Environmental Engineering Reference
In-Depth Information
In this subsection we have been investigating how vectors make manifest the
necessary invariance in the form of the equations of physics as we move between
frames of reference which differ by a rotation. In the language of many text-
books, we have been considering 'passive' rotations of the co-ordinate axes. We
have not however made any statement as to whether or not the physical world
possesses a rotational symmetry. To explore this question requires a somewhat
different approach: we need to ask what happens if we rotate the actual position
vectors corresponding to all parts of our experiment? If we want to insist that the
physical world is rotationally symmetric then performing such a rotation should not
alter the form of the equations of motion for this is the mathematical expression
of the statement that the results of an experiment do not depend upon the orienta-
tion of the experiment. Again things should become clearer if we pick a specific
example.
Let us consider a particular experiment in which a charged particle is moving in
a magnetic field. The particle moves according to
m
x
=
q
x
×
B
.
(11.15)
Now suppose that the magnetic field is generated by the apparatus of our experiment
(for example by a solenoid). We can ask what happens if we rotate all elements of
our experiment, including the solenoid, by the same amount? The new vectors can
all be obtained from the old vectors through the action of some rotation matrix
R
,
and in particular
x
=
R
x
,
(11.16)
B
=
R
B
.
(11.17)
Using Eq. (11.15) we thus have
m
R
−
1
x
=
q(
R
−
1
x
)
(
R
−
1
B
)
×
(11.18)
q
R
−
1
(
x
×
B
)
=
(11.19)
where the second line is necessary if the vector product is to be a vector quantity.
We can pre-multiply each side by
R
to get
x
=
x
×
B
.
m
q
(11.20)
Thus, we see that the Lorentz force law is invariant under 'active' rotations of all
parts of the system. Now we change the situation somewhat and move to a ficti-
tious universe in which there exists a universal uniform magnetic field
which
permeates the whole of space. Clearly this universe is not rotationally symmet-
ric since the magnetic field picks out a very special direction. Charged particles
travelling parallel to this direction would feel no force whereas those travelling
in any other direction would be deflected. Clearly, the results of experiments will
now depend upon orientation. We expect that this feature should express itself in
B
