Biomedical Engineering Reference
In-Depth Information
+
Q
-
Q
Figure 1.4
Field lines for point charges
1.5 Work
Look again at Figure 1.3, and suppose we move point charge
q
whilst keeping
Q
fixed
in position. When a force acts to make something move, energy is transferred. There is a
useful phrase in physical science that is to do with the energy transferred, and it is
work
.
Work measures the energy transferred in any change, and can be calculated from the change
in energy of a body when it moves through a distance under the influence of a force.
We have to be careful to take account of the energy balance. If a body gains energy, this
energy has to come from somewhere, and that somewhere must lose energy. What we do
is to divide the universe into two parts: the bits we are interested in called the
system
and
the rest of the universe that we call the
surroundings
.
Some texts focus on the work done
by
the system, some concern themselves with the
work done
on
the system. According to the law of conservation of energy, one is exactly
the equal and opposite of the other, but we have to be clear which is being discussed. I am
going to write
w
on
for the work done on our system.
If the system gains energy, then
w
on
will be positive. If the system loses energy then
w
on
will negative.
We also have to be careful about the phrase 'through a distance'. The phrase means
'through a distance that is the projection of the force vector on the displacement vector',
and you should instantly recognize a vector scalar product (see Appendix A).
A useful formula that relates to the energy gained by a system (i.e.
w
on
) when a constant
force
F
moves its point of application through
l
is
F
.
l
(1.6)
In the case where the force is not constant, we have to divide up the motion into differential
elements d
l
. The energy transferred is then given by the sum of all the corresponding
differential elements d
w
on
. The corresponding formulae are
d
w
on
=−
w
on
=−
F
.d
l
w
on
=−
F
.d
l
(1.7)
We now move
q
by an infinitesimal vector displacement d
l
, as shown in Figure 1.5, so
that it ends up at point
r
+
d
l
. The work done on the system in that differential change is
d
w
on
=−
F
.d
l
If the angle between the vectors
r
I
and d
l
is θ , then we have
d
w
on
=−
F
d
l
cos θ
