Biomedical Engineering Reference
In-Depth Information
1.6 Charge Distributions
So far I have concentrated on point charges, and carefully skirted round the question as to
howwe deal with continuous distributions of charge. Figure 1.7 shows a charge distribution
Q
A
. The density of charge need not be constant through space, and we normally write ρ(
r
)
for the density at the point whose position vector is
r
. The charge contained within the
volume element dτ at
r
is therefore ρ(
r
)dτ and the relationship between ρ(
r
) and
Q
A
is
discussed in Appendix A. It is
Q
A
=
ρ (
r
) dτ
(1.9)
In order to find the force between the charge distribution and the point charge
Q
B
we simply
extend our ideas about the force between two point charges; one of the point charges being
ρ(
r
)dτ and the other
Q
B
.
Q
A
ρ
A
τ
d
R
A
R
AB
Origin
R
B
Q
B
Figure 1.7
Charge distribution
The total force is given by the sum of all possible contributions from the elements of the
continuous charge distribution
Q
A
with point charge
Q
B
. The practical calculation of such
a force can be a nightmare, even for simple charge distributions. One of the reasons for the
nightmare is that forces are vector quantities; we need to know about both their magnitude
and their direction.
In the next section, I am going to tell you about a very useful scalar field called the
mutual potential energy
U
. This field has the great advantage that it is a scalar field, and so
we do not need to worry about direction in our calculations.
1.7 Mutual Potential Energy,
U
Suppose nowwe start with charge
q
at infinity, and move it up to a point with vector position
r
, as shown in Figure 1.3. The work done is
1
4πε
0
Qq
r
w
on
=
(1.10)
