Biomedical Engineering Reference
In-Depth Information
and this represents the energy change on building up the charge distribution, with the
charges initially at infinite separation. It turns out that this energy change is an important
property, and we give it a special name (the
mutual potential energy
) and a special symbol
U
(occasionally Φ).
Comparison of the equations for force, work and mutual potential energy given above
suggests that there might be a link between the force and the mutual potential energy; at
first sight, one expression looks like the derivative of the other.
I am going to derive a relationship between force and mutual potential energy. The
relationship is perfectly general; it applies to all forces provided that they are constant
in time.
1.8 Relationship between Force and Mutual Potential Energy
Consider a body of mass
m
that moves in (say) the
x
-direction under the influence of a
constant force. Suppose that at some instant its speed is
v
. The kinetic energy is
2
mv
2
.You
are probably aware of the law of conservation of energy, and know that when I add the
potential energy
U
to the kinetic energy, I will get a constant energy that I will denote ε
(sorry about the clash of symbols with electric permittivity, but that is life):
1
2
mv
2
ε
=
+
U
(1.11)
I want to show you how to relate
U
to the force
F
. If the energy ε is constant in time,
then dε/d
t
=
0. Differentiation of Equation (1.11) with respect to time gives
dε
d
t
=
mv
d
v
d
U
d
t
d
t
+
and so, by the chain rule
d
x
d
t
If the energy ε is constant then its first differential with respect to time is zero, and
v
is just
d
x
/d
t
. Likewise d
v
/d
t
is the acceleration and so
dε
d
t
=
mv
d
v
d
U
d
x
d
t
+
m
d
2
x
d
t
2
d
x
d
t
d
U
d
x
0
=
+
(1.12)
Equation (1.12) is true if the speed is zero, or if the term in brackets is zero. According to
Newton's second law of motion, mass times acceleration is force, and so
d
U
d
x
F
=−
(1.13)
which gives us the link between force and mutual potential energy.
When working in three dimensions, we have to be careful to distinguish between vectors
and scalars. We treat a body of mass
m
whose position vector is
r
. The velocity is
v
=
d
r
/d
t
and the kinetic energy is
2
m
(d
r
/d
t
)(d
r
/d
t
).
Analysis along the lines given above shows that the force
F
and
U
are related by
F
=−
grad
U
(1.14)
