Biomedical Engineering Reference
In-Depth Information
1.9.2 Electric Second Moment
The electric dipole moment of an array of point charges is defined by the three sums
n
n
n
Q
i
X
i
,
Q
i
Y
i
and
Q
i
Z
i
i
=
1
i
=
1
i
=
1
and we can collect them into a column vector in an obvious way as
⎛
⎝
⎞
⎠
i
=
1
Q
i
X
i
n
n
i
=
1
Q
i
Y
i
n
p
e
=
(1.19)
i
=
1
Q
i
Z
i
The six independent quantities
n
n
n
n
Q
i
X
i
,
Q
i
Z
i
Q
i
X
i
Y
i
,
Q
i
X
i
Z
i
, ...,
i
=
1
i
=
1
i
=
1
i
=
1
are said to define the
electric second moment
of the charge distribution. We usually collect
them into a real symmetric 3
×
3 matrix
q
e
:
⎛
⎞
i
=
1
Q
i
X
i
i
=
1
Q
i
X
i
Y
i
i
=
1
Q
i
X
i
Z
i
n
n
n
⎝
⎠
i
=
1
Q
i
Y
i
X
i
i
=
1
Q
i
Y
i
i
=
1
Q
i
Y
i
Z
i
n
n
n
q
e
=
(1.20)
i
=
1
Q
i
Z
i
X
i
i
=
1
Q
i
Z
i
Y
i
i
=
1
Q
i
Z
i
n
n
n
The matrix is symmetric because of the obvious equalities of the off-diagonal sums such as
n
n
Q
i
X
i
Y
i
and
Q
i
Y
i
X
i
i
=
1
i
=
1
There are unfortunately many different definitions related to the second (and higher)
moments in the literature. There is little uniformity of usage, and it is necessary to be
crystal clear about the definition and choice of origin when dealing with these quantities.
Most authors prefer to work with a quantity called the
electric quadrupole moment
rather
than the second moment, but even then there are several different conventions. A common
choice is to use the symbol
e
and the definition
⎛
⎞
i
=
1
Q
i
3
X
i
R
i
i
=
1
Q
i
X
i
Y
i
i
=
1
Q
i
X
i
Z
i
n
n
n
−
3
3
⎝
⎠
i
=
1
Q
i
Y
i
X
i
i
=
1
Q
i
3
Y
i
R
i
i
=
1
Q
i
Y
i
Z
i
1
2
n
n
n
e
=
3
−
3
(1.21)
i
=
1
Q
i
Z
i
X
i
i
=
1
Q
i
Z
i
Y
i
i
=
1
Q
i
3
Z
i
R
i
n
n
n
3
3
−
Notice that the diagonal elements of this matrix sum to zero and so the matrix has zero trace
(the
trace
being the sum of the diagonal elements; see Appendix A). Some authors do not
