In previous chapters, we have talked about conservation of energy and momentum, which took some work to understand. For water, there is another, obvious conservation law that applies: conservation of amount of water. If we put a given amount of water in a reservoir and don't let it out, we expect to come back later and find the same amount of water still in the reservoir. If instead we let it out, how fast the amount changes is simply determined by the size of the out-flowing current, by definition. If we consider the amount of water in both reservoirs at either end of a current, though, we see that since water flowing into one end of the channel has to come out the other, the amount of water lost by one reservoir is just the same as the amount gained by the other. Thus not only is the amount of water in one reservoir conserved if we forbid currents, but as long as we consider all the reservoirs that are connected, the total amount of water is conserved, whether there are currents or not. For water, all of these properties are already familiar.
More complicated currents are possible, though, than just those in channels and pipes. Our understanding that water is conserved allows us to say several powerful things about them, despite their complexity. For instance, suppose we have a pipe that feeds into an otherwise closed place, and that we track the motion of the water around the mouth of the pipe. We can draw the speed of each motion we measure as an arrow. Where we make the measurement tells us where we start the arrow, how fast the flow is determines how long it will be, and the direction of the flow tells which way it points. If we agree to space our measurements so that each one measures flow perpendicular an area of the same size, the length of the arrows is just proportional to the current through that area, and what we are mapping is just the currents at every place in the fluid.
An example of one possible such flow is shown in
fig. 7.6.
In
mapping the sizes of currents, we have just made a map that
gives precise mathematical meaning to what we have been
referring to as the flow field.
Single numbers are not
enough to describe the whole complex motion of the field, even
at a single point, because it can have a direction as well as a
size. Therefore we have had to use arrows, which are a
convenient way to represent all those numbers at once.
The flow field we have measured can be very complicated, involving turbulent eddies and swirls, fast and slow places, in potentially endless configurations. However, no matter how complex it becomes, it must have certain properties. Suppose we draw an imaginary spherical shell around the mouth of the pipe, and measure how much water is flowing out through that sphere in each unit of its area. Since the arrows told how much current there was in each direction, this other question, how much water flows out through each area, is just the question of how big an outward current exists at each place.
The thing we know is that the pipe is delivering water into an
enclosed volume inside the sphere at some given rate, determined
by its overall current. Since the water can't be compressed,
whatever flows in from the pipe must push an identical amount
out through the sphere in each unit time, to make room for
the new influx. Thus, while we may not know ahead of time how
any of the little arrows on the sphere will point, we do know
that the sum of the outward components of all of them will
be equal to the current in from the pipe. This sum, of the
lengths of the outward components of a current on some surface,
is called the divergence
of that current from the volume
contained by the surface. The lengths are added positively if
the flow is out, and negatively if it is in, the same way the
components of the arrows point. We see that, since which sphere
we chose to use didn't matter, not only is the divergence
through any one sphere the same as the current in the pipe; it
is also the same as the divergence out through any other
sphere. This one element of constancy in what can otherwise be
very complicated flow fields can be very useful.
Divergence becomes a powerful notion when we consider completely closed surfaces. To do this, we would not consider just the part of a sphere surrounding the pipe, but a whole sphere, including the part that crosses the pipe. In that case, we must add the flow fields outward on all the rest of the sphere to the flow fields inward in the interior of the the pipe, because to empty inside the sphere, the pipe must transport that water in across its boundary. Of course, the amount of inflow is just the pipe's current, which exactly equals the amount of outflow everywhere else, so the total divergence out through this sphere is zero. Of course, for any flow we might induce in water, with pipes or otherwise, the total divergence will always be exactly zero, which is just another way to say that whatever flows in someplace must flow out someplace else, because the water cannot be compressed and there is nowhere else for it to go. In this way, the divergence of a current is closely related to the conservation of the amount of water.
For flow fields in water, the vanishing of the divergence out through every sphere is somewhat trivial, because it is always true. When we examine the behavior of the electric field, we will find that properties of its divergence are similarly useful for overcoming the complexity that can arise, but there they will not be trivial. The divergence of the electric field will still be related to a conservation law, but will not always be zero, and will tell us about the thing, called electric charge, that is conserved. We will also find that divergences of electric fields have important relations to divergences of electric currents, though the two are quite different kinds of fields. The thing that provides the relation between them is precisely this charge.