Next: Scalar and Vector Products
Many processes in the real world involve continuous media with their
properties varying continuously in 3-D space and time. An example
would be the density of a gas or fluid which is written as
. This is different from MT2003 where the dynamics section
only considered point masses or particles. Examples of continuous
media are (i) fluids, such as water, beer,, (ii) gases, for
example air, clouds,, (iii) electromagnetic fields, namely
electric field, magnetic field and (iv) plasmas like the solar
atmosphere, magnetosphere. A plasma is the fourth state of matter
consisting of an ionised gas. In a plasma the temperature is so high
that the electrons are no longer bound to a particular atom and the
gas consists of electrons and positive ions.
A field is a function that describes a physical quantity at
all points in space. For a scalar field this quantity is
specified by a single variables at each point such as
For a vector field the physical quantity is specified by a
vector, giving a direction as well as a magnitude. Examples are
Note that we use bold face type for vectors but when writing a
vector you MUST remember to underline vectors. Thus,
If you forget to underline vectors, I will mark you wrong!
Example 1. .1If
A vector field line, such as a magnetic field line (for
B) or a streamline (fluid or plasma velocity v) is
drawn such that the tangent to the vector field line at any point is
in the direction of the vector.
Thus, in 2-D
as illustrated in Figure 1.1.
An alternative derivation of the equation of the field lines is given
by the use of similar triangles, as shown in Figure 1.2.
Here the distance along the curve is and the triangle on the left
shows how the small distance is related to the small horizontal
and vertical distances and respectively. Using similar
triangles we get
The field line and the tangent to the field line produces a
vector in the direction of the magnetic field
as in Equation (1.1). Now, we may use the other relationships
to get the parametric form of the field line as
In using the form (1.2) we make use of
Using similar triangles we can obtain the equation of field
lines in parametric form.
In 3-D the equation of a field line is obtained by solving the
The field line separation gives an indication of the strength of the
field. When the field lines are closer together then the field is
stronger and sketches of the field lines should reflect this.
Example 1. .2For
, so that , and , the
field lines are given by
Hence, the field lines are given by simply
So the field is stronger for larger values of y. Note that the
direction of the field lines is easily worked out by looking at the
field components. We see that so that is
positive when y is positive and negative when y is negative. This is
shown in Figure 1.3.
Sketch of the field lines in Example 1.2.
Example 1. .3For a fluid velocity
streamlines are given by
Integrating we obtain
So the streamlines are given by concentric circles as shown in
Note the direction of the flow is given by the components of .
For and positive, we have and .
The streamlines of the fluid velocity given in Example 1.3.
We can calculate how individual fluid elements move in response to
this flow from the fact that and so on. Hence,
The solution is
There are some problems where a change of coordinate system
simplifies the problem. For example, in 2-D polar coordinates the magnetic field lines are given by
or in terms of the parameter we have
where is the distance along a field line.
Next: Scalar and Vector Products
Prof. Alan Hood