Biomedical Engineering Reference
In-Depth Information
the inverse distance. Hopefully, as
R
increases, the magnitude of the successive terms will
become less and eventually the mutual potential energy will be dominated by the first few
terms in the expansion.
In the more general case where we replace
Q
A
and
Q
B
with an arbitrary array of point
charges
Q
1
,
Q
2
,...,
Q
n
whose position vectors are
R
1
,
R
2
,...,
R
n
(or for that matter a
continuous charge distribution), it turns out that we can always write the mutual interaction
potential with
Q
as
n
1
R
−
n
.grad
1
R
higher order terms
Q
4πε
0
=
+
U
Q
i
Q
i
R
i
(2.6)
i
=
1
i
=
1
The first summation on the right-hand side gives the overall charge of the charge distribution,
the second term involves the electric dipole moment, the third term involves the electric
quadrupole moment, and so on.
2.4 Dipole-Dipole Interaction
Consider now a slightly more realistic model for the interaction of two simple (diatomic)
molecules (Figure 2.4). Molecule A consists of two point charges,
Q
1A
and
Q
2A
. Molecule
B consists of two point charges
Q
1B
and
Q
2B
. The overall charge on molecule A is therefore
Q
A
=
Q
2A
with a similar expression for molecule B. The electric dipole moments of
A and B are written
p
A
,
p
B
in an obvious notation, and their scalar magnitudes are written
p
A
,
p
B
. The second moments of the two molecules are each determined by a scalar value
q
A
and
q
B
, simply because they are linear.
Q
1A
+
Q
2A
Q
2B
φ
θ
A
θ
B
R
Q
1B
Q
1A
Figure 2.4
Multipole expansion for a pair of diatomics
Molecule A is centred at the origin, whilst molecule B has its centre a distance
R
away
along the horizontal axis. The inclinations to the axis are θ
A
and θ
B
, and φ gives the relative
orientation of the two molecules. The distance between the two molecules is much less than
their separation, so we can make convenient approximations such as for the small dipole.
After some standard analysis we find that the mutual potential energy of A and B is
