Biomedical Engineering Reference
In-Depth Information
I have joined up the charges with lines in order to focus attention on the charge systems
involved; there is no implication of a 'bond'. We do not normally discuss the electric
dipole due to a point charge (1). Examination of the charge distributions (2) through (6)
and calculation of their electric dipole moment for different coordinate origins suggest the
general result; neutral arrays of point charges have a unique electric dipole moment that
does not depend on where we take the coordinate origin. Otherwise, we have to state the
coordinate origin when we discuss the electric dipole moment.
I can prove this from Equation (1.16), generalized to
n
point charges:
n
p
e
=
Q
i
R
i
(1.17)
i
=
1
Suppose that we move the coordinate origin so that each point charge
Q
i
has a position
vector
R
i
where
R
i
+
R
i
=
with
a constant vector. From the definition of electric dipole moment we have
n
p
e
=
Q
i
R
i
i
=
1
and so, with respect to the new coordinate origin
n
p
e
=
Q
i
R
i
i
=
1
n
=
Q
i
(
R
i
−
)
i
=
1
n
=
p
e
−
Q
i
i
=
1
The two definitions only give the same vector if the sum of charges is zero. We often
use the phrase
gauge invariant
to describe quantities that do not depend on the choice of
coordinate origin.
Arrays (5) and (6) each have a centre of symmetry. There is a general result that any
charge distribution having no overall charge but a centre of symmetry must have a zero
dipole moment, and similar results follow for other highly symmetrical arrays of charges.
1.9.1 Continuous Charge Distributions
In order to extend the definition of an electric dipole to a continuous charge distribution
such as that shown in Figure 1.7, we first of all divide the region of space into differential
elements dτ .Ifρ(
r
) is the charge density then the change in volume element dτ is ρ(
r
)dτ .
We then treat each of these volume elements as point charges and add (i.e. integrate). The
electric dipole moment becomes
p
e
=
r
ρ (
r
) dτ
(1.18)
