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When we began to study electric forces we introduced the concept of
the electric field, . In particular, we had the fundamental relation
between the force on a charge q in an electric field:
= q.
We will introduce a magnetic field in a similar manner.
Before doing this, however, we define what is known as
the cross product
x
between two vectors and . Consider two
such vectors in the plane of this page with an angle between them,
as in Fig. 1.1.
Figure 1.1:
Two vectors

x , which itself
is a vector, is defined as follows:
As the cross product of two vectors is three dimensional
in nature, it is convenient to introduce the following notation for use
on our two dimensional paper. We represent a vector perpendicular to the
plane of this page which is directed out of the page as a circle with a
black dot in the center  this can be thought of as the tip of an arrow
coming towards you. On the other hand, a vector perpendicular to the
plane of this page which is directed into the page is represented
as a circle with a cross through the center  this can be thought of as the
tail of an arrow heading away from you. Examples of this are in
Fig. 1.2.
Figure 1.2:
Cross product of two vectors

We note two general features of the cross product:

x =


x = 0 if and
are parallel or antiparallel, as in these cases the angle between the
two vectors is either 0^{ o } or 180^{ o }, for which the magnitude
of
x given by Eq.(1.1) vanishes.
We now consider a charge q moving at velocity .
If this charge is in
a magnetic field , then it experiences
a force given by
From this we see that the units of are N s/(m C), which
are given the special name Tesla (T).
We note that, analogous to the electric field,
for a given velocity and a given magnetic field the force
on a positive charge is in the opposite direction to the force on a
negative charge.
Next: Motion of a Charged
Up: Magnetism
Previous: Magnetism
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10/9/1997