Next: Curl , , Triple
Up: Introduction
Previous: The Gradient Operator,
Consider a vector that passes through a specified surface. The vector
could be the velocity of a fluid, then the flux of
would be the amount of fluid passing through that surface in unit
time. The flux, , of a vector through a surface
element is defined as

(1.15) 
where
is normal to the surface element as
shown in Figure 1.10
Figure 1.10:
The normal to the surface element and the vector .

Hence,

(1.16) 
and is the component of normal to the
surface. Note that the largest value of occurs when
and if
so that in the latter case
does not cross the surface at all but is instead tangent to the
surface. If the vector is tangent to the surface, then the flux
through the surface is obviously zero.
Example 1. .6Consider
. This is the MASS FLUX of a fluid or
plasma and is the RATE at which fluid crosses the surface .
Let us consider the units of in this example.
Consider
, the magnetic field, then is the
amount of magnetic flux crossing .
The total flux of a vector across a surface (rather than just the
surface element) is

(1.17) 
There are two types of surfaces that we consider, namely open surfaces
and closed surfaces. A closed surface encloses a volume , as shown
in Figure 1.11.
Figure 1.11:
Two type of surface with an open surface on the left and a
closed surface, enclosing a volume, on the right.

If bounds a finite volume, as in the closed surface case, then
points outwards from the volume. In this case the
Divergence Theorem of Gauss (Carl Gauss, 17771855) gives

(1.18) 
where

(1.19) 
is the divergence of . Hence,
represents the outgoing flux per unit volume and we have two
important deductions, namely
 If
, a flow is diverging (fluid is
flowing out of a volume).
 If
, a flow is converging (fluid is
flowing into a volume).
Example 1. .7
 Consider a fluid of uniform density, , flowing with
a velocity given by
. What is the mass flux
across the square , , ? The
situation is shown in Figure 1.12.
Here
and so
 As in the above example but with a density given by
. Hence,
Magnetic Flux Tubes are the building blocks of a magnetic
field. A magnetic flux tube (for the magnetic field ) or a
flow tube (for the mass flux ) is the surface generated
by a set of field lines that intersect a simple closed curve,
as shown in Figure 1.13.
Figure 1.13:
A magnetic flux tube.

The magnetic flux crossing is
and the total magnetic flux is
.
Since the flux tube is defined by the magnetic field lines, the walls
of the flux tube are parallel to . Hence, if there is
no flux created inside the tube, then we may state that
So the magnetic flux is constant along the tube. To prove this,
consider a closed surface defined by , and the walls.
Hence, the flux through this closed surface is
The last integral is zero since is parallel to the walls.
The first integral on the right hand side has a negative sign. This
occurs because the flux through into the tube has
a direction that is opposite to the direction of the unit normal that
is out of the closed surface. Now we make use of Guass's
divergence theorem so that
However, the basic equation for magnetic fields, see later, is
Hence, we have
and so

(1.20) 
If the crosssectional area is small, so that is
approximately constant, then the flux is approximately
where is the crosssectional area and is the field strength.
Thus, if increases, the area decreases such that remains
constant.
Example 1. .8For the magnetic field given by
, the field
lines are given by
Therefore,
The flux tube passing through , and
has flux
We should check the flux passing through . We need to know where
the field lines at have gone to at .
Table 1.1:

Therefore,
Thus, the flux through equals the flux through .
Next: Curl , , Triple
Up: Introduction
Previous: The Gradient Operator,
Prof. Alan Hood
20001106