Biomedical Engineering Reference
In-Depth Information
Q
R
A
R
B
R
θ
Q
A
origin
Q
B
Figure 2.3
Charge-dipole interaction
The two point charges
Q
A
and
Q
B
have a mutual potential energy of
1
4πε
0
Q
A
Q
B
2
d
(2.2)
but we are going to investigate what happens to the mutual potential energy of the system
as we change the position vector of
Q
and so we ignore this term since it remains constant.
The mutual potential energy
U
(charge-dipole) of the point charge and the electric dipole
is given exactly by
4πε
0
Q
Q
A
1
Q
B
R
B
U
(
charge-dipole
)
=
R
A
+
(2.3)
This can also be written in terms of
R
and θ as
4πε
0
Q
1
Q
A
Q
B
U
(charge-dipole)
=
√
(
R
2
2
dR
cos θ )
+
√
(
R
2
+
d
2
+
+
d
2
−
2
dR
cos θ )
(2.4)
and, once again, this is an exact expression.
In the case where the point charge
Q
gets progressively far away from the coordinate
origin, we can usefully expand the two denominators using the binomial theorem to give
4πε
0
Q
(
Q
A
+
1
Q
B
)
(
Q
B
−
Q
A
)
d
R
2
U
(charge-dipole)
=
+
cos θ
R
3 cos
2
θ
1
+···
(
Q
A
+
Q
B
)
d
2
2
R
3
+
−
(2.5)
The first term on the right-hand side contains the sum of the two charges making up the
dipole. Very often, we deal with simple dipoles that carry no overall charge, and this term
is zero because
Q
A
=−
Q
B
.
The second term on the right-hand side obviously involves the electric dipole moment,
whose magnitude is (
Q
B
−
Q
A
)
d
. The third term involves the electric second moment whose
magnitude is (
Q
B
+
Q
A
)
d
2
, and so on. The mutual potential energy is therefore seen to be
a sum of terms; each is a product of a moment of the electric charge, and a function of
