zero everywhere
except at the origin. It has zero total charge, zero dipole moment, and
a
nonzero
quadrapole moment.
Express the elements
of
Q
in this system.
